{"id":911,"date":"2016-02-15T20:59:28","date_gmt":"2016-02-15T20:59:28","guid":{"rendered":"https:\/\/courses.candelalearning.com\/nrocarithmetic\/?post_type=chapter&#038;p=911"},"modified":"2018-01-03T23:58:15","modified_gmt":"2018-01-03T23:58:15","slug":"4-2-2-adding-and-subtracting-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/beginalgebra\/chapter\/4-2-2-adding-and-subtracting-polynomials\/","title":{"raw":"Operations on Polynomials","rendered":"Operations on Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add polynomials\r\n<ul>\r\n \t<li>Use horizontal and vertical organization to add polynomials<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Find the opposite of a polynomial\r\n<ul>\r\n \t<li>Subtract polynomials using both horizontal and vertical organization<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiply Polynomials<\/li>\r\n \t<li>\r\n<ul>\r\n \t<li>Find the product of monomials<\/li>\r\n \t<li>Find the product of polynomials and monomials<\/li>\r\n \t<li>Find the product of two binomials<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiply Binomials<\/li>\r\n \t<li>\r\n<ul>\r\n \t<li>Apply the FOIL method to multiply two binomials<\/li>\r\n \t<li>Use a table to multiply two binomials<\/li>\r\n \t<li>Simplify the product of two binomials given a wide variety of variables, constants, signs, and arrangement of terms in the binomial<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Divide Polynomials<\/li>\r\n \t<li>\r\n<ul>\r\n \t<li>Divide a binomial by a monomial<\/li>\r\n \t<li>Divide a trinomial by a monomial<\/li>\r\n \t<li>Apply polynomial long division to divide by a binomial<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nAdding and subtracting <b>polynomials<\/b> may sound complicated, but it\u2019s really not much different from the addition and subtraction that you do every day. The main thing to remember is to look for and combine <b>like terms<\/b>.\r\n\r\nYou can add two (or more) polynomials as you have added algebraic expressions. You can remove the parentheses and combine like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(3b+5\\right)+\\left(2b+4\\right)[\/latex]\r\n\r\n[reveal-answer q=\"379821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"379821\"]Regroup\r\n<p style=\"text-align: center;\">[latex]\\left(3b+2b\\right)+\\left(5+4\\right)[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center;\">[latex]5b + 9[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3b+5\\right)+\\left(2b+4\\right)=5b+9[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you are adding polynomials that have subtraction,\u00a0it is important to remember to keep the sign on each term as you are collecting like terms. \u00a0 The next example will show you how to regroup terms that are subtracted when you are collecting like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(-5x^{2}\u201310x+2\\right)+\\left(3x^{2}+7x\u20134\\right)[\/latex]\r\n\r\n[reveal-answer q=\"486380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"486380\"]\r\n\r\nCollect like terms, making sure you keep the sign on each term. For example, when you collect\u00a0the [latex]x^2[\/latex] terms, make sure to keep the negative sign on [latex]-5x^2[\/latex].\r\n\r\nHelpful Hint: We find that it is easier to put the terms with a negative sign on the right of the terms that are positive. This would mean\u00a0that the\u00a0[latex]x^2[\/latex] terms would be grouped as\u00a0[latex]\\left(3x^{2}-5x^{2}\\right)[\/latex]. If both terms are negative, then it doesn't matter which is on the left or right.\r\n\r\nThe polynomial now looks like this, with like terms collected:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\underbrace{\\left(3x^{2}-5x^{2}\\right)}+\\underbrace{\\left(7x-10x\\right)}+\\underbrace{\\left(2-4\\right)}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x^2\\text{ terms }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\text{ terms}\\,\\,\\,\\,\\,\\,\\,\\,\\text{ constants }\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The [latex]x^2[\/latex] terms will simplify to [latex]-2x^{2}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The constant terms will simplify to [latex]-2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">\u00a0Rewrite the polynomial with it's simplified terms, keeping the sign on each term.<\/p>\r\n<p style=\"text-align: center;\">[latex]-2x^{2}-3x-2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">As a matter of convention, we write polynomials in descending order based on degree. \u00a0Notice how we put the\u00a0[latex]x^2[\/latex] term first, the\u00a0[latex]x[\/latex] term second and the constant term last.<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(-5x^{2}-10x+2\\right)+\\left(3x^{2}+7x-4\\right)=-2x^{2}-3x-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this \u201cvertical\u201d method of adding polynomials:\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(3x^{2}+2x-7\\right)+\\left(7x^{2}-4x+8\\right)[\/latex]\r\n\r\n[reveal-answer q=\"425224\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"425224\"]Write one polynomial below the other, making sure to line up like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\+7x^{2}-4x+8\\end{array}[\/latex]<\/p>\r\nCombine like terms, paying close attention to the signs.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\\\underline{+7x^{2}-4x+8}\\\\10x^{2}-2x+1\\end{array}[\/latex]<b>\u00a0<\/b><\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3x^{2}+3x-7\\right)+\\left(7x^{2}-4x+8\\right)=10x^{2}-2x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn't so tidy. When there isn't a matching like term for every term, there will be empty places in the vertical arrangement.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)[\/latex]\r\n\r\n[reveal-answer q=\"232680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232680\"]Write one polynomial below the other, lining up like terms vertically.\r\n\r\nTo keep track of like terms, you can insert zeros where there aren't any shared like terms. This is optional, but some find it helpful.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+2\\\\+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10\\end{array}[\/latex]<\/p>\r\nCombine like terms, paying close attention to the signs.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+\\,\\,\\,2\\\\\\underline{+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10}\\\\4x^{3}\\,+\\,\\,x^{2}-6x+12\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)=4x^{3}+x^{2}-6x+12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may be thinking, how is this different than combining like terms, which we did in the last section? The answer is, it's not really. We just added a layer to combining like terms by adding more terms to combine. :) Polynomials are a useful tool for describing the behavior of anything that isn't linear, and sometimes you may need to add them.\r\n\r\nIn the following video, you will see more examples of combining like terms by adding polynomials.\r\n\r\nhttps:\/\/youtu.be\/KYZR7g7QcF4\r\n<h2>Find the opposite of a polynomial<\/h2>\r\n[caption id=\"attachment_4552\" align=\"aligncenter\" width=\"385\"]<img class=\" wp-image-4552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/02201910\/Screen-Shot-2016-06-02-at-1.14.57-PM-300x198.png\" alt=\"Man in a leather jacket and slicked back hair looking very rebellious sitting next to a woman in a pink dress looking very proper.\" width=\"385\" height=\"254\" \/> Opposites[\/caption]\r\n\r\n<img class=\" wp-image-4554 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/02202350\/Screen-Shot-2016-06-02-at-1.22.59-PM.png\" alt=\"SCale with a(b+c) on one side and ab+ac on the other adn an equal sign in between the two sides of the scale\" width=\"165\" height=\"123\" \/>When you are solving equations, it may come up that you need to subtract polynomials. This means subtracting each term of a polynomial, which requires\u00a0changing the sign of each term in a polynomial. Recall that changing the sign\u00a0of 3 gives [latex]\u22123[\/latex], and changing the sign\u00a0of [latex]\u22123[\/latex] gives 3. Just as changing the sign\u00a0of a number is found by multiplying the number by [latex]\u22121[\/latex], we can change the sign\u00a0of a polynomial by multiplying it by [latex]\u22121[\/latex]. Think of this in the same way as you would the distributive property. \u00a0You are distributing [latex]\u22121[\/latex] to each term in the polynomial. \u00a0Changing the sign of a polynomial is also called finding the opposite.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the opposite of [latex]9x^{2}+10x+5[\/latex].\r\n\r\n[reveal-answer q=\"161313\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"161313\"]Find the opposite by multiplying by [latex]\u22121[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\r\nDistribute [latex]\u22121[\/latex] to each term in the polynomial.\r\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)9x^{2}+\\left(-1\\right)10x+\\left(-1\\right)5[\/latex]<\/p>\r\nYour new terms all have the opposite sign:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(-1\\right)9x^{2}=-9x^{2}\\\\\\text{ }\\\\\\left(-1\\right)10x=-10x\\\\\\text{ }\\\\\\left(-1\\right)5=-5\\end{array}[\/latex]<\/p>\r\nNow you can rewrite the polynomial with the new sign on each term:\r\n<p style=\"text-align: center;\">[latex]-9x^{2}-10x-5[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe opposite of [latex]9x^{2}+10x+5[\/latex] is [latex]-9x^{2}-10x-5[\/latex]\r\n\r\nYou can also write:\r\n\r\n[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)=-9x^{2}-10x-5[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"46\" height=\"41\" \/>Be careful when there are negative terms\u00a0or subtractions in the polynomial already. \u00a0Just remember that you are changing the sign, so if it is negative, it will become positive.<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the opposite of [latex]3p^{2}\u20135p+7[\/latex].\r\n\r\n[reveal-answer q=\"278382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"278382\"]Find the opposite by multiplying by [latex]-1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)\\left(3p^{2}-5p+7\\right)[\/latex]<\/p>\r\nDistribute [latex]-1[\/latex] to each term in the polynomial by multiplying each coefficient by [latex]-1[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)3p^{2}+\\left(-1\\right)\\left(-5p\\right)+\\left(-1\\right)7[\/latex]<\/p>\r\nYour new terms all have the opposite sign:\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(-1\\right)3p^{2}=-3p^{2}\\\\\\text{ }\\\\\\left(-1\\right)\\left(-5p\\right)=5p\\\\\\text{ }\\\\\\left(-1\\right)7=-7\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Now you can rewrite the polynomial with the new sign on each term:<\/p>\r\n<p style=\"text-align: center;\">[latex]-3p^{2}+5p-7[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nThe opposite of [latex]3p^{2}-5p+7[\/latex] is [latex]-3p^{2}+5p-7[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that in finding the opposite of a polynomial, you change the sign of <i>each term<\/i> in the polynomial, then rewrite the polynomial with the new signs on each term.\r\n<h2 id=\"title3\">Subtract polynomials<\/h2>\r\nWhen you subtract one polynomial from another, you will first find the opposite of the polynomial being subtracted, then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]\r\n\r\n[reveal-answer q=\"267023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"267023\"]Change\u00a0the sign of <i>each<\/i> term in the polynomial [latex]9x^{2}+10x+5[\/latex]! All the terms are positive, so they will all become negative.\r\n<p style=\"text-align: center;\">[latex]\\left(15x^{2}+12x+20\\right)-9x^{2}-10x-5[\/latex]<\/p>\r\nRegroup to match like terms, remember to check\u00a0the sign of each term.\r\n<p style=\"text-align: center;\">[latex]15x^{2}-9x^{2}+12x-10x+20-5[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center;\">[latex]6x^{2}+2x+15[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(15x^{2}+12x+20\\right)-\\left(9x^{2}+10x+5\\right)=6x^{2}+2x+15[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"52\" height=\"47\" \/>When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]\r\n\r\n[reveal-answer q=\"783926\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"783926\"]Change the sign of each term in the polynomial [latex]7x^{3}+5x^{2}\u20138x+10[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-7x^{3}-5x^{2}+8x-10[\/latex]<\/p>\r\nRegroup to put like terms together and combine like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\underbrace{14x^{3}-7x^{3}}+\\underbrace{3x^{2}-5x^{2}}-\\underbrace{5x+8x}+\\underbrace{14-10}\\\\=7x^{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Write the resulting polynomial with each term's sign in front.<\/p>\r\n<p style=\"text-align: center;\">[latex]7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you have many terms,\u00a0like in the example above, try the vertical approach from the previous page to keep your terms organized. \u00a0However you choose to combine polynomials is up to you\u2014the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]\r\n\r\n[reveal-answer q=\"29114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"29114\"]Reorganizing using the vertical approach.\r\n<p style=\"text-align: center;\">[latex]14x^{3}+3x^{2}-5x+14-\\left(7x^{3}+5x^{2}-8x+10\\right)[\/latex]<\/p>\r\nChange the signs, and combine like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}14x^{3}+3x^{2}-5x+14\\,\\,\\,\\,\\\\\\underline{-7x^{3}-5x^{2}+8x-10}\\\\=7x^{3}-2x^{2}+3x+4\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see more examples of subtracting polynomials.\r\n\r\nhttps:\/\/youtu.be\/xq-zVm25VC0\r\n<h2 id=\"title1\">Find the product of monomials<\/h2>\r\nMultiplying <strong>polynomials<\/strong> involves applying the rules of exponents and the distributive property to simplify the product. Polynomial\u00a0multiplication can be useful in modeling real world situations. Understanding polynomial products is an important step in learning to solve algebraic equations involving polynomials. There are many, varied uses for polynomials including the generation of 3D graphics for entertainment and industry, as in the image below.\r\n\r\n[caption id=\"attachment_4567\" align=\"aligncenter\" width=\"431\"]<img class=\" wp-image-4567\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03193513\/Screen-Shot-2016-06-03-at-12.33.29-PM-300x154.png\" alt=\"Bumpy, irregular surfaces on a grid made with AutoCAD\" width=\"431\" height=\"221\" \/> Surfaces made from polynomials with AutoCAD[\/caption]\r\n\r\nIn the exponents section, we\u00a0practiced\u00a0multiplying\u00a0monomials together, like we did with this expression: [latex]24{x}^{8}2{x}^{5}[\/latex]. The only thing different between that section and this one is that we called it simplifying, and now we are calling it polynomial multiplication. \u00a0Remember that simplifying a mathematical expression means performing as many operations as we can until there are no more to perform, including multiplication. \u00a0In this section we will show examples of how to multiply more than just monomials. \u00a0We will multiply monomials with\u00a0binomials and trinomials. We will also learn some techniques for multiplying two binomials together.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. [latex]-9x^{3}\\cdot 3x^{2}[\/latex]\r\n\r\n[reveal-answer q=\"322242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"322242\"]\r\n\r\nRearrange the factors.\r\n<p style=\"text-align: center;\">[latex]-9\\cdot3\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\r\nMultiply constants. Remember that a positive number times a negative number yields a negative number.\r\n<p style=\"text-align: center;\">[latex]-27\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\r\nMultiply variable terms. Remember to add the exponents when multiplying exponents with the same base.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}-27\\cdot x^{3+2}\\\\-27\\cdot x^{5}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]-9x^{3}\\cdot 3x^{2}=-27x^{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThat\u2019s it! When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents, like this:\r\n<p style=\"text-align: center;\">[latex] \\displaystyle 5{{a}^{4}}\\cdot 7{{a}^{6}}=35{{a}^{10}}[\/latex]<\/p>\r\nThe following video provides more examples of multiplying monomials with different exponents.\r\n\r\nhttps:\/\/youtu.be\/30x8hY32B0o\r\n<h2 id=\"title2\">Find the product of polynomials and monomials<\/h2>\r\nThe distributive property can be used to multiply a monomial and a binomial. Just remember that the monomial must be multiplied by each term in the binomial. In the next example, you will see how to multiply a second degree monomial with a binomial. \u00a0Note the use of exponent rules.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]5x^2\\left(4x^{2}+3x\\right)[\/latex]\r\n[reveal-answer q=\"176215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"176215\"]Distribute the monomial to each term of the binomial. Multiply coefficients and variables separately.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5x^2\\left(4x^{2}\\right)+5x^2\\left(3x\\right)\\\\\\text{ }\\\\=20x^{2+2}+15x^{2+1}\\\\\\text{ }\\\\=20x^{4}+15x^{3}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5x^2\\left(4x^{2}+3x\\right)=20x^{4}+15x^{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's add another layer by multiplying a monomial by a trinomial. Consider the expression [latex]2x\\left(2x^{2}+5x+10\\right)[\/latex].\r\n\r\nThis expression can be modeled with a sketch like the one below.\r\n\r\n<img class=\"aligncenter wp-image-2204 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/24212311\/Screen-Shot-2016-03-24-at-2.22.48-PM.png\" alt=\"2x times 2x squared equals 4x cubed. 2x times 5x equals 10x squared. 2x times 10 equals 20x.\" width=\"508\" height=\"79\" \/>\r\n<p style=\"text-align: left;\">The only difference between this example and the previous one is there is one more term to distribute the monomial to.<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x\\left(2x^{2}+5x+10\\right)=2x\\left(2x^{2}\\right)+2x\\left(5x\\right)=2x\\left(10\\right)\\\\=4x^{3}+10x^{2}+20x\\end{array}[\/latex]<\/p>\r\nYou will always need to pay attention to negative signs when you are multiplying. Watch\u00a0what happens to the sign on the terms in the trinomial when it is multiplied by a negative monomial in the next example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]-7x\\left(2x^{2}-5x+1\\right)[\/latex]\r\n[reveal-answer q=\"590272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"590272\"]\r\n\r\nDistribute the monomial to each term in the trinomial.\r\n<p style=\"text-align: center;\">[latex]-7x\\left(2x^{2}\\right)-7x\\left(-5x\\right)-7x\\left(1\\right)[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}-14x^{1+2}+35x^{1+1}-7x\\\\\\text{ }\\\\-14x^{3}+35x^{2}-7x\\end{array}[\/latex]<\/p>\r\nRewrite addition of terms with negative coefficients as subtraction.\r\n<h4>Answer<\/h4>\r\n[latex]7x^{2}\\left(2x^{2}-5x+1\\right)=14x^{4}-35^{3}+7x^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p class=\"no-indent\" style=\"text-align: left;\">The following video provides more examples of multiplying a monomial and a polynomial.<\/p>\r\nhttps:\/\/youtu.be\/bwTmApTV_8o\r\n<h2 id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">Find the product of two binomials<\/h2>\r\nNow let's explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You'll use each binomial as one of the dimensions of a rectangle, and their product as the area.\r\n\r\nThe model below shows [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]:\r\n\r\n[caption id=\"attachment_4607\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-4607\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/04191552\/Screen-Shot-2016-06-04-at-12.15.06-PM-300x290.png\" alt=\"Visual representation of multiplying two binomials.\" width=\"300\" height=\"290\" \/> Visual representation of multiplying two binomials.[\/caption]\r\n\r\nEach binomial is expanded into variable terms and constants, [latex]x+4[\/latex], along the top of the model and [latex]x+2[\/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[\/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[\/latex].\r\n\r\nYou can use the distributive property to determine the product of two binomials.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]\r\n[reveal-answer q=\"186797\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186797\"]Distribute the [latex]x[\/latex] over [latex]x+2[\/latex], then distribute 4 over [latex]x+2[\/latex].\r\n<p style=\"text-align: center;\">[latex]x\\left(x\\right)+x\\left(2\\right)+4\\left(x\\right)+4\\left(2\\right)[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center;\">[latex]x^{2}+2x+4x+8[\/latex]<\/p>\r\nCombine like terms [latex]\\left(2x+4x\\right)[\/latex].\r\n<p style=\"text-align: center;\">[latex]x^{2}+6x+8[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+4\\right)\\left(2x+2\\right)=x^{2}+6x+8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLook back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[\/latex] comes from. Can you see where you multiply [latex]x[\/latex] by [latex]x + 2[\/latex], and where you get [latex]x^{2}[\/latex]\u00a0from [latex]x\\left(x\\right)[\/latex]?\r\n\r\nAnother way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[\/latex] in [latex]x+4[\/latex] gets multiplied by both the [latex]x[\/latex] and the 2 from [latex]x+2[\/latex], and the 4 gets multiplied by both the [latex]x[\/latex] and the 2.\r\n\r\nThe following video provides an example of multiplying two binomials using an area model as well as repeated distribution.\r\n\r\nhttps:\/\/youtu.be\/u4Hgl0BrUlo\r\n\r\nIn the next section we will explore other methods for multiplying two binomials, and\u00a0become aware of the different forms that binomials can have.\r\n<h2>FOIL<\/h2>\r\n[caption id=\"attachment_4589\" align=\"aligncenter\" width=\"335\"]<img class=\" wp-image-4589\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03212853\/Screen-Shot-2016-06-03-at-2.08.13-PM-243x300.png\" alt=\"Crane made from aluminum foil\" width=\"335\" height=\"414\" \/> Foil Crane[\/caption]\r\n\r\nIn the last section we finished with an example of multiplying two binomials,[latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. In this section we will provide examples of how to use two different methods to multiply to binomials. Keep in mind as you read through the page that simplify and multiply are used interchangeably.\r\n\r\nSome people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for <strong>First, Outer, Inner, Last.<\/strong> Let's go back to the example from the previous page, where we were asked to multiply the two binomials: [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. \u00a0The following steps show you how to apply this method to multiplying two binomials.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}\\text{ term in each binomial}: \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(2\\right)=2x\\\\\\text{Inner terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(x\\right)=4x\\\\\\text{Last terms in each binomial}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(2\\right)=8\\end{array}[\/latex]<\/p>\r\nWhen you add the four results, you get the same answer,\u00a0[latex]x^{2}+2x+4x+8=x^{2}+6x+8[\/latex].\r\nThe last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.\r\n<div class=\"textbox shaded\"><img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"41\" height=\"36\" \/>Caution! Note that the FOIL method only works for multiplying two binomials together. It sill not work for multiplying a binomial and a trinomial, or two trinomials.<\/div>\r\n\r\n[caption id=\"attachment_4595\" align=\"alignleft\" width=\"138\"]<img class=\" wp-image-4595\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03231013\/Screen-Shot-2016-06-03-at-4.09.32-PM-299x300.png\" alt=\"two tomatoes sitting next to each other with two different phonetic pronunciations for the word tomato underneath\" width=\"138\" height=\"139\" \/> Order Doesn't Matter When You Multiply[\/caption]\r\n\r\nOne of the neat things about multiplication is that\u00a0terms can be multiplied in either order. The expression [latex]\\left(x+2\\right)\\left(x+4\\right)[\/latex] has the same product as [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex], [latex]x^{2}+6x+8[\/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.\r\n\r\nPolynomials can take many forms. \u00a0So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\\left(2r-3\\right)[\/latex]. \u00a0Variables may also be on the right of the constant term, as in this binomial [latex]\\left(5+r\\right)[\/latex]. \u00a0In the next example, we will show that multiplying binomials in this form requires one extra step at the end.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the product.[latex]\\left(3\u2013s\\right)\\left(1-s\\right)[\/latex]\r\n[reveal-answer q=\"531601\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"531601\"]\r\n\r\nNotice how the binomials have the variable on the right instead of the left. \u00a0There is nothing different in the way you find the product. \u00a0At the end we will reorganize terms so they are in descending order as a matter of convention.\r\n\r\n[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)[\/latex]\r\n\r\nUse a table this time.\r\n<table style=\"width: 40%;\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<th>[latex]3[\/latex]<\/th>\r\n<th>[latex]-s[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]1[\/latex]<\/th>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]-s[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<th>[latex]-s[\/latex]<\/th>\r\n<td>[latex]-3s[\/latex]<\/td>\r\n<td>[latex]s^2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice how the <em>s<\/em> term is now positive. Collect the terms and simplify.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=3-3s-s+s^2\\\\\\text{ }\\\\=3-4s+s^2\\end{array}[\/latex]<\/p>\r\nAs a matter of convention, we will organize the terms so the one with greatest degree comes first. Pay close attention to the signs on the terms when you reorganize them. The 3 is positive, so we will use a plus in front of it, and the 4 is negative so we use a minus in front of it.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=s^{2}-4s+3\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)=s^2-4s+3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example, you will see that sometimes there are constants in front of the variable. They will get multiplied together just as we have done before.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]\\left(4x\u201310\\right)\\left(2x+3\\right)[\/latex] using the FOIL acronym.\r\n\r\n[reveal-answer q=\"930433\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"930433\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,4x\\left(2x\\right)=8x^{2}\\\\\\text{Outer}:\\,\\,\\,4x\\left(3\\right)=12x\\\\\\text{Inner}:\\,\\,\\,\u221210\\left(2x\\right)=-20x\\\\\\text{Last}:\\,\\,\\,\\,\\,-10\\left(3\\right)=-30\\end{array}[\/latex]<\/p>\r\nBe careful about including the negative sign on the [latex]\u201110[\/latex], since 10 is subtracted.\r\n\r\nCombine like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}8x^{2}+12x\u201320x\u201330\\\\\\text{ }\\\\=8x^{2}-8x\u201330\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(4x\u201310\\right)\\left(2x+3\\right)=8x^{2}\u20138x\u201330[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe video that follows gives another example of multiplying two binomials using the FOIL acronym. Remember this method only works when you are multiplying two binomials.\r\n\r\nhttps:\/\/youtu.be\/_MrdEFnXNGA\r\n<h2>The Table Method<\/h2>\r\nYou may see a binomial multiplied by itself written as\u00a0[latex]{\\left(x+3\\right)}^{2}[\/latex] instead of\u00a0[latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. \u00a0To find this product, let's use another method. We will place the terms of each binomial along the top row and first column of a table, like this:\r\n<table class=\"lines\" style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.\r\n<table style=\"width: 20%;\">\r\n<thead>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td><span style=\"color: #0000ff;\">[latex]x\\cdot{x}=x^2[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #ff0000;\">[latex]3\\cdot{x}=+3x[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<td><span style=\"color: #ff0000;\">[latex]x\\cdot{3}=+3x[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #ff00ff;\">\u00a0[latex]3\\cdot{3}=+9[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can write the terms of the polynomial from the entries in the table:\r\n<p style=\"text-align: center;\">[latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">=\u00a0<span style=\"color: #0000ff;\">[latex]x^2[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span><\/p>\r\n<p style=\"text-align: center;\">= <span style=\"color: #0000ff;\">[latex]x^{2}[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]6x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span>.<\/p>\r\n<p style=\"text-align: center;\">Pretty cool, huh?<\/p>\r\nSo far, we have shown two\u00a0methods for multiplying two binomials together. \u00a0Why are we focusing\u00a0so much on binomials? \u00a0They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSquare the binomial difference\u00a0[latex]\\left(x\u20137\\right)[\/latex]\r\n[reveal-answer q=\"293164\"]Show solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"293164\"]\r\n\r\nWrite the product of the binomial.\r\n<p style=\"text-align: center;\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Let's use the table method, just because. Note how we carry the negative sign with the\u00a0[latex]7[\/latex].<\/p>\r\n\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>\u00a0[latex]x[\/latex]<\/td>\r\n<td>\u00a0[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0[latex]x[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]x^2[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\u00a0\u00a0[latex]-7[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\r\n<td>\u00a0\u00a0[latex]49[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCollect the terms, and simplify. Note how we keep the sign on each term.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\r\nAnswer\r\n[latex]x^2-14x+49[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:\r\n<p style=\"text-align: center;\">You can't move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\r\n<p style=\"text-align: center;\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><strong>\u00a0CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\nIn the video that follows, you will see another examples of using a table to multiply two binomials.\r\n\r\nhttps:\/\/youtu.be\/tWsLJ_pn5mQ\r\n<h2>Further Examples<\/h2>\r\nThe next couple of examples show you some different forms binomials can take. \u00a0In the first, we will square a binomial that has a coefficient in front of the variable, like the product in the first example on this page. In the second we will find\u00a0the product of two binomials that have the variable on the right instead of the left.\u00a0We will use both\u00a0the FOIL method and the table method to simplify.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the product.\u00a0[latex]\\left(2x+6\\right)^{2}[\/latex]\r\n[reveal-answer q=\"255359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"255359\"]We will use the FOIL method.\r\n[latex]\\left(2x+6\\right)^{2}=\\left(2x+6\\right)\\left(2x+6\\right)[\/latex]\r\n[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,2x\\left(2x\\right)=4x^{2}\\\\\\text{Outer}:\\,\\,\\,2x\\left(6\\right)=12x\\\\\\text{Inner}:\\,\\,\\,6\\left(2x\\right)=12x\\\\\\text{Last}:\\,\\,\\,\\,\\,6\\left(6\\right)=36\\end{array}[\/latex]\r\n\r\nNow you can collect the terms and simplify:\r\n[latex]\\begin{array}{c}4x^2+12x+12x+36\\\\\\text{ }\\\\=4x^2+24x+36\\end{array}[\/latex]\r\n\r\nAnswer\r\n\r\n[latex](2x+6)^{2}=4x^{2}+24x+36[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the last example, we want to show you another common form a binomial can take, each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply the binomials. [latex]\\left(x+8\\right)\\left(x\u20138\\right)[\/latex]\r\n[reveal-answer q=\"812247\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812247\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer}:\\,\\,\\,\\,\\,\\,x\\left(-8\\right)=-8x\\\\\\text{Inner}:\\,\\,\\,\\,\\,\\,\\,8\\left(x\\right)=+8x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\left(-8\\right)=-64\\end{array}[\/latex]<\/p>\r\nAdd the terms. Note how the two x terms are opposites, so they sum to zero.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x^{2}\\underbrace{-8x+8x}-64\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\\\=x^2-64\\\\\\text{ }\\\\\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+8\\right)\\left(x-8\\right)=x^{2}-64[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\n<p style=\"text-align: left;\">There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(a-b\\right)^{2}=\\left(a-b\\right)\\left(a-b\\right)[\/latex],<\/p>\r\n<p style=\"text-align: left;\">the resulting product, after being simplified, will look like this:<\/p>\r\n<p style=\"text-align: center;\">[latex]a^2-2ab+b^2[\/latex].<\/p>\r\n<p style=\"text-align: left;\">The product of a binomial sum will have the following predictable outcome:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(a+b\\right)^{2}=\\left(a+b\\right)\\left(a+b\\right)=a^2+2ab+b^2[\/latex].<\/p>\r\n<p style=\"text-align: left;\">Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. \u00a0You will learn more about predictable patterns from products of binomials in later math classes.<\/p>\r\n\r\n<\/div>\r\nIn this section we showed two ways to find the product of two binomials, the FOIL method, and by using a table. Some of the forms a product of two binomials can take are listed here:\r\n<ul>\r\n \t<li>[latex]\\left(x+5\\right)\\left(2x-3\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(x+7\\right)^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(x-1\\right)^{2}[\/latex]<\/li>\r\n \t<li>[latex]\\left(2-y\\right)\\left(5+y\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(x+9\\right)\\left(x-9\\right)[\/latex]<\/li>\r\n \t<li>[latex]\\left(2x-4\\right)\\left(x+3\\right)[\/latex]<\/li>\r\n<\/ul>\r\nAnd this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.\r\n<h2>Divide a polynomial by a monomial<\/h2>\r\nThe fourth arithmetic operation is division, the inverse of multiplication. Division of polynomials isn\u2019t much different from division of numbers. In the exponential section,\u00a0you were asked to simplify expressions such as:\u00a0[latex]\\displaystyle\\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]. This expression is the division of two monomials.\u00a0To simplify it, we\u00a0divided\u00a0the coefficients and then divided the variables. In this section we will add another layer to this idea by dividing polynomials by monomials, and by binomials.\r\n\r\nThe distributive property states that you can distribute a factor that is being multiplied by a sum or difference, and likewise you can distribute a <em>divisor<\/em> that is being divided into a sum or difference. In this example, you can add all the terms in the numerator, then divide by 2.\r\n<p style=\"text-align: center;\">[latex]\\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\r\nOr you can\u00a0first divide each term by 2, then simplify the result.\r\n<p style=\"text-align: center;\">[latex] \\frac{8}{2}+\\frac{4}{2}+\\frac{10}{2}=4+2+5=11[\/latex]<\/p>\r\nEither way gives you the same result. The second way is helpful when you can't combine like terms in the numerator. \u00a0Let\u2019s try something similar with a binomial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDivide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]\r\n[reveal-answer q=\"641821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641821\"]\r\n\r\nDistribute [latex]3a^2[\/latex]<i>\u00a0<\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]\r\n\r\n[latex]\\frac{9a^3}{3a^2}+\\frac{6a}{3a^2}[\/latex]\r\n\r\nDivide each term, a monomial divided by another monomial.\r\n\r\n[latex]\\begin{array}{c}3a^{3-2}+2a^{1-2}\\\\\\text{ }\\\\=3a^{1}+2a^{-1}\\\\\\text{ }\\\\=3a+2a^{-1}\\end{array}[\/latex]\r\n\r\nRewrite [latex]a^{-1}[\/latex] with positive exponents, as a matter of convention.\r\n\r\n[latex]3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{9a^3+6a}{3a^2}=3a+\\frac{2}{a}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example, you will see that the same ideas apply when you are dividing a trinomial by a monomial. You can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. As we have throughout the course, simplifying with exponents includes rewriting negative exponents as positive. Pay attention to the signs of the terms in the next example, we will divide by a negative monomial.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide. [latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]\r\n[reveal-answer q=\"324719\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"324719\"]Divide each term in the polynomial by the monomial.\r\n<p style=\"text-align: center;\">[latex] \\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\r\nNote how the term[latex]-\\frac{18}{-6y}[\/latex] does not have a <em>y<\/em> in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, 27 doesn't divide nicely by [latex]-6[\/latex], so we are left with a fraction as the coefficient on the [latex]y^3[\/latex] term.\r\n\r\nSimplify.\r\n<p style=\"text-align: center;\">[latex] -\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow, we ask you to think about what would happen if you were given a quotient like this to simplify:\u00a0[latex] \\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y+3}[\/latex]. \u00a0You may be tempted to divide each term of\u00a0[latex] {27{{y}^{4}}+6{{y}^{2}}-18}[\/latex] individually by [latex]-6y[\/latex], then [latex]3[\/latex]. \u00a0This would go against the order of operations because the division sign is a grouping symbol, and the addition in the denominator cannot be simplified anymore. The result is that\u00a0no further operations can be performed with the tools we know. \u00a0We can, however, call into use a tool that you may have learned in gradeschool: long division.\r\n<h2 id=\"title1\">Polynomial Long Division<\/h2>\r\n<img class=\"wp-image-4616 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/06192222\/Screen-Shot-2016-06-06-at-12.20.58-PM-300x169.png\" alt=\"I'm Not Allergic to Long Division\" width=\"545\" height=\"307\" \/>\r\n\r\nRecall how you can use long division to divide two whole numbers, say 900 divided by 37.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064125\/image007.jpg\" alt=\"The dividend in 900 and the divisor is 37.\" width=\"51\" height=\"18\" \/>\r\nFirst, you would think about how many 37s are in 90, as 9 is too small. (<i>Note: <\/i>you could also think, how many 40s are there in 90.)\r\n\r\n<img class=\"aligncenter wp-image-2251 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223536\/Screen-Shot-2016-03-28-at-3.35.17-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.35.17 PM\" width=\"69\" height=\"55\" \/>\r\nThere are two 37s in 90, so write 2 above the last digit of 90. Two 37s is 74; write that product below the 90.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image008.jpg\" alt=\"\" width=\"51\" height=\"57\" \/>\r\nSubtract: [latex]90\u201374[\/latex] is 16. (If the result is larger than the divisor, 37, then you need to use a larger number for the quotient.)\r\n\r\n<img class=\"aligncenter size-full wp-image-2252\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223630\/Screen-Shot-2016-03-28-at-3.36.06-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.36.06 PM\" width=\"66\" height=\"68\" \/>\r\n\r\nBring down the next digit (0) and consider how many 37s are in 160.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image009.jpg\" alt=\"\" width=\"51\" height=\"74\" \/>\r\nThere are four 37s in 160, so write the 4 next to the two in the quotient. Four 37s is 148; write that product below the 160.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image010.jpg\" alt=\"\" width=\"51\" height=\"86\" \/>\r\nSubtract: [latex]160\u2013148[\/latex] is 12. This is less than 37 so the 4 is correct. Since there are no more digits in the dividend to bring down, you\u2019re done.\r\n\r\nThe final answer is 24 R12, or [latex]24\\frac{12}{37}[\/latex]. You can check this by multiplying the quotient (without the remainder) by the divisor, and then adding in the remainder. The result should be the dividend:\r\n<p style=\"text-align: center;\">[latex]24\\cdot37+12=888+12=900[\/latex]<\/p>\r\nTo divide polynomials, use the same process. This example shows how to do this when dividing by a <strong>binomial<\/strong>.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide:\u00a0[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}[\/latex]\r\n[reveal-answer q=\"455187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"455187\"]How many <i>x<\/i>\u2019s are there in [latex]x^{2}[\/latex]? That is, what is [latex] \\frac{{{x}^{2}}}{x}[\/latex]?\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image011.jpg\" alt=\"\" width=\"118\" height=\"18\" \/>\r\n\r\n[latex] \\frac{{{x}^{2}}}{x}=x[\/latex]<i>. <\/i>Put <i>x<\/i> in the quotient above the [latex]-4x[\/latex]<i>\u00a0<\/i>term. (These are like terms, which helps to organize the problem.)\r\n\r\nWrite the product of the divisor and the part of the quotient you just found under the dividend. Since [latex]x\\left(x+2\\right)=x^{2}+2x[\/latex],\u00a0write this underneath, and get ready to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image012.jpg\" alt=\"\" width=\"118\" height=\"51\" \/>\r\n\r\nRewrite [latex]\u2013\\left(x^{2} + 2x\\right)[\/latex]\u00a0as its opposite [latex]\u2013x^{2}\u20132x[\/latex]\u00a0so that you can add the opposite. (Adding the opposite is the same as subtracting, and it is easier to do.)\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image013.jpg\" alt=\"\" width=\"118\" height=\"48\" \/>\r\n\r\nAdd\u00a0[latex]-x^{2}[\/latex] to [latex]x^{2}[\/latex], and [latex]-2x[\/latex] to [latex]-4x[\/latex].\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064128\/image014.jpg\" alt=\"\" width=\"118\" height=\"60\" \/>\r\n\r\nBring down [latex]-12[\/latex].\r\n\r\n<img class=\"aligncenter size-full wp-image-2253\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232007\/Screen-Shot-2016-03-28-at-4.19.50-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.19.50 PM\" width=\"137\" height=\"75\" \/>\r\n\r\nRepeat the process. How many times does <i>x<\/i> go into [latex]-6x[\/latex]? In other words, what is [latex] \\frac{-6x}{x}[\/latex]?\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image015.jpg\" alt=\"\" width=\"118\" height=\"60\" \/>\r\n\r\nSince [latex] \\frac{-6x}{x}=-6[\/latex], write [latex]-6[\/latex] in the quotient. Multiply [latex]-6[\/latex] and [latex]x+2[\/latex]\u00a0and prepare to subtract the product.\r\n\r\n<img class=\"aligncenter size-full wp-image-2254\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232512\/Screen-Shot-2016-03-28-at-4.24.52-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.24.52 PM\" width=\"165\" height=\"113\" \/>\r\n\r\nRewrite [latex]\u2013\\left(-6x\u201312\\right)[\/latex] as [latex]6x+12[\/latex], so that you can add the opposite.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image016.jpg\" alt=\"\" width=\"118\" height=\"80\" \/>\r\n\r\nAdd. In this case, there is no remainder, so you\u2019re done.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image017.jpg\" alt=\"\" width=\"118\" height=\"90\" \/>\r\n<h4>Answer<\/h4>\r\n[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}=x\u20136[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nCheck this by multiplying:\r\n<p style=\"text-align: center;\">[latex]\\left(x-6\\right)\\left(x+2\\right)=x^{2}+2x-6x-12=x^{2}-4x-12[\/latex]<\/p>\r\nPolynomial long division involves many steps. \u00a0Hopefully this video will help you determine what step to do next when you are using polynomial long division.\r\n\r\nhttps:\/\/youtu.be\/KUPFg__Djzw\r\n\r\nLet\u2019s try another example. In this example, a term is \u201cmissing\u201d from the dividend.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nDivide: [latex]\\frac{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}[\/latex]\r\n[reveal-answer q=\"523374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"523374\"]In setting up this problem, notice that there is an [latex]x^{3}[\/latex]\u00a0term but no [latex]x^{2}[\/latex]\u00a0term. Add [latex]0x^{2}[\/latex]\u00a0as a \u201cplace holder\u201d for this term. (Since 0 times anything is 0, you\u2019re not changing the value of the dividend.)\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image018.jpg\" alt=\"\" width=\"153\" height=\"18\" \/>\r\n\r\nFocus on the first terms again: how many <i>x<\/i>\u2019s are there in [latex]x^{3}[\/latex]? Since [latex] \\frac{{{x}^{3}}}{x}=x^{2}[\/latex], put [latex]x^{2}[\/latex]\u00a0in the quotient.\r\n\r\nMultiply [latex]x^{2}\\left(x\u20133\\right)=x^{3}\u20133x^{2}[\/latex], write this underneath the dividend, and prepare to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064131\/image019.jpg\" alt=\"\" width=\"153\" height=\"49\" \/>\r\n\r\nRewrite the subtraction using the opposite of the expression [latex]x^{3}-3x^{2}[\/latex]. Then add.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064132\/image020.jpg\" alt=\"\" width=\"153\" height=\"59\" \/>\r\n\r\nBring down the rest of the expression in the dividend. It\u2019s helpful to bring down <i>all<\/i> of the remaining terms.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064133\/image021.jpg\" alt=\"\" width=\"153\" height=\"59\" \/>\r\n\r\nNow, repeat the process with the remaining expression, [latex]3x^{2}-6x\u201310[\/latex], as the dividend.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image022.jpg\" alt=\"\" width=\"153\" height=\"79\" \/>\r\n\r\nRemember to watch the signs!\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image023.jpg\" alt=\"\" width=\"153\" height=\"89\" \/>\r\n\r\nHow many <i>x<\/i>\u2019s are there in 3<i>x<\/i>? Since there are 3, multiply [latex]3\\left(x\u20133\\right)=3x\u20139[\/latex], write this underneath the dividend, and prepare to subtract.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064135\/image024.jpg\" alt=\"\" width=\"157\" height=\"118\" \/>\r\n\r\nContinue until the <strong>degree<\/strong> of the remainder is <i>less <\/i>than the degree of the divisor. In this case the degree of the remainder, [latex]-1[\/latex], is 0, which is less than the degree of [latex]x-3[\/latex], which is 1.\r\n\r\nAlso notice that you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem.\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064221\/image025.jpg\" alt=\"\" width=\"153\" height=\"118\" \/>\r\nYou can write the remainder using the symbol R, or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite.\r\n<h4>Answer<\/h4>\r\n[latex]\\begin{array}{r}{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}=x^{2}+3x+3+R-1,\\\\x^{2}+3x+3+\\frac{-1}{x-3}, \\text{ or }\\\\x^{2}+3x+3-\\frac{1}{x-3}\\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nCheck the result:\r\n<p style=\"text-align: center;\">[latex]\\left(x\u20133\\right)\\left(x^{2}+3x+3\\right)\\,\\,\\,=\\,\\,\\,x\\left(x^{2}+3x+3\\right)\u20133\\left(x^{2}+3x+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}+3x^{2}+3x\u20133x^{2}\u20139x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,x^{3}\u20136x\u20139+\\left(-1\\right)\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u201310[\/latex]<\/p>\r\nThe video that follows shows another example of dividing a third degree trinomial by a first degree binomial.\r\n\r\nhttps:\/\/youtu.be\/Rxds7Q_UTeo\r\n\r\nThe last video example shows how to divide a third degree trinomial by a second degree binomial.\r\n\r\nhttps:\/\/youtu.be\/P6OTbUf8f60\r\n<h2>Summary<\/h2>\r\nTo divide a monomial by a monomial, divide the coefficients (or simplify them as you would a fraction) and divide the variables with like bases by subtracting their exponents. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Be sure to watch the signs! Final answers should be written without any negative exponents. \u00a0Dividing polynomials by polynomials of more than one term can be done using a process very much like long division of whole numbers. You must be careful to subtract entire expressions, not just the first term. Stop when the degree of the remainder is less than the degree of the divisor. The remainder can be written using R notation, or as a fraction added to the quotient with the remainder in the numerator and the divisor in the denominator.\r\n<h2>Summary<\/h2>\r\nWe have seen that subtracting a polynomial means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are alike. \u00a0How you organize this process is up to you, but we have shown two ways here. \u00a0One method is to place the terms next to each other horizontally, putting like terms next to each other to make combining them easier. \u00a0The other method was to place the polynomial being subtracted underneath the other after changing the signs of each term. In this method it is important to align like terms and use a blank space when there is no like term.\r\n\r\nMultiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.\r\n<h2><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add polynomials\n<ul>\n<li>Use horizontal and vertical organization to add polynomials<\/li>\n<\/ul>\n<\/li>\n<li>Find the opposite of a polynomial\n<ul>\n<li>Subtract polynomials using both horizontal and vertical organization<\/li>\n<\/ul>\n<\/li>\n<li>Multiply Polynomials<\/li>\n<li>\n<ul>\n<li>Find the product of monomials<\/li>\n<li>Find the product of polynomials and monomials<\/li>\n<li>Find the product of two binomials<\/li>\n<\/ul>\n<\/li>\n<li>Multiply Binomials<\/li>\n<li>\n<ul>\n<li>Apply the FOIL method to multiply two binomials<\/li>\n<li>Use a table to multiply two binomials<\/li>\n<li>Simplify the product of two binomials given a wide variety of variables, constants, signs, and arrangement of terms in the binomial<\/li>\n<\/ul>\n<\/li>\n<li>Divide Polynomials<\/li>\n<li>\n<ul>\n<li>Divide a binomial by a monomial<\/li>\n<li>Divide a trinomial by a monomial<\/li>\n<li>Apply polynomial long division to divide by a binomial<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>Adding and subtracting <b>polynomials<\/b> may sound complicated, but it\u2019s really not much different from the addition and subtraction that you do every day. The main thing to remember is to look for and combine <b>like terms<\/b>.<\/p>\n<p>You can add two (or more) polynomials as you have added algebraic expressions. You can remove the parentheses and combine like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(3b+5\\right)+\\left(2b+4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q379821\">Show Solution<\/span><\/p>\n<div id=\"q379821\" class=\"hidden-answer\" style=\"display: none\">Regroup<\/p>\n<p style=\"text-align: center;\">[latex]\\left(3b+2b\\right)+\\left(5+4\\right)[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]5b + 9[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3b+5\\right)+\\left(2b+4\\right)=5b+9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you are adding polynomials that have subtraction,\u00a0it is important to remember to keep the sign on each term as you are collecting like terms. \u00a0 The next example will show you how to regroup terms that are subtracted when you are collecting like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(-5x^{2}\u201310x+2\\right)+\\left(3x^{2}+7x\u20134\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q486380\">Show Solution<\/span><\/p>\n<div id=\"q486380\" class=\"hidden-answer\" style=\"display: none\">\n<p>Collect like terms, making sure you keep the sign on each term. For example, when you collect\u00a0the [latex]x^2[\/latex] terms, make sure to keep the negative sign on [latex]-5x^2[\/latex].<\/p>\n<p>Helpful Hint: We find that it is easier to put the terms with a negative sign on the right of the terms that are positive. This would mean\u00a0that the\u00a0[latex]x^2[\/latex] terms would be grouped as\u00a0[latex]\\left(3x^{2}-5x^{2}\\right)[\/latex]. If both terms are negative, then it doesn&#8217;t matter which is on the left or right.<\/p>\n<p>The polynomial now looks like this, with like terms collected:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\underbrace{\\left(3x^{2}-5x^{2}\\right)}+\\underbrace{\\left(7x-10x\\right)}+\\underbrace{\\left(2-4\\right)}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x^2\\text{ terms }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\text{ terms}\\,\\,\\,\\,\\,\\,\\,\\,\\text{ constants }\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">The [latex]x^2[\/latex] terms will simplify to [latex]-2x^{2}[\/latex]<\/p>\n<p style=\"text-align: left;\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex]<\/p>\n<p style=\"text-align: left;\">The constant terms will simplify to [latex]-2[\/latex]<\/p>\n<p style=\"text-align: left;\">\u00a0Rewrite the polynomial with it&#8217;s simplified terms, keeping the sign on each term.<\/p>\n<p style=\"text-align: center;\">[latex]-2x^{2}-3x-2[\/latex]<\/p>\n<p style=\"text-align: left;\">As a matter of convention, we write polynomials in descending order based on degree. \u00a0Notice how we put the\u00a0[latex]x^2[\/latex] term first, the\u00a0[latex]x[\/latex] term second and the constant term last.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(-5x^{2}-10x+2\\right)+\\left(3x^{2}+7x-4\\right)=-2x^{2}-3x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this \u201cvertical\u201d method of adding polynomials:<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(3x^{2}+2x-7\\right)+\\left(7x^{2}-4x+8\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q425224\">Show Solution<\/span><\/p>\n<div id=\"q425224\" class=\"hidden-answer\" style=\"display: none\">Write one polynomial below the other, making sure to line up like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\+7x^{2}-4x+8\\end{array}[\/latex]<\/p>\n<p>Combine like terms, paying close attention to the signs.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\\\underline{+7x^{2}-4x+8}\\\\10x^{2}-2x+1\\end{array}[\/latex]<b>\u00a0<\/b><\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3x^{2}+3x-7\\right)+\\left(7x^{2}-4x+8\\right)=10x^{2}-2x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn&#8217;t so tidy. When there isn&#8217;t a matching like term for every term, there will be empty places in the vertical arrangement.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232680\">Show Solution<\/span><\/p>\n<div id=\"q232680\" class=\"hidden-answer\" style=\"display: none\">Write one polynomial below the other, lining up like terms vertically.<\/p>\n<p>To keep track of like terms, you can insert zeros where there aren&#8217;t any shared like terms. This is optional, but some find it helpful.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+2\\\\+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10\\end{array}[\/latex]<\/p>\n<p>Combine like terms, paying close attention to the signs.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+\\,\\,\\,2\\\\\\underline{+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10}\\\\4x^{3}\\,+\\,\\,x^{2}-6x+12\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)=4x^{3}+x^{2}-6x+12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may be thinking, how is this different than combining like terms, which we did in the last section? The answer is, it&#8217;s not really. We just added a layer to combining like terms by adding more terms to combine. :) Polynomials are a useful tool for describing the behavior of anything that isn&#8217;t linear, and sometimes you may need to add them.<\/p>\n<p>In the following video, you will see more examples of combining like terms by adding polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Adding Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KYZR7g7QcF4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Find the opposite of a polynomial<\/h2>\n<div id=\"attachment_4552\" style=\"width: 395px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4552\" class=\"wp-image-4552\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/02201910\/Screen-Shot-2016-06-02-at-1.14.57-PM-300x198.png\" alt=\"Man in a leather jacket and slicked back hair looking very rebellious sitting next to a woman in a pink dress looking very proper.\" width=\"385\" height=\"254\" \/><\/p>\n<p id=\"caption-attachment-4552\" class=\"wp-caption-text\">Opposites<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4554 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/02202350\/Screen-Shot-2016-06-02-at-1.22.59-PM.png\" alt=\"SCale with a(b+c) on one side and ab+ac on the other adn an equal sign in between the two sides of the scale\" width=\"165\" height=\"123\" \/>When you are solving equations, it may come up that you need to subtract polynomials. This means subtracting each term of a polynomial, which requires\u00a0changing the sign of each term in a polynomial. Recall that changing the sign\u00a0of 3 gives [latex]\u22123[\/latex], and changing the sign\u00a0of [latex]\u22123[\/latex] gives 3. Just as changing the sign\u00a0of a number is found by multiplying the number by [latex]\u22121[\/latex], we can change the sign\u00a0of a polynomial by multiplying it by [latex]\u22121[\/latex]. Think of this in the same way as you would the distributive property. \u00a0You are distributing [latex]\u22121[\/latex] to each term in the polynomial. \u00a0Changing the sign of a polynomial is also called finding the opposite.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the opposite of [latex]9x^{2}+10x+5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q161313\">Show Solution<\/span><\/p>\n<div id=\"q161313\" class=\"hidden-answer\" style=\"display: none\">Find the opposite by multiplying by [latex]\u22121[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\n<p>Distribute [latex]\u22121[\/latex] to each term in the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)9x^{2}+\\left(-1\\right)10x+\\left(-1\\right)5[\/latex]<\/p>\n<p>Your new terms all have the opposite sign:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(-1\\right)9x^{2}=-9x^{2}\\\\\\text{ }\\\\\\left(-1\\right)10x=-10x\\\\\\text{ }\\\\\\left(-1\\right)5=-5\\end{array}[\/latex]<\/p>\n<p>Now you can rewrite the polynomial with the new sign on each term:<\/p>\n<p style=\"text-align: center;\">[latex]-9x^{2}-10x-5[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The opposite of [latex]9x^{2}+10x+5[\/latex] is [latex]-9x^{2}-10x-5[\/latex]<\/p>\n<p>You can also write:<\/p>\n<p>[latex]\\left(-1\\right)\\left(9x^{2}+10x+5\\right)=-9x^{2}-10x-5[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"46\" height=\"41\" \/>Be careful when there are negative terms\u00a0or subtractions in the polynomial already. \u00a0Just remember that you are changing the sign, so if it is negative, it will become positive.<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the opposite of [latex]3p^{2}\u20135p+7[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q278382\">Show Solution<\/span><\/p>\n<div id=\"q278382\" class=\"hidden-answer\" style=\"display: none\">Find the opposite by multiplying by [latex]-1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)\\left(3p^{2}-5p+7\\right)[\/latex]<\/p>\n<p>Distribute [latex]-1[\/latex] to each term in the polynomial by multiplying each coefficient by [latex]-1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\left(-1\\right)3p^{2}+\\left(-1\\right)\\left(-5p\\right)+\\left(-1\\right)7[\/latex]<\/p>\n<p>Your new terms all have the opposite sign:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(-1\\right)3p^{2}=-3p^{2}\\\\\\text{ }\\\\\\left(-1\\right)\\left(-5p\\right)=5p\\\\\\text{ }\\\\\\left(-1\\right)7=-7\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Now you can rewrite the polynomial with the new sign on each term:<\/p>\n<p style=\"text-align: center;\">[latex]-3p^{2}+5p-7[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>The opposite of [latex]3p^{2}-5p+7[\/latex] is [latex]-3p^{2}+5p-7[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that in finding the opposite of a polynomial, you change the sign of <i>each term<\/i> in the polynomial, then rewrite the polynomial with the new signs on each term.<\/p>\n<h2 id=\"title3\">Subtract polynomials<\/h2>\n<p>When you subtract one polynomial from another, you will first find the opposite of the polynomial being subtracted, then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q267023\">Show Solution<\/span><\/p>\n<div id=\"q267023\" class=\"hidden-answer\" style=\"display: none\">Change\u00a0the sign of <i>each<\/i> term in the polynomial [latex]9x^{2}+10x+5[\/latex]! All the terms are positive, so they will all become negative.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(15x^{2}+12x+20\\right)-9x^{2}-10x-5[\/latex]<\/p>\n<p>Regroup to match like terms, remember to check\u00a0the sign of each term.<\/p>\n<p style=\"text-align: center;\">[latex]15x^{2}-9x^{2}+12x-10x+20-5[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]6x^{2}+2x+15[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(15x^{2}+12x+20\\right)-\\left(9x^{2}+10x+5\\right)=6x^{2}+2x+15[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"52\" height=\"47\" \/>When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q783926\">Show Solution<\/span><\/p>\n<div id=\"q783926\" class=\"hidden-answer\" style=\"display: none\">Change the sign of each term in the polynomial [latex]7x^{3}+5x^{2}\u20138x+10[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-7x^{3}-5x^{2}+8x-10[\/latex]<\/p>\n<p>Regroup to put like terms together and combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\underbrace{14x^{3}-7x^{3}}+\\underbrace{3x^{2}-5x^{2}}-\\underbrace{5x+8x}+\\underbrace{14-10}\\\\=7x^{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Write the resulting polynomial with each term&#8217;s sign in front.<\/p>\n<p style=\"text-align: center;\">[latex]7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you have many terms,\u00a0like in the example above, try the vertical approach from the previous page to keep your terms organized. \u00a0However you choose to combine polynomials is up to you\u2014the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q29114\">Show Solution<\/span><\/p>\n<div id=\"q29114\" class=\"hidden-answer\" style=\"display: none\">Reorganizing using the vertical approach.<\/p>\n<p style=\"text-align: center;\">[latex]14x^{3}+3x^{2}-5x+14-\\left(7x^{3}+5x^{2}-8x+10\\right)[\/latex]<\/p>\n<p>Change the signs, and combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}14x^{3}+3x^{2}-5x+14\\,\\,\\,\\,\\\\\\underline{-7x^{3}-5x^{2}+8x-10}\\\\=7x^{3}-2x^{2}+3x+4\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see more examples of subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Subtracting Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xq-zVm25VC0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title1\">Find the product of monomials<\/h2>\n<p>Multiplying <strong>polynomials<\/strong> involves applying the rules of exponents and the distributive property to simplify the product. Polynomial\u00a0multiplication can be useful in modeling real world situations. Understanding polynomial products is an important step in learning to solve algebraic equations involving polynomials. There are many, varied uses for polynomials including the generation of 3D graphics for entertainment and industry, as in the image below.<\/p>\n<div id=\"attachment_4567\" style=\"width: 441px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4567\" class=\"wp-image-4567\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03193513\/Screen-Shot-2016-06-03-at-12.33.29-PM-300x154.png\" alt=\"Bumpy, irregular surfaces on a grid made with AutoCAD\" width=\"431\" height=\"221\" \/><\/p>\n<p id=\"caption-attachment-4567\" class=\"wp-caption-text\">Surfaces made from polynomials with AutoCAD<\/p>\n<\/div>\n<p>In the exponents section, we\u00a0practiced\u00a0multiplying\u00a0monomials together, like we did with this expression: [latex]24{x}^{8}2{x}^{5}[\/latex]. The only thing different between that section and this one is that we called it simplifying, and now we are calling it polynomial multiplication. \u00a0Remember that simplifying a mathematical expression means performing as many operations as we can until there are no more to perform, including multiplication. \u00a0In this section we will show examples of how to multiply more than just monomials. \u00a0We will multiply monomials with\u00a0binomials and trinomials. We will also learn some techniques for multiplying two binomials together.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. [latex]-9x^{3}\\cdot 3x^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q322242\">Show Solution<\/span><\/p>\n<div id=\"q322242\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rearrange the factors.<\/p>\n<p style=\"text-align: center;\">[latex]-9\\cdot3\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\n<p>Multiply constants. Remember that a positive number times a negative number yields a negative number.<\/p>\n<p style=\"text-align: center;\">[latex]-27\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\n<p>Multiply variable terms. Remember to add the exponents when multiplying exponents with the same base.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}-27\\cdot x^{3+2}\\\\-27\\cdot x^{5}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]-9x^{3}\\cdot 3x^{2}=-27x^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>That\u2019s it! When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents, like this:<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle 5{{a}^{4}}\\cdot 7{{a}^{6}}=35{{a}^{10}}[\/latex]<\/p>\n<p>The following video provides more examples of multiplying monomials with different exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Multiplying Monomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/30x8hY32B0o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">Find the product of polynomials and monomials<\/h2>\n<p>The distributive property can be used to multiply a monomial and a binomial. Just remember that the monomial must be multiplied by each term in the binomial. In the next example, you will see how to multiply a second degree monomial with a binomial. \u00a0Note the use of exponent rules.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]5x^2\\left(4x^{2}+3x\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q176215\">Show Solution<\/span><\/p>\n<div id=\"q176215\" class=\"hidden-answer\" style=\"display: none\">Distribute the monomial to each term of the binomial. Multiply coefficients and variables separately.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}5x^2\\left(4x^{2}\\right)+5x^2\\left(3x\\right)\\\\\\text{ }\\\\=20x^{2+2}+15x^{2+1}\\\\\\text{ }\\\\=20x^{4}+15x^{3}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5x^2\\left(4x^{2}+3x\\right)=20x^{4}+15x^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now let&#8217;s add another layer by multiplying a monomial by a trinomial. Consider the expression [latex]2x\\left(2x^{2}+5x+10\\right)[\/latex].<\/p>\n<p>This expression can be modeled with a sketch like the one below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2204 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/24212311\/Screen-Shot-2016-03-24-at-2.22.48-PM.png\" alt=\"2x times 2x squared equals 4x cubed. 2x times 5x equals 10x squared. 2x times 10 equals 20x.\" width=\"508\" height=\"79\" \/><\/p>\n<p style=\"text-align: left;\">The only difference between this example and the previous one is there is one more term to distribute the monomial to.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2x\\left(2x^{2}+5x+10\\right)=2x\\left(2x^{2}\\right)+2x\\left(5x\\right)=2x\\left(10\\right)\\\\=4x^{3}+10x^{2}+20x\\end{array}[\/latex]<\/p>\n<p>You will always need to pay attention to negative signs when you are multiplying. Watch\u00a0what happens to the sign on the terms in the trinomial when it is multiplied by a negative monomial in the next example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]-7x\\left(2x^{2}-5x+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q590272\">Show Solution<\/span><\/p>\n<div id=\"q590272\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute the monomial to each term in the trinomial.<\/p>\n<p style=\"text-align: center;\">[latex]-7x\\left(2x^{2}\\right)-7x\\left(-5x\\right)-7x\\left(1\\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}-14x^{1+2}+35x^{1+1}-7x\\\\\\text{ }\\\\-14x^{3}+35x^{2}-7x\\end{array}[\/latex]<\/p>\n<p>Rewrite addition of terms with negative coefficients as subtraction.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]7x^{2}\\left(2x^{2}-5x+1\\right)=14x^{4}-35^{3}+7x^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p class=\"no-indent\" style=\"text-align: left;\">The following video provides more examples of multiplying a monomial and a polynomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Multiplying Using the Distributive Property\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"video2\" class=\"no-indent\" style=\"text-align: left;\">Find the product of two binomials<\/h2>\n<p>Now let&#8217;s explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You&#8217;ll use each binomial as one of the dimensions of a rectangle, and their product as the area.<\/p>\n<p>The model below shows [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]:<\/p>\n<div id=\"attachment_4607\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4607\" class=\"size-medium wp-image-4607\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/04191552\/Screen-Shot-2016-06-04-at-12.15.06-PM-300x290.png\" alt=\"Visual representation of multiplying two binomials.\" width=\"300\" height=\"290\" \/><\/p>\n<p id=\"caption-attachment-4607\" class=\"wp-caption-text\">Visual representation of multiplying two binomials.<\/p>\n<\/div>\n<p>Each binomial is expanded into variable terms and constants, [latex]x+4[\/latex], along the top of the model and [latex]x+2[\/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[\/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[\/latex].<\/p>\n<p>You can use the distributive property to determine the product of two binomials.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186797\">Show Solution<\/span><\/p>\n<div id=\"q186797\" class=\"hidden-answer\" style=\"display: none\">Distribute the [latex]x[\/latex] over [latex]x+2[\/latex], then distribute 4 over [latex]x+2[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]x\\left(x\\right)+x\\left(2\\right)+4\\left(x\\right)+4\\left(2\\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center;\">[latex]x^{2}+2x+4x+8[\/latex]<\/p>\n<p>Combine like terms [latex]\\left(2x+4x\\right)[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]x^{2}+6x+8[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+4\\right)\\left(2x+2\\right)=x^{2}+6x+8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Look back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[\/latex] comes from. Can you see where you multiply [latex]x[\/latex] by [latex]x + 2[\/latex], and where you get [latex]x^{2}[\/latex]\u00a0from [latex]x\\left(x\\right)[\/latex]?<\/p>\n<p>Another way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[\/latex] in [latex]x+4[\/latex] gets multiplied by both the [latex]x[\/latex] and the 2 from [latex]x+2[\/latex], and the 4 gets multiplied by both the [latex]x[\/latex] and the 2.<\/p>\n<p>The following video provides an example of multiplying two binomials using an area model as well as repeated distribution.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Multiply Binomials Using An Area Model and Using Repeated Distribution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/u4Hgl0BrUlo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next section we will explore other methods for multiplying two binomials, and\u00a0become aware of the different forms that binomials can have.<\/p>\n<h2>FOIL<\/h2>\n<div id=\"attachment_4589\" style=\"width: 345px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4589\" class=\"wp-image-4589\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03212853\/Screen-Shot-2016-06-03-at-2.08.13-PM-243x300.png\" alt=\"Crane made from aluminum foil\" width=\"335\" height=\"414\" \/><\/p>\n<p id=\"caption-attachment-4589\" class=\"wp-caption-text\">Foil Crane<\/p>\n<\/div>\n<p>In the last section we finished with an example of multiplying two binomials,[latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. In this section we will provide examples of how to use two different methods to multiply to binomials. Keep in mind as you read through the page that simplify and multiply are used interchangeably.<\/p>\n<p>Some people use the FOIL method to keep track of which pairs of terms have been multiplied when you are multiplying two binomials. This is not the same thing you use to wrap up leftovers, but an acronym for <strong>First, Outer, Inner, Last.<\/strong> Let&#8217;s go back to the example from the previous page, where we were asked to multiply the two binomials: [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]. \u00a0The following steps show you how to apply this method to multiplying two binomials.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}\\text{ term in each binomial}: \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(2\\right)=2x\\\\\\text{Inner terms}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(x\\right)=4x\\\\\\text{Last terms in each binomial}:\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\left(x+4\\right)\\left(x+2\\right)\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,4\\left(2\\right)=8\\end{array}[\/latex]<\/p>\n<p>When you add the four results, you get the same answer,\u00a0[latex]x^{2}+2x+4x+8=x^{2}+6x+8[\/latex].<br \/>\nThe last step in multiplying polynomials is to combine like terms. Remember that a polynomial is simplified only when there are no like terms remaining.<\/p>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"41\" height=\"36\" \/>Caution! Note that the FOIL method only works for multiplying two binomials together. It sill not work for multiplying a binomial and a trinomial, or two trinomials.<\/div>\n<div id=\"attachment_4595\" style=\"width: 148px\" class=\"wp-caption alignleft\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4595\" class=\"wp-image-4595\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/03231013\/Screen-Shot-2016-06-03-at-4.09.32-PM-299x300.png\" alt=\"two tomatoes sitting next to each other with two different phonetic pronunciations for the word tomato underneath\" width=\"138\" height=\"139\" \/><\/p>\n<p id=\"caption-attachment-4595\" class=\"wp-caption-text\">Order Doesn&#8217;t Matter When You Multiply<\/p>\n<\/div>\n<p>One of the neat things about multiplication is that\u00a0terms can be multiplied in either order. The expression [latex]\\left(x+2\\right)\\left(x+4\\right)[\/latex] has the same product as [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex], [latex]x^{2}+6x+8[\/latex]. (Work it out and see.) The order in which you multiply binomials does not matter. What matters is that you multiply each term in one binomial by each term in the other binomial.<\/p>\n<p>Polynomials can take many forms. \u00a0So far we have seen examples of binomials with variable terms on the left and constant terms on the right, such as this binomial [latex]\\left(2r-3\\right)[\/latex]. \u00a0Variables may also be on the right of the constant term, as in this binomial [latex]\\left(5+r\\right)[\/latex]. \u00a0In the next example, we will show that multiplying binomials in this form requires one extra step at the end.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the product.[latex]\\left(3\u2013s\\right)\\left(1-s\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q531601\">Show Solution<\/span><\/p>\n<div id=\"q531601\" class=\"hidden-answer\" style=\"display: none\">\n<p>Notice how the binomials have the variable on the right instead of the left. \u00a0There is nothing different in the way you find the product. \u00a0At the end we will reorganize terms so they are in descending order as a matter of convention.<\/p>\n<p>[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)[\/latex]<\/p>\n<p>Use a table this time.<\/p>\n<table style=\"width: 40%;\">\n<tbody>\n<tr>\n<td><\/td>\n<th>[latex]3[\/latex]<\/th>\n<th>[latex]-s[\/latex]<\/th>\n<\/tr>\n<tr>\n<th>[latex]1[\/latex]<\/th>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]-s[\/latex]<\/td>\n<\/tr>\n<tr>\n<th>[latex]-s[\/latex]<\/th>\n<td>[latex]-3s[\/latex]<\/td>\n<td>[latex]s^2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice how the <em>s<\/em> term is now positive. Collect the terms and simplify.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=3-3s-s+s^2\\\\\\text{ }\\\\=3-4s+s^2\\end{array}[\/latex]<\/p>\n<p>As a matter of convention, we will organize the terms so the one with greatest degree comes first. Pay close attention to the signs on the terms when you reorganize them. The 3 is positive, so we will use a plus in front of it, and the 4 is negative so we use a minus in front of it.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\left(3\u2013s\\right)\\left(1\u2013s\\right)\\\\\\text{ }\\\\=s^{2}-4s+3\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3\u2013s\\right)\\left(1\u2013s\\right)=s^2-4s+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, you will see that sometimes there are constants in front of the variable. They will get multiplied together just as we have done before.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]\\left(4x\u201310\\right)\\left(2x+3\\right)[\/latex] using the FOIL acronym.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q930433\">Show Solution<\/span><\/p>\n<div id=\"q930433\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,4x\\left(2x\\right)=8x^{2}\\\\\\text{Outer}:\\,\\,\\,4x\\left(3\\right)=12x\\\\\\text{Inner}:\\,\\,\\,\u221210\\left(2x\\right)=-20x\\\\\\text{Last}:\\,\\,\\,\\,\\,-10\\left(3\\right)=-30\\end{array}[\/latex]<\/p>\n<p>Be careful about including the negative sign on the [latex]\u201110[\/latex], since 10 is subtracted.<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}8x^{2}+12x\u201320x\u201330\\\\\\text{ }\\\\=8x^{2}-8x\u201330\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(4x\u201310\\right)\\left(2x+3\\right)=8x^{2}\u20138x\u201330[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The video that follows gives another example of multiplying two binomials using the FOIL acronym. Remember this method only works when you are multiplying two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"Multiply Binomials Using the FOIL Acronym\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_MrdEFnXNGA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Table Method<\/h2>\n<p>You may see a binomial multiplied by itself written as\u00a0[latex]{\\left(x+3\\right)}^{2}[\/latex] instead of\u00a0[latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. \u00a0To find this product, let&#8217;s use another method. We will place the terms of each binomial along the top row and first column of a table, like this:<\/p>\n<table class=\"lines\" style=\"width: 20%;\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]+3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]+3[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.<\/p>\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<td><\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]+3[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td><span style=\"color: #0000ff;\">[latex]x\\cdot{x}=x^2[\/latex]<\/span><\/td>\n<td><span style=\"color: #ff0000;\">[latex]3\\cdot{x}=+3x[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]+3[\/latex]<\/td>\n<td><span style=\"color: #ff0000;\">[latex]x\\cdot{3}=+3x[\/latex]<\/span><\/td>\n<td><span style=\"color: #ff00ff;\">\u00a0[latex]3\\cdot{3}=+9[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can write the terms of the polynomial from the entries in the table:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\n<p style=\"text-align: center;\">=\u00a0<span style=\"color: #0000ff;\">[latex]x^2[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span><\/p>\n<p style=\"text-align: center;\">= <span style=\"color: #0000ff;\">[latex]x^{2}[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]6x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span>.<\/p>\n<p style=\"text-align: center;\">Pretty cool, huh?<\/p>\n<p>So far, we have shown two\u00a0methods for multiplying two binomials together. \u00a0Why are we focusing\u00a0so much on binomials? \u00a0They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Square the binomial difference\u00a0[latex]\\left(x\u20137\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q293164\">Show solution<\/span><\/p>\n<div id=\"q293164\" class=\"hidden-answer\" style=\"display: none\">\n<p>Write the product of the binomial.<\/p>\n<p style=\"text-align: center;\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">Let&#8217;s use the table method, just because. Note how we carry the negative sign with the\u00a0[latex]7[\/latex].<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td><\/td>\n<td>\u00a0[latex]x[\/latex]<\/td>\n<td>\u00a0[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0[latex]x[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]x^2[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>\u00a0\u00a0[latex]-7[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]-7x[\/latex]<\/td>\n<td>\u00a0\u00a0[latex]49[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Collect the terms, and simplify. Note how we keep the sign on each term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\n<p>Answer<br \/>\n[latex]x^2-14x+49[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:<\/p>\n<p style=\"text-align: center;\">You can&#8217;t move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\n<p style=\"text-align: center;\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\n<p style=\"text-align: center;\"><strong>\u00a0CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\n<\/div>\n<p>In the video that follows, you will see another examples of using a table to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" title=\"Multiply Binomials Using a Table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tWsLJ_pn5mQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Further Examples<\/h2>\n<p>The next couple of examples show you some different forms binomials can take. \u00a0In the first, we will square a binomial that has a coefficient in front of the variable, like the product in the first example on this page. In the second we will find\u00a0the product of two binomials that have the variable on the right instead of the left.\u00a0We will use both\u00a0the FOIL method and the table method to simplify.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the product.\u00a0[latex]\\left(2x+6\\right)^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q255359\">Show Solution<\/span><\/p>\n<div id=\"q255359\" class=\"hidden-answer\" style=\"display: none\">We will use the FOIL method.<br \/>\n[latex]\\left(2x+6\\right)^{2}=\\left(2x+6\\right)\\left(2x+6\\right)[\/latex]<br \/>\n[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,2x\\left(2x\\right)=4x^{2}\\\\\\text{Outer}:\\,\\,\\,2x\\left(6\\right)=12x\\\\\\text{Inner}:\\,\\,\\,6\\left(2x\\right)=12x\\\\\\text{Last}:\\,\\,\\,\\,\\,6\\left(6\\right)=36\\end{array}[\/latex]<\/p>\n<p>Now you can collect the terms and simplify:<br \/>\n[latex]\\begin{array}{c}4x^2+12x+12x+36\\\\\\text{ }\\\\=4x^2+24x+36\\end{array}[\/latex]<\/p>\n<p>Answer<\/p>\n<p>[latex](2x+6)^{2}=4x^{2}+24x+36[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the last example, we want to show you another common form a binomial can take, each of the terms in the two binomials is the same, but the signs are different. You will see that in this case, the middle term will disappear.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply the binomials. [latex]\\left(x+8\\right)\\left(x\u20138\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q812247\">Show Solution<\/span><\/p>\n<div id=\"q812247\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\text{First}:\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\left(x\\right)=x^{2}\\\\\\text{Outer}:\\,\\,\\,\\,\\,\\,x\\left(-8\\right)=-8x\\\\\\text{Inner}:\\,\\,\\,\\,\\,\\,\\,8\\left(x\\right)=+8x\\\\\\text{Last}:\\,\\,\\,\\,\\,\\,\\,\\,\\,8\\left(-8\\right)=-64\\end{array}[\/latex]<\/p>\n<p>Add the terms. Note how the two x terms are opposites, so they sum to zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x^{2}\\underbrace{-8x+8x}-64\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\text{ }\\\\=x^2-64\\\\\\text{ }\\\\\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+8\\right)\\left(x-8\\right)=x^{2}-64[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p style=\"text-align: left;\">There are predictable outcomes when you square a binomial sum or difference. In general terms, for a binomial difference,<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a-b\\right)^{2}=\\left(a-b\\right)\\left(a-b\\right)[\/latex],<\/p>\n<p style=\"text-align: left;\">the resulting product, after being simplified, will look like this:<\/p>\n<p style=\"text-align: center;\">[latex]a^2-2ab+b^2[\/latex].<\/p>\n<p style=\"text-align: left;\">The product of a binomial sum will have the following predictable outcome:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(a+b\\right)^{2}=\\left(a+b\\right)\\left(a+b\\right)=a^2+2ab+b^2[\/latex].<\/p>\n<p style=\"text-align: left;\">Note that a and b in these generalizations could be integers, fractions, or variables with any kind of constant. \u00a0You will learn more about predictable patterns from products of binomials in later math classes.<\/p>\n<\/div>\n<p>In this section we showed two ways to find the product of two binomials, the FOIL method, and by using a table. Some of the forms a product of two binomials can take are listed here:<\/p>\n<ul>\n<li>[latex]\\left(x+5\\right)\\left(2x-3\\right)[\/latex]<\/li>\n<li>[latex]\\left(x+7\\right)^{2}[\/latex]<\/li>\n<li>[latex]\\left(x-1\\right)^{2}[\/latex]<\/li>\n<li>[latex]\\left(2-y\\right)\\left(5+y\\right)[\/latex]<\/li>\n<li>[latex]\\left(x+9\\right)\\left(x-9\\right)[\/latex]<\/li>\n<li>[latex]\\left(2x-4\\right)\\left(x+3\\right)[\/latex]<\/li>\n<\/ul>\n<p>And this is just a small list, the possible combinations are endless. For each of the products in the list, using one of the two methods presented here will work to simplify.<\/p>\n<h2>Divide a polynomial by a monomial<\/h2>\n<p>The fourth arithmetic operation is division, the inverse of multiplication. Division of polynomials isn\u2019t much different from division of numbers. In the exponential section,\u00a0you were asked to simplify expressions such as:\u00a0[latex]\\displaystyle\\frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[\/latex]. This expression is the division of two monomials.\u00a0To simplify it, we\u00a0divided\u00a0the coefficients and then divided the variables. In this section we will add another layer to this idea by dividing polynomials by monomials, and by binomials.<\/p>\n<p>The distributive property states that you can distribute a factor that is being multiplied by a sum or difference, and likewise you can distribute a <em>divisor<\/em> that is being divided into a sum or difference. In this example, you can add all the terms in the numerator, then divide by 2.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{dividend}\\rightarrow}{\\text{divisor}\\rightarrow}\\,\\,\\,\\,\\,\\, \\frac{8+4+10}{2}=\\frac{22}{2}=11[\/latex]<\/p>\n<p>Or you can\u00a0first divide each term by 2, then simplify the result.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{8}{2}+\\frac{4}{2}+\\frac{10}{2}=4+2+5=11[\/latex]<\/p>\n<p>Either way gives you the same result. The second way is helpful when you can&#8217;t combine like terms in the numerator. \u00a0Let\u2019s try something similar with a binomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\frac{9a^3+6a}{3a^2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q641821\">Show Solution<\/span><\/p>\n<div id=\"q641821\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute [latex]3a^2[\/latex]<i>\u00a0<\/i>over the polynomial by dividing each term by [latex]3a^2[\/latex]<\/p>\n<p>[latex]\\frac{9a^3}{3a^2}+\\frac{6a}{3a^2}[\/latex]<\/p>\n<p>Divide each term, a monomial divided by another monomial.<\/p>\n<p>[latex]\\begin{array}{c}3a^{3-2}+2a^{1-2}\\\\\\text{ }\\\\=3a^{1}+2a^{-1}\\\\\\text{ }\\\\=3a+2a^{-1}\\end{array}[\/latex]<\/p>\n<p>Rewrite [latex]a^{-1}[\/latex] with positive exponents, as a matter of convention.<\/p>\n<p>[latex]3a+2a^{-1}=3a+\\frac{2}{a}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{9a^3+6a}{3a^2}=3a+\\frac{2}{a}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example, you will see that the same ideas apply when you are dividing a trinomial by a monomial. You can distribute the divisor to each term in the trinomial and simplify using the rules for exponents. As we have throughout the course, simplifying with exponents includes rewriting negative exponents as positive. Pay attention to the signs of the terms in the next example, we will divide by a negative monomial.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide. [latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q324719\">Show Solution<\/span><\/p>\n<div id=\"q324719\" class=\"hidden-answer\" style=\"display: none\">Divide each term in the polynomial by the monomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{27{{y}^{4}}}{-6y}+\\frac{6{{y}^{2}}}{-6y}-\\frac{18}{-6y}[\/latex]<\/p>\n<p>Note how the term[latex]-\\frac{18}{-6y}[\/latex] does not have a <em>y<\/em> in the numerator, so division is only applied to the numbers [latex]18, -6[\/latex]. Also, 27 doesn&#8217;t divide nicely by [latex]-6[\/latex], so we are left with a fraction as the coefficient on the [latex]y^3[\/latex] term.<\/p>\n<p>Simplify.<\/p>\n<p style=\"text-align: center;\">[latex]-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y}=-\\frac{9}{2}{{y}^{3}}-y+\\frac{3}{y}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now, we ask you to think about what would happen if you were given a quotient like this to simplify:\u00a0[latex]\\frac{27{{y}^{4}}+6{{y}^{2}}-18}{-6y+3}[\/latex]. \u00a0You may be tempted to divide each term of\u00a0[latex]{27{{y}^{4}}+6{{y}^{2}}-18}[\/latex] individually by [latex]-6y[\/latex], then [latex]3[\/latex]. \u00a0This would go against the order of operations because the division sign is a grouping symbol, and the addition in the denominator cannot be simplified anymore. The result is that\u00a0no further operations can be performed with the tools we know. \u00a0We can, however, call into use a tool that you may have learned in gradeschool: long division.<\/p>\n<h2 id=\"title1\">Polynomial Long Division<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4616 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/06\/06192222\/Screen-Shot-2016-06-06-at-12.20.58-PM-300x169.png\" alt=\"I'm Not Allergic to Long Division\" width=\"545\" height=\"307\" \/><\/p>\n<p>Recall how you can use long division to divide two whole numbers, say 900 divided by 37.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064125\/image007.jpg\" alt=\"The dividend in 900 and the divisor is 37.\" width=\"51\" height=\"18\" \/><br \/>\nFirst, you would think about how many 37s are in 90, as 9 is too small. (<i>Note: <\/i>you could also think, how many 40s are there in 90.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2251 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223536\/Screen-Shot-2016-03-28-at-3.35.17-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.35.17 PM\" width=\"69\" height=\"55\" \/><br \/>\nThere are two 37s in 90, so write 2 above the last digit of 90. Two 37s is 74; write that product below the 90.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image008.jpg\" alt=\"\" width=\"51\" height=\"57\" \/><br \/>\nSubtract: [latex]90\u201374[\/latex] is 16. (If the result is larger than the divisor, 37, then you need to use a larger number for the quotient.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2252\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28223630\/Screen-Shot-2016-03-28-at-3.36.06-PM.png\" alt=\"Screen Shot 2016-03-28 at 3.36.06 PM\" width=\"66\" height=\"68\" \/><\/p>\n<p>Bring down the next digit (0) and consider how many 37s are in 160.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image009.jpg\" alt=\"\" width=\"51\" height=\"74\" \/><br \/>\nThere are four 37s in 160, so write the 4 next to the two in the quotient. Four 37s is 148; write that product below the 160.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image010.jpg\" alt=\"\" width=\"51\" height=\"86\" \/><br \/>\nSubtract: [latex]160\u2013148[\/latex] is 12. This is less than 37 so the 4 is correct. Since there are no more digits in the dividend to bring down, you\u2019re done.<\/p>\n<p>The final answer is 24 R12, or [latex]24\\frac{12}{37}[\/latex]. You can check this by multiplying the quotient (without the remainder) by the divisor, and then adding in the remainder. The result should be the dividend:<\/p>\n<p style=\"text-align: center;\">[latex]24\\cdot37+12=888+12=900[\/latex]<\/p>\n<p>To divide polynomials, use the same process. This example shows how to do this when dividing by a <strong>binomial<\/strong>.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide:\u00a0[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q455187\">Show Solution<\/span><\/p>\n<div id=\"q455187\" class=\"hidden-answer\" style=\"display: none\">How many <i>x<\/i>\u2019s are there in [latex]x^{2}[\/latex]? That is, what is [latex]\\frac{{{x}^{2}}}{x}[\/latex]?<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064126\/image011.jpg\" alt=\"\" width=\"118\" height=\"18\" \/><\/p>\n<p>[latex]\\frac{{{x}^{2}}}{x}=x[\/latex]<i>. <\/i>Put <i>x<\/i> in the quotient above the [latex]-4x[\/latex]<i>\u00a0<\/i>term. (These are like terms, which helps to organize the problem.)<\/p>\n<p>Write the product of the divisor and the part of the quotient you just found under the dividend. Since [latex]x\\left(x+2\\right)=x^{2}+2x[\/latex],\u00a0write this underneath, and get ready to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image012.jpg\" alt=\"\" width=\"118\" height=\"51\" \/><\/p>\n<p>Rewrite [latex]\u2013\\left(x^{2} + 2x\\right)[\/latex]\u00a0as its opposite [latex]\u2013x^{2}\u20132x[\/latex]\u00a0so that you can add the opposite. (Adding the opposite is the same as subtracting, and it is easier to do.)<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064127\/image013.jpg\" alt=\"\" width=\"118\" height=\"48\" \/><\/p>\n<p>Add\u00a0[latex]-x^{2}[\/latex] to [latex]x^{2}[\/latex], and [latex]-2x[\/latex] to [latex]-4x[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064128\/image014.jpg\" alt=\"\" width=\"118\" height=\"60\" \/><\/p>\n<p>Bring down [latex]-12[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2253\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232007\/Screen-Shot-2016-03-28-at-4.19.50-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.19.50 PM\" width=\"137\" height=\"75\" \/><\/p>\n<p>Repeat the process. How many times does <i>x<\/i> go into [latex]-6x[\/latex]? In other words, what is [latex]\\frac{-6x}{x}[\/latex]?<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image015.jpg\" alt=\"\" width=\"118\" height=\"60\" \/><\/p>\n<p>Since [latex]\\frac{-6x}{x}=-6[\/latex], write [latex]-6[\/latex] in the quotient. Multiply [latex]-6[\/latex] and [latex]x+2[\/latex]\u00a0and prepare to subtract the product.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2254\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/28232512\/Screen-Shot-2016-03-28-at-4.24.52-PM.png\" alt=\"Screen Shot 2016-03-28 at 4.24.52 PM\" width=\"165\" height=\"113\" \/><\/p>\n<p>Rewrite [latex]\u2013\\left(-6x\u201312\\right)[\/latex] as [latex]6x+12[\/latex], so that you can add the opposite.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064129\/image016.jpg\" alt=\"\" width=\"118\" height=\"80\" \/><\/p>\n<p>Add. In this case, there is no remainder, so you\u2019re done.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image017.jpg\" alt=\"\" width=\"118\" height=\"90\" \/><\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\frac{\\left(x^{2}\u20134x\u201312\\right)}{\\left(x+2\\right)}=x\u20136[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Check this by multiplying:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x-6\\right)\\left(x+2\\right)=x^{2}+2x-6x-12=x^{2}-4x-12[\/latex]<\/p>\n<p>Polynomial long division involves many steps. \u00a0Hopefully this video will help you determine what step to do next when you are using polynomial long division.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" title=\"Ex 1:  Divide a Trinomial by a Binomial Using Long Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KUPFg__Djzw?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>Let\u2019s try another example. In this example, a term is \u201cmissing\u201d from the dividend.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Divide: [latex]\\frac{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q523374\">Show Solution<\/span><\/p>\n<div id=\"q523374\" class=\"hidden-answer\" style=\"display: none\">In setting up this problem, notice that there is an [latex]x^{3}[\/latex]\u00a0term but no [latex]x^{2}[\/latex]\u00a0term. Add [latex]0x^{2}[\/latex]\u00a0as a \u201cplace holder\u201d for this term. (Since 0 times anything is 0, you\u2019re not changing the value of the dividend.)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064130\/image018.jpg\" alt=\"\" width=\"153\" height=\"18\" \/><\/p>\n<p>Focus on the first terms again: how many <i>x<\/i>\u2019s are there in [latex]x^{3}[\/latex]? Since [latex]\\frac{{{x}^{3}}}{x}=x^{2}[\/latex], put [latex]x^{2}[\/latex]\u00a0in the quotient.<\/p>\n<p>Multiply [latex]x^{2}\\left(x\u20133\\right)=x^{3}\u20133x^{2}[\/latex], write this underneath the dividend, and prepare to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064131\/image019.jpg\" alt=\"\" width=\"153\" height=\"49\" \/><\/p>\n<p>Rewrite the subtraction using the opposite of the expression [latex]x^{3}-3x^{2}[\/latex]. Then add.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064132\/image020.jpg\" alt=\"\" width=\"153\" height=\"59\" \/><\/p>\n<p>Bring down the rest of the expression in the dividend. It\u2019s helpful to bring down <i>all<\/i> of the remaining terms.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064133\/image021.jpg\" alt=\"\" width=\"153\" height=\"59\" \/><\/p>\n<p>Now, repeat the process with the remaining expression, [latex]3x^{2}-6x\u201310[\/latex], as the dividend.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image022.jpg\" alt=\"\" width=\"153\" height=\"79\" \/><\/p>\n<p>Remember to watch the signs!<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064134\/image023.jpg\" alt=\"\" width=\"153\" height=\"89\" \/><\/p>\n<p>How many <i>x<\/i>\u2019s are there in 3<i>x<\/i>? Since there are 3, multiply [latex]3\\left(x\u20133\\right)=3x\u20139[\/latex], write this underneath the dividend, and prepare to subtract.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064135\/image024.jpg\" alt=\"\" width=\"157\" height=\"118\" \/><\/p>\n<p>Continue until the <strong>degree<\/strong> of the remainder is <i>less <\/i>than the degree of the divisor. In this case the degree of the remainder, [latex]-1[\/latex], is 0, which is less than the degree of [latex]x-3[\/latex], which is 1.<\/p>\n<p>Also notice that you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend. These are other ways to check whether you have completed the problem.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064221\/image025.jpg\" alt=\"\" width=\"153\" height=\"118\" \/><br \/>\nYou can write the remainder using the symbol R, or as a fraction added to the rest of the quotient with the remainder in the numerator and the divisor in the denominator. In this case, since the remainder is negative, you can also subtract the opposite.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\begin{array}{r}{\\left(x^{3}\u20136x\u201310\\right)}{\\left(x\u20133\\right)}=x^{2}+3x+3+R-1,\\\\x^{2}+3x+3+\\frac{-1}{x-3}, \\text{ or }\\\\x^{2}+3x+3-\\frac{1}{x-3}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Check the result:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x\u20133\\right)\\left(x^{2}+3x+3\\right)\\,\\,\\,=\\,\\,\\,x\\left(x^{2}+3x+3\\right)\u20133\\left(x^{2}+3x+3\\right)\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}+3x^{2}+3x\u20133x^{2}\u20139x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u20139\\\\\\,\\,\\,\\,\\,\\,\\,\\,x^{3}\u20136x\u20139+\\left(-1\\right)\\,\\,\\,=\\,\\,\\,x^{3}\u20136x\u201310[\/latex]<\/p>\n<p>The video that follows shows another example of dividing a third degree trinomial by a first degree binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/Rxds7Q_UTeo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The last video example shows how to divide a third degree trinomial by a second degree binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Ex 6:  Divide a Polynomial by a Degree Two Binomial Using Long Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/P6OTbUf8f60?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>To divide a monomial by a monomial, divide the coefficients (or simplify them as you would a fraction) and divide the variables with like bases by subtracting their exponents. To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. Be sure to watch the signs! Final answers should be written without any negative exponents. \u00a0Dividing polynomials by polynomials of more than one term can be done using a process very much like long division of whole numbers. You must be careful to subtract entire expressions, not just the first term. Stop when the degree of the remainder is less than the degree of the divisor. The remainder can be written using R notation, or as a fraction added to the quotient with the remainder in the numerator and the divisor in the denominator.<\/p>\n<h2>Summary<\/h2>\n<p>We have seen that subtracting a polynomial means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are alike. \u00a0How you organize this process is up to you, but we have shown two ways here. \u00a0One method is to place the terms next to each other horizontally, putting like terms next to each other to make combining them easier. \u00a0The other method was to place the polynomial being subtracted underneath the other after changing the signs of each term. In this method it is important to align like terms and use a blank space when there is no like term.<\/p>\n<p>Multiplication of binomials and polynomials requires an understanding of the distributive property, rules for exponents, and a keen eye for collecting like terms. Whether the polynomials are monomials, binomials, or trinomials, carefully multiply each term in one polynomial by each term in the other polynomial. Be careful to watch the addition and subtraction signs and negative coefficients. A product is written in simplified form if all of its like terms have been combined.<\/p>\n<h2><\/h2>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-911\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Ex: Adding Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KYZR7g7QcF4\">https:\/\/youtu.be\/KYZR7g7QcF4<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Opposites. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Image: Caution. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Distributive Property Scales. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Subtracting Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xq-zVm25VC0\">https:\/\/youtu.be\/xq-zVm25VC0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Multiplying Monomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"\"><\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Multiplying Using the Distributive Property. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bwTmApTV_8o\">https:\/\/youtu.be\/bwTmApTV_8o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Multiply Binomials Using An Area Model and Using Repeated Distribution. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/u4Hgl0BrUlo\">https:\/\/youtu.be\/u4Hgl0BrUlo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot I&#039;m Not Allergic to Long Division. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 1: Divide a Trinomial by a Binomial Using Long Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KUPFg__Djzw\">https:\/\/youtu.be\/KUPFg__Djzw<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Rxds7Q_UTeo\">https:\/\/youtu.be\/Rxds7Q_UTeo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex 6: Divide a Polynomial by a Degree Two Binomial Using Long Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/P6OTbUf8f60\">https:\/\/youtu.be\/P6OTbUf8f60<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":20,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Ex: Adding Polynomials\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KYZR7g7QcF4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot: 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