{"id":998,"date":"2016-06-01T20:47:43","date_gmt":"2016-06-01T20:47:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=998"},"modified":"2016-10-03T21:15:58","modified_gmt":"2016-10-03T21:15:58","slug":"read-define-and-evaluate-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/chapter\/read-define-and-evaluate-polynomials\/","title":{"raw":"Algebraic Operations on Polynomials","rendered":"Algebraic Operations on Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Anatomy of a polynomial\r\n<ul>\r\n \t<li>Identify the degree and leading coefficient\u00a0of a polynomial<\/li>\r\n \t<li>Evaluate a polynomial for given values<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Sums and Products of Polynomials<\/li>\r\n \t<li>\r\n<ul>\r\n \t<li>Add and subtract polynomials<\/li>\r\n \t<li>Find the product of polynomials<\/li>\r\n \t<li>Find the product of two binomials using the FOIL method<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiply a Trinomial and a Binomial<\/li>\r\n \t<li>Divide Polynomials<\/li>\r\n \t<li>\r\n<ul>\r\n \t<li>Divide a polynomials using long division<\/li>\r\n \t<li>Divide\u00a0polynomials\u00a0using synthetic division<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the example on the previous page, we saw how combining the formulas for different shapes provides a way to accurately predict the amount of paint needed for a construction project. The result was a\u00a0<strong>polynomial<\/strong>.\r\n\r\nA polynomial function is a function consisting of sum or difference of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and variables with an\u00a0non-negative integer exponents. Non negative integers are 0, 1, 2, 3, 4, ...\r\nYou may see a resemblance between expressions and polynomials, which we have been studying in this course. \u00a0Polynomials are a special sub-group of mathematical expressions and equations.\r\n\r\nThe following table is intended to help you tell the difference between what is a polynomial and what is not.\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>IS a Polynomial<\/td>\r\n<td>Is NOT a Polynomial<\/td>\r\n<td>Because<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\r\n<td>[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\r\n<td>Polynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{y}{4}-y^3[\/latex]<\/td>\r\n<td>[latex]\\frac{2}{y}+4[\/latex]<\/td>\r\n<td>Polynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\r\n<td>\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\r\n<td>Roots are equivalent to rational exponents, and polynomials only have integer exponents<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe basic building block of a polynomial is a <b>monomial<\/b>. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the <b>coefficient<\/b>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183539\/image003.jpg\" alt=\"The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent.\" width=\"183\" height=\"82\" \/>\r\n\r\nA polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <strong>binomial<\/strong>. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <strong>trinomial<\/strong>.\r\n\r\nWe can find the <strong>degree<\/strong> of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the <strong>leading term<\/strong> because it is usually written first. The coefficient of the leading term is called the <strong>leading coefficient<\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents.\r\n\r\n<img class=\"wp-image-2550 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15150341\/Screen-Shot-2016-07-15-at-8.03.13-AM-300x150.png\" alt=\"4x^3 - 9x^2 + 6x, with the text &quot;degree = 3&quot; and an arrow pointing at the exponent on x^3, and the text &quot;leading term =4&quot; with an arrow pointing at the 4. \" width=\"504\" height=\"252\" \/>\r\n<div class=\"textbox\">\r\n<h4>Given a polynomial expression, identify the degree and leading coefficient.<\/h4>\r\n<ol>\r\n \t<li>Find the highest power of <em>x<\/em> to determine the degree.<\/li>\r\n \t<li>Identify the term containing the highest power of <em>x<\/em> to find the leading term.<\/li>\r\n \t<li>Identify the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFor the following polynomials, identify the degree, the leading term, and the leading coefficient.\r\n<ol>\r\n \t<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\r\n \t<li>[latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\r\n \t<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"753071\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"753071\"]\r\n<ol>\r\n \t<li>The highest power of <em>x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\r\n \t<li>The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\r\n \t<li>The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the following video example, we will identify the terms, leading coefficient, and degree of a polynomial.\r\n\r\nhttps:\/\/youtu.be\/3u16B2PN9zk\r\n\r\nThe table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.\r\n<table style=\"border-spacing: 0px;\" border=\"1\" cellpadding=\"0\">\r\n<tbody>\r\n<tr>\r\n<td><b>Monomials<\/b><\/td>\r\n<td><b>Binomials<\/b><\/td>\r\n<td><b>Trinomials<\/b><\/td>\r\n<td><b>Other Polynomials<\/b><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>15<\/td>\r\n<td>[latex]3y+13[\/latex]<\/td>\r\n<td>[latex]x^{3}-x^{2}+1[\/latex]<\/td>\r\n<td>[latex]5x^{4}+3x^{3}-6x^{2}+2x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\displaystyle \\frac{1}{2}x[\/latex]<\/td>\r\n<td>[latex]4p-7[\/latex]<\/td>\r\n<td>[latex]3x^{2}+2x-9[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{3}x^{5}-2x^{4}+\\frac{2}{9}x^{3}-x^{2}+4x-\\frac{5}{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-4y^{3}[\/latex]<\/td>\r\n<td>[latex]3x^{2}+\\frac{5}{8}x[\/latex]<\/td>\r\n<td>[latex]3y^{3}+y^{2}-2[\/latex]<\/td>\r\n<td>[latex]3t^{3}-3t^{2}-3t-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]16n^{4}[\/latex]<\/td>\r\n<td>[latex]14y^{3}+3y[\/latex]<\/td>\r\n<td>[latex]a^{7}+2a^{5}-3a^{3}[\/latex]<\/td>\r\n<td>[latex]q^{7}+2q^{5}-3q^{3}+q[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhen the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn\u2019t change the value). The last binomial above could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[\/latex].\r\n\r\nA term without a variable is called a <b>constant <\/b>term, and the degree of that term is 0. For example 13 is the constant term in [latex]3y+13[\/latex]. You would usually say that [latex]14y^{3}+3y[\/latex] has no constant term or that the constant term is 0.\r\n<h2>Evaluate a polynomial<\/h2>\r\nYou can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable <i>every time<\/i> it appears. Then use the order of operations to find the resulting value for the expression.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex]3x^{2}-2x+1[\/latex] for [latex]x=-1[\/latex].\r\n\r\n[reveal-answer q=\"280466\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"280466\"]Substitute [latex]-1[\/latex] for each <i>x<\/i> in the polynomial.\r\n<p style=\"text-align: center;\">[latex]3\\left(-1\\right)^{2}-2\\left(-1\\right)+1[\/latex]<\/p>\r\nFollowing the order of operations, evaluate exponents first.\r\n<p style=\"text-align: center;\">[latex]3\\left(1\\right)-2\\left(-1\\right)+1[\/latex]<\/p>\r\nMultiply 3 times 1, and then multiply [latex]-2[\/latex] times [latex]-1[\/latex].\r\n<p style=\"text-align: center;\">[latex]3+\\left(-2\\right)\\left(-1\\right)+1[\/latex]<\/p>\r\nChange the subtraction to addition of the opposite.\r\n<p style=\"text-align: center;\">[latex]3+2+1[\/latex]<\/p>\r\nFind the sum.\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}-2x+1=6[\/latex], for [latex]x=-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nEvaluate [latex] \\displaystyle -\\frac{2}{3}p^{4}+2^{3}-p[\/latex] for [latex]p = 3[\/latex].\r\n\r\n[reveal-answer q=\"745542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"745542\"]Substitute 3 for each <i>p<\/i> in the polynomial.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle -\\frac{2}{3}\\left(3\\right)^{4}+2\\left(3\\right)^{3}-3[\/latex]<\/p>\r\nFollowing the order of operations, evaluate exponents first and then multiply.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle -\\frac{2}{3}\\left(81\\right)+2\\left(27\\right)-3[\/latex]<\/p>\r\nAdd and then subtract to get [latex]-3[\/latex].\r\n<p style=\"text-align: center;\">[latex]-54 + 54 \u2013 3[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex] \\displaystyle -\\frac{2}{3}p^{4}+2p^{3}-p=-3[\/latex], for [latex]p = 3[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p class=\"no-indent\" style=\"text-align: left;\">\u00a0IN the following video we show more examples of evaluating polynomials for given values of the variable.<\/p>\r\nhttps:\/\/youtu.be\/2EeFrgQP1hM\r\n<h2>Add and Subtract Polynomials<\/h2>\r\nWe can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms, and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms, and therefore cannot be added.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the sum.\r\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"222892\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"222892\"]<\/p>\r\n<p style=\"text-align: left;\">[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nHere\u00a0is a summary of some helpful steps for adding and subtracting polynomials.\r\n<div class=\"textbox\">\r\n<h3>\u00a0Given multiple polynomials, add or subtract them to simplify the expressions.<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Combine like terms.<\/li>\r\n \t<li>Simplify and write in standard form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div>\r\n\r\nWhen you subtract polynomials you will still be looking for like terms to combine, but you will need to pay attention to the sign of the terms you are combining. In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the difference.\r\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"279648\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"279648\"]<\/p>\r\n<p style=\"text-align: left;\">[latex]\\begin{array}{cc}7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<h3>Analysis of the Solution<\/h3>\r\nNote that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.\r\n\r\nIn the following video we show more examples of adding and subtracting polynomials.\r\n\r\nhttps:\/\/youtu.be\/jiq3toC7wGM\r\n<h2>Multiplying Polynomials<\/h2>\r\nMultiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms.\r\n\r\nYou may have used the distributive property to help you solve linear equations such as\u00a0[latex]2\\left(x+7\\right)=21[\/latex]. We can distribute the [latex]2[\/latex] in [latex]2\\left(x+7\\right)[\/latex] to obtain the equivalent expression [latex]2x+14[\/latex]. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.\r\n\r\nThe following video will provide you with examples of using the distributive property to find the product of\u00a0monomials and polynomials.\r\n\r\nhttps:\/\/youtu.be\/bwTmApTV_8o\r\n\r\nBelow is a summary of the steps we used to find the product of two polynomials using the distributive property.\r\n<div class=\"textbox\">\r\n<h3>How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression.<\/h3>\r\n<ol>\r\n \t<li>Multiply each term of the first polynomial by each term of the second.<\/li>\r\n \t<li>Combine like terms.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Using FOIL to Multiply Binomials<\/h2>\r\nWe can also use a shortcut called the FOIL method when multiplying binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200224\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" data-media-type=\"image\/jpg\" \/>\r\n\r\nThe FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse FOIL to find the product. [latex](2x-18)(3x+3)[\/latex]\r\n[reveal-answer q=\"787670\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"787670\"]\r\n\r\nFind the product of the first terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200225\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/>\r\n\r\nFind the product of the outer terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200227\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/>\r\n\r\nFind the product of the inner terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200228\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/>\r\n\r\nFind the product of the last terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200229\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill &amp; \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this video, we show an example of how to use the FOIL method to multiply two binomials.\r\n\r\nhttps:\/\/youtu.be\/_MrdEFnXNGA\r\n\r\nThe following steps summarize the process for using FOIL to multiply two binomials. \u00a0It is very important to note that this process only works for the product of two binomials. If you are multiplying a binomial\u00a0and a trinomial, it is better to use a table to keep track of your terms.\r\n<div class=\"textbox\">\r\n<h3>How To: Given two binomials, use FOIL to simplify the expression.<\/h3>\r\n<ol>\r\n \t<li>Multiply the first terms of each binomial.<\/li>\r\n \t<li>Multiply the outer terms of the binomials.<\/li>\r\n \t<li>Multiply the inner terms of the binomials.<\/li>\r\n \t<li>Multiply the last terms of each binomial.<\/li>\r\n \t<li>Add the products.<\/li>\r\n \t<li>Combine like terms and simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>Multiply a Trinomial and a Binomial<\/h2>\r\nAnother type of polynomial multiplication problem is the product of a binomial and trinomial. Although the FOIL method can not be used since there are more than two terms in a trinomial, you still use the Distributive Property to organize the individual products. Using the distributive property, each term in the binomial must be multiplied by each of the terms in the trinomial.\r\n\r\nFor our first examples, we will show you two ways to organize all of the terms that result from multiplying polynomials with more than two terms. The most important part of the process is finding a way to organize terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nFind the product. \u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex].\r\n[reveal-answer q=\"637359\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"637359\"]Distribute the trinomial to each term in the binomial.\r\n\r\n[latex]3x\\left(5x^{2}+3x+10\\right)+6\\left(5x2+3x+10\\right)[\/latex]\r\n\r\nUse the distributive property to distribute the monomials to each term in the trinomials.\r\n\r\n[latex]3x\\left(5x^{2}\\right)+3x\\left(3x\\right)+3x\\left(10\\right)+6\\left(5x^{2}\\right)+6\\left(3x\\right)+6\\left(10\\right)[\/latex]\r\n\r\nMultiply.\r\n\r\n[latex]15x^{3}9x^{2}+30x^{2}+18x+60[\/latex]\r\n\r\nGroup like terms.\r\n\r\n[latex]15x^{3}+\\left(9x^{2}+30x^{2}\\right)+\\left(30x+18x\\right)+60[\/latex]\r\n\r\nCombine like terms.\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)=15x^{3}+39x^{2}+48x+60[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAs you can see, multiplying a binomial by a trinomial leads to a lot of individual terms! Using the same problem as above, we will show another way to organize all the terms produced by multiplying two polynomials with more than two terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply.\u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex]\r\n[reveal-answer q=\"262750\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"262750\"]Set up the problem in a vertical form, and begin by multiplying [latex]3x+6[\/latex] by [latex]+10[\/latex]. Place the products underneath, as shown.\r\n\r\n[latex]\\begin{array}{r}3x+\\,\\,\\,6\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}+\\,\\,3x+10}\\\\+30x+60\\,\\end{array}[\/latex]\r\n\r\nNow multiply [latex]3x+6[\/latex] by [latex]+3x[\/latex]. Notice that [latex]\\left(6\\right)\\left(3x\\right)=18x[\/latex]; since this term is like [latex]30x[\/latex], place it directly beneath it.\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]\r\n\r\nFinally, multiply [latex]3x+6[\/latex] by [latex]5x^{2}[\/latex]. Notice that [latex]30x^{2}[\/latex]\u00a0is placed underneath [latex]9x^{2}[\/latex].\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\end{array}[\/latex]\r\n\r\nNow add like terms.\r\n\r\n[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}\\,\\,\\,\\,\\,\\,+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\+15x^{3}\\,\\,\\,\\,\\,\\,+39x^{2}\\,\\,\\,\\,+48x\\,\\,\\,\\,\\,+60\\end{array}[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]15x^{3}+39x^{2}+48x+60[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that although the two problems were solved using different strategies, the product is the same. Both the horizontal and vertical methods apply the Distributive Property to multiply a binomial by a trinomial.\r\n\r\nIn our next example we will multiply a binomial and a trinomial that contains subtraction. Pay attention to the signs on the terms. \u00a0Forgetting a negative sign is the easiest mistake to make in this case.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the product.\r\n\r\n[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]\r\n\r\n[reveal-answer q=\"485882\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"485882\"]\r\n\r\n[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div>\r\n<h3>Analysis of the Solution<\/h3>\r\nAnother way to keep track of all the terms involved in this product is to use a table, as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify. Notice how we kept the sign on each term, for example we are subtracting [latex]x[\/latex] from [latex]3x^2[\/latex], so we place [latex]-x[\/latex] in the table.\r\n<table style=\"width: 30%;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\r\n<thead>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3{x}^{2}[\/latex]<\/td>\r\n<td>[latex]-x[\/latex]<\/td>\r\n<td>[latex]+4[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2x[\/latex]<\/td>\r\n<td>[latex]6{x}^{3}\\\\[\/latex]<\/td>\r\n<td>[latex]-2{x}^{2}[\/latex]<\/td>\r\n<td>[latex]8x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+1[\/latex]<\/td>\r\n<td>[latex]3{x}^{2}[\/latex]<\/td>\r\n<td>[latex]-x[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nMultiply. \u00a0[latex]\\left(2p-1\\right)\\left(3p^{2}-3p+1\\right)[\/latex]\r\n[reveal-answer q=\"654814\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"654814\"]\r\n\r\nDistribute 2p and -1 to each term in the trinomial.\r\n<p style=\"text-align: center;\">[latex]2p\\left(3p^{2}-3p+1\\right)-1\\left(3p^{2}-3p+1\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]2p\\left(3p^{2}\\right)+2p\\left(-3p\\right)+2p\\left(1\\right)-1\\left(3p^{2}\\right)-1\\left(-3p\\right)-1\\left(1\\right)[\/latex]<\/p>\r\nMultiply. (Notice that the subtracted 1 and the subtracted 3<em>p<\/em> have a positive product that is added.)\r\n<p style=\"text-align: center;\">[latex]6p^{3}-6p^{2}+2p-3p^{2}+3p-1[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center;\">[latex]6p^{3}-9p^{2}+5p-1[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]6p^{3}-9p^{2}+5p-1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show more examples of multiplying polynomials.\r\n\r\nhttps:\/\/youtu.be\/bBKbldmlbqI\r\n<h2>Divide a polynomial by a binomial<\/h2>\r\nDividing a <strong>polynomial<\/strong> by a monomial can be handled by dividing each term in the polynomial separately. This can\u2019t be done when the divisor has more than one term. However, the process of long division can be very helpful with polynomials.\r\n\r\nRecall how you can use long division to divide two whole numbers, say 900 divided by 37.\r\n<p style=\"text-align: center;\">[latex]37\\overline{)900}[\/latex]<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152919\/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<p style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\,\\,\\,74\\end{array}[\/latex]<\/p>\r\n<img class=\"aligncenter wp-image-2251 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152920\/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<p style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\underline{-74}\\\\\\,\\,\\,\\,\\,\\,\\,16\\end{array}[\/latex]<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152921\/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<p style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\underline{-74}\\\\\\,\\,\\,\\,\\,\\,\\,160\\end{array}[\/latex]<\/p>\r\n<img class=\"aligncenter size-full wp-image-2252\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152922\/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<p style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\underline{-74}\\\\\\,\\,\\,\\,\\,\\,\\,160\\\\\\,\\,\\,\\underline{-148}\\end{array}[\/latex]<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152923\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152924\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152924\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152925\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152926\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152927\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152928\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152929\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152931\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152932\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152933\/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\nIn this video we show another example of dividing a degree two trinomial by a degree one binomial.\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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152934\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152935\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152936\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152938\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152939\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152940\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152941\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152942\/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\nIn the video that follows, we show another example of dividing a degree three trinomial by a binomial, not the \"missing\" term and how we work with it.\r\n\r\nhttps:\/\/youtu.be\/Rxds7Q_UTeo\r\n\r\nThe process above works for dividing any polynomials, no matter how many terms are in the divisor or the dividend. The main things to remember are:\r\n<ul>\r\n \t<li>When subtracting, be sure to subtract the whole expression, not just the first term. <i>This is very easy to forget, so be careful!<\/i><\/li>\r\n \t<li>Stop when the degree of the remainder is less than the degree of the dividend, or when you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend.<\/li>\r\n<\/ul>\r\nIn this video we present one more example of polynomial long division.\r\n\r\nhttps:\/\/youtu.be\/P6OTbUf8f60\r\n<h2>Synthetic Division<\/h2>\r\nAs we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a polynomial\u00a0whose leading coefficient is 1.\r\n<div class=\"textbox\">\r\n<h3 class=\"title\" data-type=\"title\">\u00a0Synthetic Division<\/h3>\r\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form <em>x<\/em> \u2013\u00a0<em>k, <\/em>for a real number k.\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\r\n\r\n<\/div>\r\n<p class=\"title\" data-type=\"title\"><span style=\"font-size: 16px; line-height: 1.5;\">To illustrate the process, d<\/span><span style=\"font-size: 16px; line-height: 1.5;\">ivide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/span><\/p>\r\n<p id=\"fs-id1165137932636\"><span id=\"eip-id1163740536072\" data-type=\"media\" data-alt=\".\" data-display=\"block\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201527\/CNX_Precalc_revised_eq_42.png\" alt=\".\" width=\"250\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137932377\">There is a lot of repetition in this process.\u00a0If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign, we already have a simpler version of the entire problem.<\/p>\r\n<span id=\"fs-id1165134305375\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201529\/CNX_Precalc_Figure_03_05_0042.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-media-type=\"image\/jpg\" \/><\/span>\r\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the \"divisor\" to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<span id=\"fs-id1165137696374\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201531\/CNX_Precalc_Figure_03_05_0112.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\r\n<p id=\"fs-id1165137696388\">We then multiply it by the \"divisor\" and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[\/latex]\u00a0and the remainder is \u201331.\u00a0The process will be made more clear in the following example.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse synthetic division to divide [latex]5{x}^{2}-3x - 36[\/latex]\u00a0by [latex]x - 3[\/latex].\r\n[reveal-answer q=\"152802\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"152802\"]\r\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write <i>3<\/i>\u00a0and the coefficients of the polynomial.<\/p>\r\n<span id=\"fs-id1165135177629\" data-type=\"media\" data-alt=\"A collapsed version of the previous synthetic division.\" data-display=\"block\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201532\/CNX_Precalc_Figure_03_05_0052.jpg\" alt=\"A collapsed version of the previous synthetic division.\" data-media-type=\"image\/jpg\" \/><\/span>\r\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by <i>3\u00a0<\/i>and place the result in the second column.<\/p>\r\n<span id=\"fs-id1165135439966\" data-type=\"media\" data-alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" data-display=\"block\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201533\/CNX_Precalc_Figure_03_05_0062.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" data-media-type=\"image\/jpg\" \/><\/span>\r\n<p id=\"fs-id1165135179942\">Continue by adding [latex]-3+15[\/latex]\u00a0in the second column. Multiply the resulting number, [latex]12[\/latex] by <i>3<\/i>.\u00a0Write the result, [latex]36[\/latex] in the next column. Then add the numbers in the third column.<\/p>\r\n<span id=\"fs-id1165135179966\" data-type=\"media\" data-alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" data-display=\"block\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201535\/CNX_Precalc_Figure_03_05_0072.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" data-media-type=\"image\/jpg\" \/><\/span>\r\n<p id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].<\/p>\r\nWe can check our work by multiplying the result by the original divisor [latex]x-3[\/latex], if we get\u00a0[latex]5{x}^{2}-3x - 36[\/latex], we have used the method correctly.\r\n\r\nCheck:\u00a0[latex](5x+12)(x-3)[\/latex]\r\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{array}{cc}(5x+12)(x-3)\\\\=5x^2-15x+12x-36\\\\=5x^2-3x-36\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Because we got a result of\u00a0[latex]5{x}^{2}-3x - 36[\/latex] when we multiplied the divisor and our answer, we can be sure that we have used synthetic division correctly.<\/p>\r\n\r\n<h4 style=\"text-align: left;\">Answer<\/h4>\r\n[latex]5{x}^{2}-3x - 36\\div{x-3}=5x+12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nAnalysis of the solution\r\n\r\nIt is important to note that the result, [latex]5x+12[\/latex], of [latex]5{x}^{2}-3x - 36\\div{x-3}[\/latex] is one degree less than[latex]5{x}^{2}-3x - 36[\/latex]. Why is that? Think about how you would have solved this using long division. The first thing you would ask yourself is how many x's are in [latex]5x^2[\/latex]?\r\n<p style=\"text-align: center;\">[latex]x-3\\overline{)5{x}^{2}-3x - 36}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">To get a result of [latex]5x^2[\/latex], you need to multiply [latex]x[\/latex] by [latex]5x[\/latex]. \u00a0The next step in long division is to subtract this result from [latex]5x^2[\/latex]. \u00a0This leaves us with no [latex]x^2[\/latex] term in the result.<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nReflect on this idea - if you multiply two polynomials and get a result whose degree is 2, what are the possible degrees of the two polynomials that were multiplied? Write your ideas in the box below before looking at the discussion.[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"962896\"]Show Discussion[\/reveal-answer]\r\n[hidden-answer a=\"962896\"]\r\n\r\nA degree two polynomial will have a leading term with [latex]x^2[\/latex]. \u00a0Let's use [latex]2x^2-2x-24[\/latex] as an example. We can write\u00a0two products\u00a0that will give this as a result of multiplication:\r\n\r\n[latex]2(x^2-x-12) =2x^2-2x-24[\/latex]\r\n\r\n[latex](2x+6)(x-4)=2x^2-2x-24[\/latex]\r\n\r\nIf we work backward, starting from [latex]2x^2-2x-24[\/latex] if we divide by a binomial with degree one, such as [latex](x-4)[\/latex], our result will also have degree one.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn this video example, you will see another example of using synthetic division for division of a degree two polynomial by a degree one binomial.\r\n\r\nhttps:\/\/youtu.be\/KeZ_zMOYu9o\r\n<div id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\r\n<h3 id=\"fs-id1165135393414\">How To: Given two polynomials, use synthetic division to divide.<\/h3>\r\n<ol id=\"fs-id1165135393418\" data-number-style=\"arabic\">\r\n \t<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\r\n \t<li>Write the coefficients of the dividend.<\/li>\r\n \t<li>Bring the lead coefficient down.<\/li>\r\n \t<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Add the terms of the second column.<\/li>\r\n \t<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\r\n \t<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\r\n \t<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.<\/li>\r\n<\/ol>\r\n<\/div>\r\nIn the next example we will use synthetic division to divide a third-degree polynomial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].\r\n[reveal-answer q=\"153403\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"153403\"]The binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\" data-type=\"media\" data-alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\">\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201536\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" data-media-type=\"image\/jpg\" \/><\/span>\r\n<p id=\"fs-id1165134433356\">The result is [latex]4{x}^{2}+2x - 10[\/latex]. Again notice\u00a0the degree of the result is less than the degree of the quotient,\u00a0[latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\r\nWe can check that we are correct by multiplying the result with the divisor:\r\n\r\n[latex](x+2)(4{x}^{2}+2x - 10)=4x^3+2x^2-10x+8x^2+4x-20=4x^3+10x^2-6x-20[\/latex]\r\n<h4>Answer<\/h4>\r\n[latex]4{x}^{3}+10{x}^{2}-6x - 20\\div{x+2}=4{x}^{2}+2x - 10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next example we will show division of a fourth degree polynomial by a binomial. \u00a0Note how there is no x term in the fourth degree polynomial, so we need to use a placeholder of 0 to ensure proper alignment of terms.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse synthetic division to divide [latex]-9{x}^{4}+10{x}^{3}+7{x}^{2}-6[\/latex]\u00a0by [latex]x - 1[\/latex].\r\n[reveal-answer q=\"76281\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"76281\"]\r\n<p id=\"fs-id1165135571794\">Notice there is no <em data-effect=\"italics\">x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\" data-type=\"media\" data-alt=\".\" data-display=\"block\">\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201539\/CNX_Precalc_revised_eq_52.png\" alt=\".\" width=\"230\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\r\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our last video example we show another example of how to use synthetic division to divide a degree three polynomial by a degree one binomial.\r\n\r\nhttps:\/\/youtu.be\/h1oSCNuA9i0\r\n<h2>Summary<\/h2>\r\nMultiplication of binomials and polynomials requires use of the distributive property as well as the commutative and associative properties of multiplication. 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\r\nDividing 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<h3><\/h3>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Anatomy of a polynomial\n<ul>\n<li>Identify the degree and leading coefficient\u00a0of a polynomial<\/li>\n<li>Evaluate a polynomial for given values<\/li>\n<\/ul>\n<\/li>\n<li>Sums and Products of Polynomials<\/li>\n<li>\n<ul>\n<li>Add and subtract polynomials<\/li>\n<li>Find the product of polynomials<\/li>\n<li>Find the product of two binomials using the FOIL method<\/li>\n<\/ul>\n<\/li>\n<li>Multiply a Trinomial and a Binomial<\/li>\n<li>Divide Polynomials<\/li>\n<li>\n<ul>\n<li>Divide a polynomials using long division<\/li>\n<li>Divide\u00a0polynomials\u00a0using synthetic division<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>In the example on the previous page, we saw how combining the formulas for different shapes provides a way to accurately predict the amount of paint needed for a construction project. The result was a\u00a0<strong>polynomial<\/strong>.<\/p>\n<p>A polynomial function is a function consisting of sum or difference of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and variables with an\u00a0non-negative integer exponents. Non negative integers are 0, 1, 2, 3, 4, &#8230;<br \/>\nYou may see a resemblance between expressions and polynomials, which we have been studying in this course. \u00a0Polynomials are a special sub-group of mathematical expressions and equations.<\/p>\n<p>The following table is intended to help you tell the difference between what is a polynomial and what is not.<\/p>\n<table>\n<thead>\n<tr>\n<td>IS a Polynomial<\/td>\n<td>Is NOT a Polynomial<\/td>\n<td>Because<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\n<td>[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\n<td>Polynomials only have variables in the numerator<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{y}{4}-y^3[\/latex]<\/td>\n<td>[latex]\\frac{2}{y}+4[\/latex]<\/td>\n<td>Polynomials only have variables in the numerator<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\n<td>\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\n<td>Roots are equivalent to rational exponents, and polynomials only have integer exponents<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The basic building block of a polynomial is a <b>monomial<\/b>. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the <b>coefficient<\/b>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01183539\/image003.jpg\" alt=\"The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent.\" width=\"183\" height=\"82\" \/><\/p>\n<p>A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <strong>binomial<\/strong>. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <strong>trinomial<\/strong>.<\/p>\n<p>We can find the <strong>degree<\/strong> of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the <strong>leading term<\/strong> because it is usually written first. The coefficient of the leading term is called the <strong>leading coefficient<\/strong>. When a polynomial is written so that the powers are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2550 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15150341\/Screen-Shot-2016-07-15-at-8.03.13-AM-300x150.png\" alt=\"4x^3 - 9x^2 + 6x, with the text &quot;degree = 3&quot; and an arrow pointing at the exponent on x^3, and the text &quot;leading term =4&quot; with an arrow pointing at the 4.\" width=\"504\" height=\"252\" \/><\/p>\n<div class=\"textbox\">\n<h4>Given a polynomial expression, identify the degree and leading coefficient.<\/h4>\n<ol>\n<li>Find the highest power of <em>x<\/em> to determine the degree.<\/li>\n<li>Identify the term containing the highest power of <em>x<\/em> to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>For the following polynomials, identify the degree, the leading term, and the leading coefficient.<\/p>\n<ol>\n<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\n<li>[latex]5{t}^{5}-2{t}^{3}+7t[\/latex]<\/li>\n<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q753071\">Show Answer<\/span><\/p>\n<div id=\"q753071\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The highest power of <em>x<\/em> is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\n<li>The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\n<li>The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the following video example, we will identify the terms, leading coefficient, and degree of a polynomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Intro to Polynomials in One Variable\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3u16B2PN9zk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.<\/p>\n<table style=\"border-spacing: 0px;\" cellpadding=\"0\">\n<tbody>\n<tr>\n<td><b>Monomials<\/b><\/td>\n<td><b>Binomials<\/b><\/td>\n<td><b>Trinomials<\/b><\/td>\n<td><b>Other Polynomials<\/b><\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>[latex]3y+13[\/latex]<\/td>\n<td>[latex]x^{3}-x^{2}+1[\/latex]<\/td>\n<td>[latex]5x^{4}+3x^{3}-6x^{2}+2x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\displaystyle \\frac{1}{2}x[\/latex]<\/td>\n<td>[latex]4p-7[\/latex]<\/td>\n<td>[latex]3x^{2}+2x-9[\/latex]<\/td>\n<td>[latex]\\frac{1}{3}x^{5}-2x^{4}+\\frac{2}{9}x^{3}-x^{2}+4x-\\frac{5}{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-4y^{3}[\/latex]<\/td>\n<td>[latex]3x^{2}+\\frac{5}{8}x[\/latex]<\/td>\n<td>[latex]3y^{3}+y^{2}-2[\/latex]<\/td>\n<td>[latex]3t^{3}-3t^{2}-3t-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]16n^{4}[\/latex]<\/td>\n<td>[latex]14y^{3}+3y[\/latex]<\/td>\n<td>[latex]a^{7}+2a^{5}-3a^{3}[\/latex]<\/td>\n<td>[latex]q^{7}+2q^{5}-3q^{3}+q[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>When the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn\u2019t change the value). The last binomial above could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[\/latex].<\/p>\n<p>A term without a variable is called a <b>constant <\/b>term, and the degree of that term is 0. For example 13 is the constant term in [latex]3y+13[\/latex]. You would usually say that [latex]14y^{3}+3y[\/latex] has no constant term or that the constant term is 0.<\/p>\n<h2>Evaluate a polynomial<\/h2>\n<p>You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable <i>every time<\/i> it appears. Then use the order of operations to find the resulting value for the expression.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]3x^{2}-2x+1[\/latex] for [latex]x=-1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q280466\">Show Solution<\/span><\/p>\n<div id=\"q280466\" class=\"hidden-answer\" style=\"display: none\">Substitute [latex]-1[\/latex] for each <i>x<\/i> in the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]3\\left(-1\\right)^{2}-2\\left(-1\\right)+1[\/latex]<\/p>\n<p>Following the order of operations, evaluate exponents first.<\/p>\n<p style=\"text-align: center;\">[latex]3\\left(1\\right)-2\\left(-1\\right)+1[\/latex]<\/p>\n<p>Multiply 3 times 1, and then multiply [latex]-2[\/latex] times [latex]-1[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]3+\\left(-2\\right)\\left(-1\\right)+1[\/latex]<\/p>\n<p>Change the subtraction to addition of the opposite.<\/p>\n<p style=\"text-align: center;\">[latex]3+2+1[\/latex]<\/p>\n<p>Find the sum.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}-2x+1=6[\/latex], for [latex]x=-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Evaluate [latex]\\displaystyle -\\frac{2}{3}p^{4}+2^{3}-p[\/latex] for [latex]p = 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q745542\">Show Solution<\/span><\/p>\n<div id=\"q745542\" class=\"hidden-answer\" style=\"display: none\">Substitute 3 for each <i>p<\/i> in the polynomial.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle -\\frac{2}{3}\\left(3\\right)^{4}+2\\left(3\\right)^{3}-3[\/latex]<\/p>\n<p>Following the order of operations, evaluate exponents first and then multiply.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle -\\frac{2}{3}\\left(81\\right)+2\\left(27\\right)-3[\/latex]<\/p>\n<p>Add and then subtract to get [latex]-3[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]-54 + 54 \u2013 3[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\displaystyle -\\frac{2}{3}p^{4}+2p^{3}-p=-3[\/latex], for [latex]p = 3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p class=\"no-indent\" style=\"text-align: left;\">\u00a0IN the following video we show more examples of evaluating polynomials for given values of the variable.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Evaluate a Polynomial in One Variable\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/2EeFrgQP1hM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Add and Subtract Polynomials<\/h2>\n<p>We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms, and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms, and therefore cannot be added.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the sum.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q222892\">Show Answer<\/span><\/p>\n<div id=\"q222892\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill & \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<p>Here\u00a0is a summary of some helpful steps for adding and subtracting polynomials.<\/p>\n<div class=\"textbox\">\n<h3>\u00a0Given multiple polynomials, add or subtract them to simplify the expressions.<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Combine like terms.<\/li>\n<li>Simplify and write in standard form.<\/li>\n<\/ol>\n<\/div>\n<div>\n<p>When you subtract polynomials you will still be looking for like terms to combine, but you will need to pay attention to the sign of the terms you are combining. In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the difference.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q279648\">Show Answer<\/span><\/p>\n<div id=\"q279648\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\begin{array}{cc}7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill & \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<h3>Analysis of the Solution<\/h3>\n<p>Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\n<p>In the following video we show more examples of adding and subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Adding and Subtracting Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jiq3toC7wGM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplying Polynomials<\/h2>\n<p>Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms.<\/p>\n<p>You may have used the distributive property to help you solve linear equations such as\u00a0[latex]2\\left(x+7\\right)=21[\/latex]. We can distribute the [latex]2[\/latex] in [latex]2\\left(x+7\\right)[\/latex] to obtain the equivalent expression [latex]2x+14[\/latex]. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\n<p>The following video will provide you with examples of using the distributive property to find the product of\u00a0monomials and polynomials.<\/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<p>Below is a summary of the steps we used to find the product of two polynomials using the distributive property.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression.<\/h3>\n<ol>\n<li>Multiply each term of the first polynomial by each term of the second.<\/li>\n<li>Combine like terms.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<h2>Using FOIL to Multiply Binomials<\/h2>\n<p>We can also use a shortcut called the FOIL method when multiplying binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200224\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial, and then combining like terms.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use FOIL to find the product. [latex](2x-18)(3x+3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q787670\">Show Answer<\/span><\/p>\n<div id=\"q787670\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the product of the first terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200225\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p>Find the product of the outer terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200227\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p>Find the product of the inner terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200228\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p>Find the product of the last terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200229\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill & \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this video, we show an example of how to use the FOIL method to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" 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<p>The following steps summarize the process for using FOIL to multiply two binomials. \u00a0It is very important to note that this process only works for the product of two binomials. If you are multiplying a binomial\u00a0and a trinomial, it is better to use a table to keep track of your terms.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two binomials, use FOIL to simplify the expression.<\/h3>\n<ol>\n<li>Multiply the first terms of each binomial.<\/li>\n<li>Multiply the outer terms of the binomials.<\/li>\n<li>Multiply the inner terms of the binomials.<\/li>\n<li>Multiply the last terms of each binomial.<\/li>\n<li>Add the products.<\/li>\n<li>Combine like terms and simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>Multiply a Trinomial and a Binomial<\/h2>\n<p>Another type of polynomial multiplication problem is the product of a binomial and trinomial. Although the FOIL method can not be used since there are more than two terms in a trinomial, you still use the Distributive Property to organize the individual products. Using the distributive property, each term in the binomial must be multiplied by each of the terms in the trinomial.<\/p>\n<p>For our first examples, we will show you two ways to organize all of the terms that result from multiplying polynomials with more than two terms. The most important part of the process is finding a way to organize terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Find the product. \u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q637359\">Show Solution<\/span><\/p>\n<div id=\"q637359\" class=\"hidden-answer\" style=\"display: none\">Distribute the trinomial to each term in the binomial.<\/p>\n<p>[latex]3x\\left(5x^{2}+3x+10\\right)+6\\left(5x2+3x+10\\right)[\/latex]<\/p>\n<p>Use the distributive property to distribute the monomials to each term in the trinomials.<\/p>\n<p>[latex]3x\\left(5x^{2}\\right)+3x\\left(3x\\right)+3x\\left(10\\right)+6\\left(5x^{2}\\right)+6\\left(3x\\right)+6\\left(10\\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p>[latex]15x^{3}9x^{2}+30x^{2}+18x+60[\/latex]<\/p>\n<p>Group like terms.<\/p>\n<p>[latex]15x^{3}+\\left(9x^{2}+30x^{2}\\right)+\\left(30x+18x\\right)+60[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)=15x^{3}+39x^{2}+48x+60[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>As you can see, multiplying a binomial by a trinomial leads to a lot of individual terms! Using the same problem as above, we will show another way to organize all the terms produced by multiplying two polynomials with more than two terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply.\u00a0[latex]\\left(3x+6\\right)\\left(5x^{2}+3x+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q262750\">Show Solution<\/span><\/p>\n<div id=\"q262750\" class=\"hidden-answer\" style=\"display: none\">Set up the problem in a vertical form, and begin by multiplying [latex]3x+6[\/latex] by [latex]+10[\/latex]. Place the products underneath, as shown.<\/p>\n<p>[latex]\\begin{array}{r}3x+\\,\\,\\,6\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}+\\,\\,3x+10}\\\\+30x+60\\,\\end{array}[\/latex]<\/p>\n<p>Now multiply [latex]3x+6[\/latex] by [latex]+3x[\/latex]. Notice that [latex]\\left(6\\right)\\left(3x\\right)=18x[\/latex]; since this term is like [latex]30x[\/latex], place it directly beneath it.<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Finally, multiply [latex]3x+6[\/latex] by [latex]5x^{2}[\/latex]. Notice that [latex]30x^{2}[\/latex]\u00a0is placed underneath [latex]9x^{2}[\/latex].<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\end{array}[\/latex]<\/p>\n<p>Now add like terms.<\/p>\n<p>[latex]\\begin{array}{r}3x\\,\\,\\,\\,\\,\\,+\\,\\,\\,6\\,\\,\\\\\\underline{\\times\\,\\,\\,\\,\\,\\,5x^{2}\\,\\,\\,\\,\\,\\,+3x\\,\\,\\,\\,\\,\\,+10}\\\\+30x\\,\\,\\,\\,\\,+60\\,\\,\\\\+9x^{2}\\,\\,\\,+18x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\underline{+15x^{3}\\,\\,\\,\\,\\,\\,+30x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}\\\\+15x^{3}\\,\\,\\,\\,\\,\\,+39x^{2}\\,\\,\\,\\,+48x\\,\\,\\,\\,\\,+60\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]15x^{3}+39x^{2}+48x+60[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that although the two problems were solved using different strategies, the product is the same. Both the horizontal and vertical methods apply the Distributive Property to multiply a binomial by a trinomial.<\/p>\n<p>In our next example we will multiply a binomial and a trinomial that contains subtraction. Pay attention to the signs on the terms. \u00a0Forgetting a negative sign is the easiest mistake to make in this case.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the product.<\/p>\n<p>[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q485882\">Show Answer<\/span><\/p>\n<div id=\"q485882\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill & \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill & \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<h3>Analysis of the Solution<\/h3>\n<p>Another way to keep track of all the terms involved in this product is to use a table, as shown below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify. Notice how we kept the sign on each term, for example we are subtracting [latex]x[\/latex] from [latex]3x^2[\/latex], so we place [latex]-x[\/latex] in the table.<\/p>\n<table style=\"width: 30%;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<thead>\n<tr>\n<td><\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]+4[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2x[\/latex]<\/td>\n<td>[latex]6{x}^{3}\\\\[\/latex]<\/td>\n<td>[latex]-2{x}^{2}[\/latex]<\/td>\n<td>[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]+1[\/latex]<\/td>\n<td>[latex]3{x}^{2}[\/latex]<\/td>\n<td>[latex]-x[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Multiply. \u00a0[latex]\\left(2p-1\\right)\\left(3p^{2}-3p+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654814\">Show Solution<\/span><\/p>\n<div id=\"q654814\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute 2p and -1 to each term in the trinomial.<\/p>\n<p style=\"text-align: center;\">[latex]2p\\left(3p^{2}-3p+1\\right)-1\\left(3p^{2}-3p+1\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]2p\\left(3p^{2}\\right)+2p\\left(-3p\\right)+2p\\left(1\\right)-1\\left(3p^{2}\\right)-1\\left(-3p\\right)-1\\left(1\\right)[\/latex]<\/p>\n<p>Multiply. (Notice that the subtracted 1 and the subtracted 3<em>p<\/em> have a positive product that is added.)<\/p>\n<p style=\"text-align: center;\">[latex]6p^{3}-6p^{2}+2p-3p^{2}+3p-1[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]6p^{3}-9p^{2}+5p-1[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]6p^{3}-9p^{2}+5p-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show more examples of multiplying polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" title=\"(New Version Available) Polynomial Multiplication Involving Binomials and Trinomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bBKbldmlbqI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Divide a polynomial by a binomial<\/h2>\n<p>Dividing a <strong>polynomial<\/strong> by a monomial can be handled by dividing each term in the polynomial separately. This can\u2019t be done when the divisor has more than one term. However, the process of long division can be very helpful with polynomials.<\/p>\n<p>Recall how you can use long division to divide two whole numbers, say 900 divided by 37.<\/p>\n<p style=\"text-align: center;\">[latex]37\\overline{)900}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152919\/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 style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\,\\,\\,74\\end{array}[\/latex]<\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152920\/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 style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\underline{-74}\\\\\\,\\,\\,\\,\\,\\,\\,16\\end{array}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152921\/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 style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\underline{-74}\\\\\\,\\,\\,\\,\\,\\,\\,160\\end{array}[\/latex]<\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152922\/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 style=\"text-align: center;\">[latex]\\begin{array}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\\\37\\overline{)900}\\\\\\,\\,\\,\\,\\,\\underline{-74}\\\\\\,\\,\\,\\,\\,\\,\\,160\\\\\\,\\,\\,\\underline{-148}\\end{array}[\/latex]<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15152923\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152924\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152924\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152925\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152926\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152927\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152928\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152929\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152931\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152932\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152933\/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>In this video we show another example of dividing a degree two trinomial by a degree one binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" 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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152934\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152935\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152936\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152938\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152939\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152940\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152941\/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\/wp-content\/uploads\/sites\/121\/2016\/07\/15152942\/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>In the video that follows, we show another example of dividing a degree three trinomial by a binomial, not the &#8220;missing&#8221; term and how we work with it.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-8\" 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 process above works for dividing any polynomials, no matter how many terms are in the divisor or the dividend. The main things to remember are:<\/p>\n<ul>\n<li>When subtracting, be sure to subtract the whole expression, not just the first term. <i>This is very easy to forget, so be careful!<\/i><\/li>\n<li>Stop when the degree of the remainder is less than the degree of the dividend, or when you have brought down all the terms in the dividend, and that the quotient extends to the right edge of the dividend.<\/li>\n<\/ul>\n<p>In this video we present one more example of polynomial long division.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" 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>Synthetic Division<\/h2>\n<p>As we\u2019ve seen, long division of polynomials can involve many steps and be quite cumbersome. <strong>Synthetic division<\/strong> is a shorthand method of dividing polynomials for the special case of dividing by a polynomial\u00a0whose leading coefficient is 1.<\/p>\n<div class=\"textbox\">\n<h3 class=\"title\" data-type=\"title\">\u00a0Synthetic Division<\/h3>\n<p id=\"fs-id1165135383649\">Synthetic division is a shortcut that can be used when the divisor is a binomial in the form <em>x<\/em> \u2013\u00a0<em>k, <\/em>for a real number k.\u00a0In <strong>synthetic division<\/strong>, only the coefficients are used in the division process.<\/p>\n<\/div>\n<p class=\"title\" data-type=\"title\"><span style=\"font-size: 16px; line-height: 1.5;\">To illustrate the process, d<\/span><span style=\"font-size: 16px; line-height: 1.5;\">ivide [latex]2{x}^{3}-3{x}^{2}+4x+5[\/latex]\u00a0by [latex]x+2[\/latex]\u00a0using the long division algorithm.<\/span><\/p>\n<p id=\"fs-id1165137932636\"><span id=\"eip-id1163740536072\" data-type=\"media\" data-alt=\".\" data-display=\"block\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201527\/CNX_Precalc_revised_eq_42.png\" alt=\".\" width=\"250\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165137932377\">There is a lot of repetition in this process.\u00a0If we don\u2019t write the variables but, instead, line up their coefficients in columns under the division sign, we already have a simpler version of the entire problem.<\/p>\n<p><span id=\"fs-id1165134305375\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201529\/CNX_Precalc_Figure_03_05_0042.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165134305388\">Synthetic division carries this simplification even a few more steps. Collapse the table by moving each of the rows up to fill any vacant spots. Also, instead of dividing by 2, as we would in division of whole numbers, then multiplying and subtracting the middle product, we change the sign of the &#8220;divisor&#8221; to \u20132, multiply and add. The process starts by bringing down the leading coefficient.<span id=\"fs-id1165137696374\" data-type=\"media\" data-alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-display=\"block\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201531\/CNX_Precalc_Figure_03_05_0112.jpg\" alt=\"Synthetic division of the polynomial 2x^3-3x^2+4x+5 by x+2 in which it only contains the coefficients of each polynomial.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165137696388\">We then multiply it by the &#8220;divisor&#8221; and add, repeating this process column by column, until there are no entries left. The bottom row represents the coefficients of the quotient; the last entry of the bottom row is the remainder. In this case, the quotient is [latex]2x{^2} -7x+18[\/latex]\u00a0and the remainder is \u201331.\u00a0The process will be made more clear in the following example.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use synthetic division to divide [latex]5{x}^{2}-3x - 36[\/latex]\u00a0by [latex]x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q152802\">Show Answer<\/span><\/p>\n<div id=\"q152802\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135177608\">Begin by setting up the synthetic division. Write <i>3<\/i>\u00a0and the coefficients of the polynomial.<\/p>\n<p><span id=\"fs-id1165135177629\" data-type=\"media\" data-alt=\"A collapsed version of the previous synthetic division.\" data-display=\"block\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201532\/CNX_Precalc_Figure_03_05_0052.jpg\" alt=\"A collapsed version of the previous synthetic division.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165135439942\">Bring down the lead coefficient. Multiply the lead coefficient by <i>3\u00a0<\/i>and place the result in the second column.<\/p>\n<p><span id=\"fs-id1165135439966\" data-type=\"media\" data-alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" data-display=\"block\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201533\/CNX_Precalc_Figure_03_05_0062.jpg\" alt=\"The set-up of the synthetic division for the polynomial 5x^2-3x-36 by x-3, which renders {5, -3, -36} by 3.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165135179942\">Continue by adding [latex]-3+15[\/latex]\u00a0in the second column. Multiply the resulting number, [latex]12[\/latex] by <i>3<\/i>.\u00a0Write the result, [latex]36[\/latex] in the next column. Then add the numbers in the third column.<\/p>\n<p><span id=\"fs-id1165135179966\" data-type=\"media\" data-alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" data-display=\"block\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201535\/CNX_Precalc_Figure_03_05_0072.jpg\" alt=\"Multiplied by the lead coefficient, 5, in the second column, and the lead coefficient is brought down to the second row.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165135628639\">The result is [latex]5x+12[\/latex].<\/p>\n<p>We can check our work by multiplying the result by the original divisor [latex]x-3[\/latex], if we get\u00a0[latex]5{x}^{2}-3x - 36[\/latex], we have used the method correctly.<\/p>\n<p>Check:\u00a0[latex](5x+12)(x-3)[\/latex]<\/p>\n<p style=\"text-align: center;\">\u00a0[latex]\\begin{array}{cc}(5x+12)(x-3)\\\\=5x^2-15x+12x-36\\\\=5x^2-3x-36\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Because we got a result of\u00a0[latex]5{x}^{2}-3x - 36[\/latex] when we multiplied the divisor and our answer, we can be sure that we have used synthetic division correctly.<\/p>\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p>[latex]5{x}^{2}-3x - 36\\div{x-3}=5x+12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Analysis of the solution<\/p>\n<p>It is important to note that the result, [latex]5x+12[\/latex], of [latex]5{x}^{2}-3x - 36\\div{x-3}[\/latex] is one degree less than[latex]5{x}^{2}-3x - 36[\/latex]. Why is that? Think about how you would have solved this using long division. The first thing you would ask yourself is how many x&#8217;s are in [latex]5x^2[\/latex]?<\/p>\n<p style=\"text-align: center;\">[latex]x-3\\overline{)5{x}^{2}-3x - 36}[\/latex]<\/p>\n<p style=\"text-align: left;\">To get a result of [latex]5x^2[\/latex], you need to multiply [latex]x[\/latex] by [latex]5x[\/latex]. \u00a0The next step in long division is to subtract this result from [latex]5x^2[\/latex]. \u00a0This leaves us with no [latex]x^2[\/latex] term in the result.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Reflect on this idea &#8211; if you multiply two polynomials and get a result whose degree is 2, what are the possible degrees of the two polynomials that were multiplied? Write your ideas in the box below before looking at the discussion.<textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q962896\">Show Discussion<\/span><\/p>\n<div id=\"q962896\" class=\"hidden-answer\" style=\"display: none\">\n<p>A degree two polynomial will have a leading term with [latex]x^2[\/latex]. \u00a0Let&#8217;s use [latex]2x^2-2x-24[\/latex] as an example. We can write\u00a0two products\u00a0that will give this as a result of multiplication:<\/p>\n<p>[latex]2(x^2-x-12) =2x^2-2x-24[\/latex]<\/p>\n<p>[latex](2x+6)(x-4)=2x^2-2x-24[\/latex]<\/p>\n<p>If we work backward, starting from [latex]2x^2-2x-24[\/latex] if we divide by a binomial with degree one, such as [latex](x-4)[\/latex], our result will also have degree one.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In this video example, you will see another example of using synthetic division for division of a degree two polynomial by a degree one binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Ex 1:  Divide a Trinomial by a Binomial Using Synthetic Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KeZ_zMOYu9o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div id=\"fs-id1165135393407\" class=\"note precalculus howto textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<h3 id=\"fs-id1165135393414\">How To: Given two polynomials, use synthetic division to divide.<\/h3>\n<ol id=\"fs-id1165135393418\" data-number-style=\"arabic\">\n<li>Write <em>k<\/em>\u00a0for the divisor.<\/li>\n<li>Write the coefficients of the dividend.<\/li>\n<li>Bring the lead coefficient down.<\/li>\n<li>Multiply the lead coefficient by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Add the terms of the second column.<\/li>\n<li>Multiply the result by <em>k<\/em>.\u00a0Write the product in the next column.<\/li>\n<li>Repeat steps 5 and 6 for the remaining columns.<\/li>\n<li>Use the bottom numbers to write the quotient. The number in the last column is the remainder and has degree 0, the next number from the right has degree 1, the next number from the right has degree 2, and so on.<\/li>\n<\/ol>\n<\/div>\n<p>In the next example we will use synthetic division to divide a third-degree polynomial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use synthetic division to divide [latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex]\u00a0by [latex]x+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q153403\">Show Answer<\/span><\/p>\n<div id=\"q153403\" class=\"hidden-answer\" style=\"display: none\">The binomial divisor is [latex]x+2[\/latex]\u00a0so [latex]k=-2[\/latex].\u00a0Add each column, multiply the result by \u20132, and repeat until the last column is reached.<span id=\"fs-id1165134176031\" data-type=\"media\" data-alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\"><br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201536\/CNX_Precalc_Figure_03_05_0082.jpg\" alt=\"Synthetic division of 4x^3+10x^2-6x-20 divided by x+2.\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165134433356\">The result is [latex]4{x}^{2}+2x - 10[\/latex]. Again notice\u00a0the degree of the result is less than the degree of the quotient,\u00a0[latex]4{x}^{3}+10{x}^{2}-6x - 20[\/latex].<\/p>\n<p>We can check that we are correct by multiplying the result with the divisor:<\/p>\n<p>[latex](x+2)(4{x}^{2}+2x - 10)=4x^3+2x^2-10x+8x^2+4x-20=4x^3+10x^2-6x-20[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]4{x}^{3}+10{x}^{2}-6x - 20\\div{x+2}=4{x}^{2}+2x - 10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example we will show division of a fourth degree polynomial by a binomial. \u00a0Note how there is no x term in the fourth degree polynomial, so we need to use a placeholder of 0 to ensure proper alignment of terms.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use synthetic division to divide [latex]-9{x}^{4}+10{x}^{3}+7{x}^{2}-6[\/latex]\u00a0by [latex]x - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q76281\">Show Answer<\/span><\/p>\n<div id=\"q76281\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135571794\">Notice there is no <em data-effect=\"italics\">x<\/em>-term. We will use a zero as the coefficient for that term.<span id=\"eip-id6273758\" data-type=\"media\" data-alt=\".\" data-display=\"block\"><br \/>\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201539\/CNX_Precalc_revised_eq_52.png\" alt=\".\" width=\"230\" data-media-type=\"image\/jpg\" \/><\/span><\/p>\n<p id=\"fs-id1165135341342\">The result is [latex]-9{x}^{3}+{x}^{2}+8x+8+\\frac{2}{x - 1}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our last video example we show another example of how to use synthetic division to divide a degree three polynomial by a degree one binomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-11\" title=\"Ex 3:  Divide a Polynomial by a Binomial Using Synthetic Division\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/h1oSCNuA9i0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Multiplication of binomials and polynomials requires use of the distributive property as well as the commutative and associative properties of multiplication. 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<p>Dividing 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<h3><\/h3>\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-998\">\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>Evaluate a Polynomial in One Variable. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/2EeFrgQP1hM\">https:\/\/youtu.be\/2EeFrgQP1hM<\/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>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: 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>Multiply Binomials Using the FOIL Acronym. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/_MrdEFnXNGA\">https:\/\/youtu.be\/_MrdEFnXNGA<\/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><li>Screenshot Polynomial Generated Images. <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><\/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. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/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>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay, et al.. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at :http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/li><li>Ex: Intro to Polynomials in One Variable. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3u16B2PN9zk\">https:\/\/youtu.be\/3u16B2PN9zk<\/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: Adding and Subtracting Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jiq3toC7wGM\">https:\/\/youtu.be\/jiq3toC7wGM<\/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: Polynomial Multiplication Involving Binomials and Trinomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bBKbldmlbqI\">https:\/\/youtu.be\/bBKbldmlbqI<\/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 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>Ex 6: Divide a Polynomial by a Degree Two Binomial Using Long Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <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><li>Ex 1: Divide a Trinomial by a Binomial Using Synthetic Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KeZ_zMOYu9o\">https:\/\/youtu.be\/KeZ_zMOYu9o<\/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 3: Divide a Polynomial by a Binomial Using Synthetic Division. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/h1oSCNuA9i0\">https:\/\/youtu.be\/h1oSCNuA9i0<\/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":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Evaluate a Polynomial in One Variable\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/2EeFrgQP1hM\",\"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\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay, et al.\",\"organization\":\"\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"Download for free at :http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\"},{\"type\":\"cc\",\"description\":\"Ex: Intro to Polynomials in One Variable\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/3u16B2PN9zk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex: Multiplying Using the Distributive Property\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/bwTmApTV_8o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Multiply Binomials Using An Area Model and Using Repeated Distribution\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/u4Hgl0BrUlo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Multiply Binomials Using the FOIL Acronym\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/_MrdEFnXNGA\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Adding and Subtracting Polynomials\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/jiq3toC7wGM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Polynomial Multiplication Involving Binomials and Trinomials\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/bBKbldmlbqI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Divide a Trinomial by a Binomial Using Long Division\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KUPFg__Djzw\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Divide a Degree 3 Polynomial by a Degree 1 Polynomial (Long Division with Missing Term)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/Rxds7Q_UTeo\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 6: Divide a Polynomial by a Degree Two Binomial Using Long Division\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/P6OTbUf8f60\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 1: Divide a Trinomial by a Binomial Using Synthetic Division\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KeZ_zMOYu9o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex 3: Divide a Polynomial by a Binomial Using Synthetic Division\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/h1oSCNuA9i0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Screenshot Polynomial Generated Images\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-998","chapter","type-chapter","status-publish","hentry"],"part":994,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/998","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/998\/revisions"}],"predecessor-version":[{"id":4442,"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/998\/revisions\/4442"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/pressbooks\/v2\/parts\/994"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/pressbooks\/v2\/chapters\/998\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/wp\/v2\/media?parent=998"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=998"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/wp\/v2\/contributor?post=998"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-suffolkccc-intermediatealgebra\/wp-json\/wp\/v2\/license?post=998"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}