{"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":"2018-05-16T23:43:23","modified_gmt":"2018-05-16T23:43:23","slug":"algebraic_operations_on_polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/algebraic_operations_on_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\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<\/ul>\r\n<\/div>\r\n<h2>Anatomy of a Polynomial<\/h2>\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.\" \/>\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=\"\" \/>\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=\"\" \/>\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=\"\" \/>\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=\"\" \/>\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\r\n&nbsp;","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\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<\/ul>\n<\/div>\n<h2>Anatomy of a Polynomial<\/h2>\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.\" \/><\/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=\"\" \/><\/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=\"\" \/><\/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=\"\" \/><\/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=\"\" \/><\/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<p>&nbsp;<\/p>\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":4,"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 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