{"id":18265,"date":"2022-04-13T00:28:56","date_gmt":"2022-04-13T00:28:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18265"},"modified":"2022-04-28T23:15:38","modified_gmt":"2022-04-28T23:15:38","slug":"cr-9-multiplying-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-9-multiplying-polynomials\/","title":{"raw":"CR.9: Multiplying Polynomials","rendered":"CR.9: Multiplying Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the power and product properties of exponents to multiply monomials<\/li>\r\n \t<li>Use the\u00a0power and product properties of exponents to simplify monomials<\/li>\r\n \t<li>Multiply a polynomial by a monomial using the distributive property<\/li>\r\n \t<li>Use the distributive property to multiply two binomials<\/li>\r\n \t<li>Multiply a trinomial by a binomial<\/li>\r\n<\/ul>\r\n<\/div>\r\n&nbsp;\r\n\r\nWe now have three properties for multiplying expressions with exponents. Let\u2019s summarize them and then we\u2019ll do some examples that use more than one of the properties.\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Exponents<\/h3>\r\nIf [latex]a,b[\/latex] are real numbers and [latex]m,n[\/latex] are whole numbers, then\r\n\r\n[latex]\\begin{array}{cccc}\\text{Product Property}\\hfill &amp; &amp; &amp; \\hfill {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\text{Power Property}\\hfill &amp; &amp; &amp; \\hfill {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\text{Product to a Power Property}\\hfill &amp; &amp; &amp; \\hfill {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left({x}^{2}\\right)}^{6}{\\left({x}^{5}\\right)}^{4}[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168467221786\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left({x}^{2}\\right)}^{6}{\\left({x}^{5}\\right)}^{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property.<\/td>\r\n<td>[latex]{x}^{12}\\cdot {x}^{20}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add the exponents.<\/td>\r\n<td>[latex]{x}^{32}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146171[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(-7{x}^{3}{y}^{4}\\right)}^{2}[\/latex].\r\n[reveal-answer q=\"412728\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"412728\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469890733\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(-7{x}^{3}{y}^{4}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Take each factor to the second power.<\/td>\r\n<td>[latex]{\\left(-7\\right)}^{2}{\\left({x}^{3}\\right)}^{2}{\\left({y}^{4}\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property.<\/td>\r\n<td>[latex]49{x}^{6}{y}^{8}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146174[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(6n\\right)}^{2}\\left(4{n}^{3}\\right)[\/latex].\r\n[reveal-answer q=\"625558\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"625558\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168465997267\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(6n\\right)}^{2}\\left(4{n}^{3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Raise [latex]6n[\/latex] to the second power.<\/td>\r\n<td>[latex]{6}^{2}{n}^{2}\\cdot 4{n}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]36{n}^{2}\\cdot 4{n}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property.<\/td>\r\n<td>[latex]36\\cdot 4\\cdot {n}^{2}\\cdot {n}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the constants and add the exponents.<\/td>\r\n<td>[latex]144{n}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the <em>n<\/em>.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146177[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{\\left(3{p}^{2}q\\right)}^{4}{\\left(2p{q}^{2}\\right)}^{3}[\/latex].\r\n[reveal-answer q=\"299315\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"299315\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468310384\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{\\left(3{p}^{2}q\\right)}^{4}{\\left(2p{q}^{2}\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power of a Product Property.<\/td>\r\n<td>[latex]{3}^{4}{\\left({p}^{2}\\right)}^{4}{q}^{4}\\cdot {2}^{3}{p}^{3}{\\left({q}^{2}\\right)}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Power Property.<\/td>\r\n<td>[latex]81{p}^{8}{q}^{4}\\cdot 8{p}^{3}{q}^{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property.<\/td>\r\n<td>[latex]81\\cdot 8\\cdot {p}^{8}\\cdot {p}^{3}\\cdot {q}^{4}\\cdot {q}^{6}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply the constants and add the exponents for\r\n\r\neach variable.<\/td>\r\n<td>[latex]648{p}^{11}{q}^{10}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146179[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h3>Multiply Monomials<\/h3>\r\nSince a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(4{x}^{2}\\right)\\left(-5{x}^{3}\\right)[\/latex].\r\n[reveal-answer q=\"762661\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"762661\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469853450\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(4{x}^{2}\\right)\\left(-5{x}^{3}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property to rearrange the factors.<\/td>\r\n<td>[latex]4\\cdot \\left(-5\\right)\\cdot {x}^{2}\\cdot {x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]-20{x}^{5}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146195[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(\\frac{3}{4}{c}^{3}d\\right)\\left(12c{d}^{2}\\right)[\/latex].\r\n[reveal-answer q=\"867589\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867589\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466307238\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(\\frac{3}{4}{c}^{3}d\\right)\\left(12c{d}^{2}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Use the Commutative Property to rearrange\r\n\r\nthe factors.<\/td>\r\n<td>[latex]\\frac{3}{4}\\cdot 12\\cdot {c}^{3}\\cdot c\\cdot d\\cdot {d}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]9{c}^{4}{d}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146196[\/ohm_question]\r\n\r\n<\/div>\r\nFor more examples of how to use the power and product rules of exponents to simplify and multiply monomials, watch the following video.\r\n\r\nhttps:\/\/youtu.be\/E_D8PO1G7gU\r\n<h2>Multiply a Monomial by a Polynomial<\/h2>\r\nIn Distributive Property you learned to use the Distributive Property to simplify expressions such as [latex]2\\left(x - 3\\right)[\/latex]. You multiplied both terms in the parentheses, [latex]x\\text{ and }3[\/latex], by [latex]2[\/latex], to get [latex]2x - 6[\/latex]. With this chapter's new vocabulary, you can say you were multiplying a binomial, [latex]x - 3[\/latex], by a monomial, [latex]2[\/latex]. Multiplying a binomial by a monomial is nothing new for you!\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]3\\left(x+7\\right)[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168468708606\" class=\"unnumbered unstyled\" summary=\"The top line shows 3 times parentheses x plus 7. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3\\left(x+7\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224419\/CNX_BMath_Figure_10_03_001_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3\\cdot x+3\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]3x+21[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146197[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]x\\left(x - 8\\right)[\/latex].\r\n[reveal-answer q=\"200462\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"200462\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468567807\" class=\"unnumbered unstyled\" summary=\"The top line shows x times parentheses x minus 8. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x(x-8)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224421\/CNX_BMath_Figure_10_03_044_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x^2-8x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]x^2-8x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146198[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]10x\\left(4x+y\\right)[\/latex].\r\n[reveal-answer q=\"346464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"346464\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468433683\" class=\"unnumbered unstyled\" summary=\"The top line shows 10x times parentheses 4x plus y. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]10x(4x+y)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224423\/CNX_BMath_Figure_10_03_045_img-02.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]10x\\cdot{4x}+10x\\cdot{y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]40x^2+10xy[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146201[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nMultiplying a monomial by a trinomial works in much the same way.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]-2x\\left(5{x}^{2}+7x - 3\\right)[\/latex].\r\n[reveal-answer q=\"218200\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"218200\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469497833\" class=\"unnumbered unstyled\" summary=\"The top line shows negative 2x times parentheses 5 x squared plus 7x minus 3. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-2x\\left(5{x}^{2}+7x - 3\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224426\/CNX_BMath_Figure_10_03_046_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]-2x\\cdot 5{x}^{2}+\\left(-2x\\right)\\cdot 7x-\\left(-2x\\right)\\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]-10{x}^{3}-14{x}^{2}+6x[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146203[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]4{y}^{3}\\left({y}^{2}-8y+1\\right)[\/latex].\r\n[reveal-answer q=\"648407\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"648407\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168466095135\" class=\"unnumbered unstyled\" summary=\"The top line shows 4 y cubed times parentheses y squared minus 8y plus 1. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4{y}^{3}\\left({y}^{2}-8y+1\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224426\/CNX_BMath_Figure_10_03_047_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]4{y}^{3}\\cdot {y}^{2}-4{y}^{3}\\cdot 8y+4{y}^{3}\\cdot 1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]4{y}^{5}-32{y}^{4}+4{y}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146204[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNow we will have the monomial as the second factor.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(x+3\\right)p[\/latex].\r\n[reveal-answer q=\"544089\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"544089\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468557799\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 3, times p. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(x+3\\right)p[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224427\/CNX_BMath_Figure_10_03_048_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x\\cdot p+3\\cdot p[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]xp+3p[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146206[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video we show more examples of how to multiply monomials with other polynomials.\r\n\r\nhttps:\/\/youtu.be\/bwTmApTV_8o\r\n<h2>Multiply Two Binomials<\/h2>\r\nJust like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.\r\n<h3>Using the Distributive Property<\/h3>\r\nWe will start by using the Distributive Property. Look again at the following example.\r\n<table id=\"eip-id1168466233256\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 3, times p, with red arrows from the p to the x and to the 3. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224428\/CNX_BMath_Figure_10_03_049_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>We distributed the [latex]p[\/latex] to get<\/td>\r\n<td>[latex]x\\color{red}{p}+3\\color{red}{p}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>What if we have [latex]\\left(x+7\\right)[\/latex] instead of [latex]p[\/latex] ?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224429\/CNX_BMath_Figure_10_03_049_img-05.png\" alt=\".\" \/><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224430\/CNX_BMath_Figure_10_03_049_img-03.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute [latex]\\left(x+7\\right)[\/latex] .<\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224432\/CNX_BMath_Figure_10_03_049_img-04.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]{x}^{2}+7x+3x+21[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]{x}^{2}+10x+21[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial\u2014four multiplications.\r\nBe careful to distinguish between a sum and a product.\r\n[latex]\\begin{array}{cccc}\\hfill \\mathbf{\\text{Sum}}\\hfill &amp; &amp; &amp; \\hfill \\mathbf{\\text{Product}}\\hfill \\\\ \\hfill x+x\\hfill &amp; &amp; &amp; \\hfill x\\cdot x\\hfill \\\\ \\hfill 2x\\hfill &amp; &amp; &amp; \\hfill {x}^{2}\\hfill \\\\ \\hfill \\text{combine like terms}\\hfill &amp; &amp; &amp; \\hfill \\text{add exponents of like bases}\\hfill \\end{array}[\/latex]\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168468281692\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 6 times parentheses x plus 8. The next line shows parentheses x plus 6 times red parentheses x plus 8, with red arrows from x plus 8 to x and to 6. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224433\/CNX_BMath_Figure_10_03_050_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute [latex]\\left(x+8\\right)[\/latex] .<\/td>\r\n<td>[latex]x\\color{red}{(x+8)}+6\\color{red}{(x+8)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]{x}^{2}+8x+6x+48[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]{x}^{2}+14x+48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146207[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNow we'll see how to multiply binomials where the variable has a coefficient.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex].\r\n[reveal-answer q=\"901421\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"901421\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467332174\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2x plus 9 times parentheses 3x plus 4. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute. [latex]\\left(3x+4\\right)[\/latex]<\/td>\r\n<td>[latex]2x\\color{red}{(3x+4)}+9\\color{red}{(3x+4)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]6{x}^{2}+8x+27x+36[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]6{x}^{2}+35x+36[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146208[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex].\r\n[reveal-answer q=\"834420\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"834420\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469625351\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 4y plus 3 times parentheses 6y minus 5. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]4y\\color{red}{(6y-5)}+3\\color{red}{(6y-5)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]24{y}^{2}-20y+18y - 15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>[latex]24{y}^{2}-2y - 15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146209[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nUp to this point, the product of two binomials has been a trinomial. This is not always the case.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply: [latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex].\r\n[reveal-answer q=\"982155\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"982155\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468254725\" class=\"unnumbered unstyled\" summary=\"The top line says parentheses x plus 2 times parentheses x minus y. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex](x+2)(x-y)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]x\\color{red}{(x-y)}+2\\color{red}{(x-y)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute again.<\/td>\r\n<td>[latex]x^2-xy+2x-2y[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Simplify.<\/td>\r\n<td>There are no like terms to combine.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146210[\/ohm_question]\r\n\r\n<\/div>\r\nTo see another example of how to visualize multiplying two binomials, watch the following video. We use an area model as well as repeated distribution to multiply two binomials.\r\n\r\nhttps:\/\/youtu.be\/u4Hgl0BrUlo\r\n<h2>The Table Method<\/h2>\r\nYou may see a binomial multiplied by itself written as [latex]{\\left(x+3\\right)}^{2}[\/latex] instead of [latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. To find this product, let\u2019s use another method. We will place the terms of each binomial along the top row and first column of a table, like this:\r\n<table class=\"lines\" style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<td><\/td>\r\n<td><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.\r\n<table style=\"width: 20%;\">\r\n<thead>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td><span style=\"color: #0000ff;\">[latex]x\\cdot{x}=x^2[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #ff0000;\">[latex]3\\cdot{x}=+3x[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]+3[\/latex]<\/td>\r\n<td><span style=\"color: #ff0000;\">[latex]x\\cdot{3}=+3x[\/latex]<\/span><\/td>\r\n<td><span style=\"color: #ff00ff;\"> [latex]3\\cdot{3}=+9[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can write the terms of the polynomial from the entries in the table:\r\n<p style=\"text-align: center;\">[latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">= <span style=\"color: #0000ff;\">[latex]x^2[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span><\/p>\r\n<p style=\"text-align: center;\">= <span style=\"color: #0000ff;\">[latex]x^{2}[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]6x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span>.<\/p>\r\n<p style=\"text-align: center;\">Pretty cool, huh?<\/p>\r\nSo far, we have shown two methods for multiplying two binomials together. Why are we focusing so much on binomials? They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSquare the binomial difference [latex]\\left(x\u20137\\right)[\/latex]\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q293164\">Show solution<\/span>\r\n<div id=\"q293164\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nWrite the product of the binomial.\r\n<p style=\"text-align: center;\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Let\u2019s use the table method, just because. Note how we carry the negative sign with the [latex]7[\/latex].<\/p>\r\n\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]-7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]x[\/latex]<\/td>\r\n<td>[latex]x^2[\/latex]<\/td>\r\n<td>[latex]-7x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-7[\/latex]<\/td>\r\n<td>[latex]-7x[\/latex]<\/td>\r\n<td>[latex]49[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nCollect the terms, and simplify. Note how we keep the sign on each term.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\r\nAnswer\r\n[latex]x^2-14x+49[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" sizes=\"(max-width: 51px) 100vw, 51px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-768x678.png 768w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-1024x903.png 1024w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-65x57.png 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-225x199.png 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-350x309.png 350w\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:\r\n<p style=\"text-align: center;\">You can\u2019t move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\r\n<p style=\"text-align: center;\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\r\n<p style=\"text-align: center;\"><strong> CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\nIn the video that follows, you will see another example of using a table to multiply two binomials.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/tWsLJ_pn5mQ?feature=oembed\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2>Multiply a Trinomial by a Binomial<\/h2>\r\nWe have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply using the Distributive Property: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex].\r\n\r\nSolution\r\n<table id=\"eip-id1168467363028\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2 x squared minus 5x plus 8 times x plus 3 is written, with a line beneath it. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224530\/CNX_BMath_Figure_10_03_061_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Distribute.<\/td>\r\n<td>[latex]x\\color{red}{(2x^2-5x+8)}+3\\color{red}{(2x^2-5x+8)}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]2{x}^{3}+{x}^{2}-7x+24[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146217[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNow let's do this same multiplication using the Vertical Method.\r\n\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nMultiply using the Vertical Method: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex].\r\n[reveal-answer q=\"80068\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80068\"]\r\n\r\nSolution\r\nIt is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.\r\n<table id=\"eip-id1168469786876\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2 x squared minus 5x plus 8 times x plus 3 is written, with a line beneath it. The next line says, \">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224533\/CNX_BMath_Figure_10_03_062_img-01.png\" alt=\".\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]3[\/latex].<\/td>\r\n<td>[latex]6x^2-15x+24[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]x[\/latex] .<\/td>\r\n<td>[latex]2x^3-5x^2+8x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Add like terms.<\/td>\r\n<td>[latex]2x^3+x^2-7x+24[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146218[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWatch the following video to see more examples of multiplying polynomials.\r\n\r\nhttps:\/\/youtu.be\/bBKbldmlbqI","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the power and product properties of exponents to multiply monomials<\/li>\n<li>Use the\u00a0power and product properties of exponents to simplify monomials<\/li>\n<li>Multiply a polynomial by a monomial using the distributive property<\/li>\n<li>Use the distributive property to multiply two binomials<\/li>\n<li>Multiply a trinomial by a binomial<\/li>\n<\/ul>\n<\/div>\n<p>&nbsp;<\/p>\n<p>We now have three properties for multiplying expressions with exponents. Let\u2019s summarize them and then we\u2019ll do some examples that use more than one of the properties.<\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Exponents<\/h3>\n<p>If [latex]a,b[\/latex] are real numbers and [latex]m,n[\/latex] are whole numbers, then<\/p>\n<p>[latex]\\begin{array}{cccc}\\text{Product Property}\\hfill & & & \\hfill {a}^{m}\\cdot {a}^{n}={a}^{m+n}\\hfill \\\\ \\text{Power Property}\\hfill & & & \\hfill {\\left({a}^{m}\\right)}^{n}={a}^{m\\cdot n}\\hfill \\\\ \\text{Product to a Power Property}\\hfill & & & \\hfill {\\left(ab\\right)}^{m}={a}^{m}{b}^{m}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left({x}^{2}\\right)}^{6}{\\left({x}^{5}\\right)}^{4}[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467221786\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left({x}^{2}\\right)}^{6}{\\left({x}^{5}\\right)}^{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property.<\/td>\n<td>[latex]{x}^{12}\\cdot {x}^{20}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add the exponents.<\/td>\n<td>[latex]{x}^{32}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146171\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146171&theme=oea&iframe_resize_id=ohm146171&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(-7{x}^{3}{y}^{4}\\right)}^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q412728\">Show Solution<\/span><\/p>\n<div id=\"q412728\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469890733\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(-7{x}^{3}{y}^{4}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Take each factor to the second power.<\/td>\n<td>[latex]{\\left(-7\\right)}^{2}{\\left({x}^{3}\\right)}^{2}{\\left({y}^{4}\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property.<\/td>\n<td>[latex]49{x}^{6}{y}^{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146174\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146174&theme=oea&iframe_resize_id=ohm146174&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(6n\\right)}^{2}\\left(4{n}^{3}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q625558\">Show Solution<\/span><\/p>\n<div id=\"q625558\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168465997267\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(6n\\right)}^{2}\\left(4{n}^{3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Raise [latex]6n[\/latex] to the second power.<\/td>\n<td>[latex]{6}^{2}{n}^{2}\\cdot 4{n}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]36{n}^{2}\\cdot 4{n}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property.<\/td>\n<td>[latex]36\\cdot 4\\cdot {n}^{2}\\cdot {n}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the constants and add the exponents.<\/td>\n<td>[latex]144{n}^{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that in the first monomial, the exponent was outside the parentheses and it applied to both factors inside. In the second monomial, the exponent was inside the parentheses and so it only applied to the <em>n<\/em>.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146177\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146177&theme=oea&iframe_resize_id=ohm146177&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{\\left(3{p}^{2}q\\right)}^{4}{\\left(2p{q}^{2}\\right)}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q299315\">Show Solution<\/span><\/p>\n<div id=\"q299315\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468310384\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{\\left(3{p}^{2}q\\right)}^{4}{\\left(2p{q}^{2}\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power of a Product Property.<\/td>\n<td>[latex]{3}^{4}{\\left({p}^{2}\\right)}^{4}{q}^{4}\\cdot {2}^{3}{p}^{3}{\\left({q}^{2}\\right)}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Power Property.<\/td>\n<td>[latex]81{p}^{8}{q}^{4}\\cdot 8{p}^{3}{q}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property.<\/td>\n<td>[latex]81\\cdot 8\\cdot {p}^{8}\\cdot {p}^{3}\\cdot {q}^{4}\\cdot {q}^{6}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply the constants and add the exponents for<\/p>\n<p>each variable.<\/td>\n<td>[latex]648{p}^{11}{q}^{10}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146179\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146179&theme=oea&iframe_resize_id=ohm146179&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>Multiply Monomials<\/h3>\n<p>Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(4{x}^{2}\\right)\\left(-5{x}^{3}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q762661\">Show Solution<\/span><\/p>\n<div id=\"q762661\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469853450\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(4{x}^{2}\\right)\\left(-5{x}^{3}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property to rearrange the factors.<\/td>\n<td>[latex]4\\cdot \\left(-5\\right)\\cdot {x}^{2}\\cdot {x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]-20{x}^{5}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146195\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146195&theme=oea&iframe_resize_id=ohm146195&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(\\frac{3}{4}{c}^{3}d\\right)\\left(12c{d}^{2}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q867589\">Show Solution<\/span><\/p>\n<div id=\"q867589\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466307238\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(\\frac{3}{4}{c}^{3}d\\right)\\left(12c{d}^{2}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Use the Commutative Property to rearrange<\/p>\n<p>the factors.<\/td>\n<td>[latex]\\frac{3}{4}\\cdot 12\\cdot {c}^{3}\\cdot c\\cdot d\\cdot {d}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]9{c}^{4}{d}^{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146196\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146196&theme=oea&iframe_resize_id=ohm146196&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>For more examples of how to use the power and product rules of exponents to simplify and multiply monomials, watch the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex 2: Exponent Properties (Product, Power Properties)\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/E_D8PO1G7gU?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiply a Monomial by a Polynomial<\/h2>\n<p>In Distributive Property you learned to use the Distributive Property to simplify expressions such as [latex]2\\left(x - 3\\right)[\/latex]. You multiplied both terms in the parentheses, [latex]x\\text{ and }3[\/latex], by [latex]2[\/latex], to get [latex]2x - 6[\/latex]. With this chapter&#8217;s new vocabulary, you can say you were multiplying a binomial, [latex]x - 3[\/latex], by a monomial, [latex]2[\/latex]. Multiplying a binomial by a monomial is nothing new for you!<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]3\\left(x+7\\right)[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468708606\" class=\"unnumbered unstyled\" summary=\"The top line shows 3 times parentheses x plus 7. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]3\\left(x+7\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224419\/CNX_BMath_Figure_10_03_001_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3\\cdot x+3\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]3x+21[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146197\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146197&theme=oea&iframe_resize_id=ohm146197&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]x\\left(x - 8\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200462\">Show Solution<\/span><\/p>\n<div id=\"q200462\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468567807\" class=\"unnumbered unstyled\" summary=\"The top line shows x times parentheses x minus 8. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x(x-8)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224421\/CNX_BMath_Figure_10_03_044_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]x^2-8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]x^2-8x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146198\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146198&theme=oea&iframe_resize_id=ohm146198&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]10x\\left(4x+y\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q346464\">Show Solution<\/span><\/p>\n<div id=\"q346464\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468433683\" class=\"unnumbered unstyled\" summary=\"The top line shows 10x times parentheses 4x plus y. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]10x(4x+y)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224423\/CNX_BMath_Figure_10_03_045_img-02.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]10x\\cdot{4x}+10x\\cdot{y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]40x^2+10xy[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146201\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146201&theme=oea&iframe_resize_id=ohm146201&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Multiplying a monomial by a trinomial works in much the same way.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]-2x\\left(5{x}^{2}+7x - 3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q218200\">Show Solution<\/span><\/p>\n<div id=\"q218200\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469497833\" class=\"unnumbered unstyled\" summary=\"The top line shows negative 2x times parentheses 5 x squared plus 7x minus 3. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]-2x\\left(5{x}^{2}+7x - 3\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224426\/CNX_BMath_Figure_10_03_046_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]-2x\\cdot 5{x}^{2}+\\left(-2x\\right)\\cdot 7x-\\left(-2x\\right)\\cdot 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]-10{x}^{3}-14{x}^{2}+6x[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146203\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146203&theme=oea&iframe_resize_id=ohm146203&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]4{y}^{3}\\left({y}^{2}-8y+1\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q648407\">Show Solution<\/span><\/p>\n<div id=\"q648407\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168466095135\" class=\"unnumbered unstyled\" summary=\"The top line shows 4 y cubed times parentheses y squared minus 8y plus 1. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]4{y}^{3}\\left({y}^{2}-8y+1\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224426\/CNX_BMath_Figure_10_03_047_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]4{y}^{3}\\cdot {y}^{2}-4{y}^{3}\\cdot 8y+4{y}^{3}\\cdot 1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]4{y}^{5}-32{y}^{4}+4{y}^{3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146204\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146204&theme=oea&iframe_resize_id=ohm146204&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now we will have the monomial as the second factor.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(x+3\\right)p[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q544089\">Show Solution<\/span><\/p>\n<div id=\"q544089\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468557799\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 3, times p. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(x+3\\right)p[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224427\/CNX_BMath_Figure_10_03_048_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]x\\cdot p+3\\cdot p[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]xp+3p[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146206\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146206&theme=oea&iframe_resize_id=ohm146206&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the following video we show more examples of how to multiply monomials with other polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Multiplying Using the Distributive Property\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiply Two Binomials<\/h2>\n<p>Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial.<\/p>\n<h3>Using the Distributive Property<\/h3>\n<p>We will start by using the Distributive Property. Look again at the following example.<\/p>\n<table id=\"eip-id1168466233256\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 3, times p, with red arrows from the p to the x and to the 3. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224428\/CNX_BMath_Figure_10_03_049_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>We distributed the [latex]p[\/latex] to get<\/td>\n<td>[latex]x\\color{red}{p}+3\\color{red}{p}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>What if we have [latex]\\left(x+7\\right)[\/latex] instead of [latex]p[\/latex] ?<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224429\/CNX_BMath_Figure_10_03_049_img-05.png\" alt=\".\" \/><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224430\/CNX_BMath_Figure_10_03_049_img-03.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute [latex]\\left(x+7\\right)[\/latex] .<\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224432\/CNX_BMath_Figure_10_03_049_img-04.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]{x}^{2}+7x+3x+21[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]{x}^{2}+10x+21[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that before combining like terms, we had four terms. We multiplied the two terms of the first binomial by the two terms of the second binomial\u2014four multiplications.<br \/>\nBe careful to distinguish between a sum and a product.<br \/>\n[latex]\\begin{array}{cccc}\\hfill \\mathbf{\\text{Sum}}\\hfill & & & \\hfill \\mathbf{\\text{Product}}\\hfill \\\\ \\hfill x+x\\hfill & & & \\hfill x\\cdot x\\hfill \\\\ \\hfill 2x\\hfill & & & \\hfill {x}^{2}\\hfill \\\\ \\hfill \\text{combine like terms}\\hfill & & & \\hfill \\text{add exponents of like bases}\\hfill \\end{array}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168468281692\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses x plus 6 times parentheses x plus 8. The next line shows parentheses x plus 6 times red parentheses x plus 8, with red arrows from x plus 8 to x and to 6. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(x+6\\right)\\left(x+8\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224433\/CNX_BMath_Figure_10_03_050_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute [latex]\\left(x+8\\right)[\/latex] .<\/td>\n<td>[latex]x\\color{red}{(x+8)}+6\\color{red}{(x+8)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]{x}^{2}+8x+6x+48[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]{x}^{2}+14x+48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146207\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146207&theme=oea&iframe_resize_id=ohm146207&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now we&#8217;ll see how to multiply binomials where the variable has a coefficient.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q901421\">Show Solution<\/span><\/p>\n<div id=\"q901421\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467332174\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 2x plus 9 times parentheses 3x plus 4. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(2x+9\\right)\\left(3x+4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute. [latex]\\left(3x+4\\right)[\/latex]<\/td>\n<td>[latex]2x\\color{red}{(3x+4)}+9\\color{red}{(3x+4)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]6{x}^{2}+8x+27x+36[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]6{x}^{2}+35x+36[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146208\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146208&theme=oea&iframe_resize_id=ohm146208&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In the previous examples, the binomials were sums. When there are differences, we pay special attention to make sure the signs of the product are correct.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q834420\">Show Solution<\/span><\/p>\n<div id=\"q834420\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469625351\" class=\"unnumbered unstyled\" summary=\"The top line shows parentheses 4y plus 3 times parentheses 6y minus 5. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]\\left(4y+3\\right)\\left(6y - 5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]4y\\color{red}{(6y-5)}+3\\color{red}{(6y-5)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]24{y}^{2}-20y+18y - 15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>[latex]24{y}^{2}-2y - 15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146209\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146209&theme=oea&iframe_resize_id=ohm146209&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Up to this point, the product of two binomials has been a trinomial. This is not always the case.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]\\left(x+2\\right)\\left(x-y\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982155\">Show Solution<\/span><\/p>\n<div id=\"q982155\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468254725\" class=\"unnumbered unstyled\" summary=\"The top line says parentheses x plus 2 times parentheses x minus y. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex](x+2)(x-y)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]x\\color{red}{(x-y)}+2\\color{red}{(x-y)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Distribute again.<\/td>\n<td>[latex]x^2-xy+2x-2y[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Simplify.<\/td>\n<td>There are no like terms to combine.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146210\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146210&theme=oea&iframe_resize_id=ohm146210&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>To see another example of how to visualize multiplying two binomials, watch the following video. We use an area model as well as repeated distribution to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Multiply Binomials Using An Area Model and Using Repeated Distribution\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/u4Hgl0BrUlo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Table Method<\/h2>\n<p>You may see a binomial multiplied by itself written as [latex]{\\left(x+3\\right)}^{2}[\/latex] instead of [latex]\\left(x+3\\right)\\left(x+3\\right)[\/latex]. To find this product, let\u2019s use another method. We will place the terms of each binomial along the top row and first column of a table, like this:<\/p>\n<table class=\"lines\" style=\"width: 20%;\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]+3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td>[latex]+3[\/latex]<\/td>\n<td><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now multiply the term in each column by the term in each row to get the terms of the resulting polynomial. Note how we keep the signs on the terms, even when they are positive, this will help us write the new polynomial.<\/p>\n<table style=\"width: 20%;\">\n<thead>\n<tr>\n<td><\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]+3[\/latex]<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td><span style=\"color: #0000ff;\">[latex]x\\cdot{x}=x^2[\/latex]<\/span><\/td>\n<td><span style=\"color: #ff0000;\">[latex]3\\cdot{x}=+3x[\/latex]<\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]+3[\/latex]<\/td>\n<td><span style=\"color: #ff0000;\">[latex]x\\cdot{3}=+3x[\/latex]<\/span><\/td>\n<td><span style=\"color: #ff00ff;\"> [latex]3\\cdot{3}=+9[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can write the terms of the polynomial from the entries in the table:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x+3\\right)^{2}[\/latex]<\/p>\n<p style=\"text-align: center;\">= <span style=\"color: #0000ff;\">[latex]x^2[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]3x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span><\/p>\n<p style=\"text-align: center;\">= <span style=\"color: #0000ff;\">[latex]x^{2}[\/latex]<\/span> + <span style=\"color: #ff0000;\">[latex]6x[\/latex]<\/span> + <span style=\"color: #ff00ff;\">[latex]9[\/latex]<\/span>.<\/p>\n<p style=\"text-align: center;\">Pretty cool, huh?<\/p>\n<p>So far, we have shown two methods for multiplying two binomials together. Why are we focusing so much on binomials? They are one of the most well studied and widely used polynomials, so there is a lot of information out there about them. In the previous example, we saw the result of squaring a binomial that was a sum of two terms. In the next example we will find the product of squaring a binomial that is the difference of two terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Square the binomial difference [latex]\\left(x\u20137\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q293164\">Show solution<\/span><\/p>\n<div id=\"q293164\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Write the product of the binomial.<\/p>\n<p style=\"text-align: center;\">[latex]{\\left(x-7\\right)}^2=\\left(x\u20137\\right)\\left(x\u20137\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">Let\u2019s use the table method, just because. Note how we carry the negative sign with the [latex]7[\/latex].<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]-7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]x[\/latex]<\/td>\n<td>[latex]x^2[\/latex]<\/td>\n<td>[latex]-7x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-7[\/latex]<\/td>\n<td>[latex]-7x[\/latex]<\/td>\n<td>[latex]49[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Collect the terms, and simplify. Note how we keep the sign on each term.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}x^2-7x-7x+49\\\\\\text{ }\\\\=x^2-14x+49\\end{array}[\/latex]<\/p>\n<p>Answer<br \/>\n[latex]x^2-14x+49[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" sizes=\"auto, (max-width: 51px) 100vw, 51px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-768x678.png 768w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-1024x903.png 1024w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-65x57.png 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-225x199.png 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-350x309.png 350w\" alt=\"Caution\" width=\"51\" height=\"45\" \/>Caution! It is VERY important to remember the caution from the exponents section about squaring a binomial:<\/p>\n<p style=\"text-align: center;\">You can\u2019t move the exponent into a grouped sum because of the order of operations!!!!!<\/p>\n<p style=\"text-align: center;\"><strong>INCORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}\\neq2^{2}+x^{2}[\/latex]<\/p>\n<p style=\"text-align: center;\"><strong> CORRECT:<\/strong> [latex]\\left(2+x\\right)^{2}=\\left(2+x\\right)\\left(2+x\\right)[\/latex]<\/p>\n<\/div>\n<p>In the video that follows, you will see another example of using a table to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/tWsLJ_pn5mQ?feature=oembed\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiply a Trinomial by a Binomial<\/h2>\n<p>We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we&#8217;re ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply using the Distributive Property: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex].<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467363028\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2 x squared minus 5x plus 8 times x plus 3 is written, with a line beneath it. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224530\/CNX_BMath_Figure_10_03_061_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Distribute.<\/td>\n<td>[latex]x\\color{red}{(2x^2-5x+8)}+3\\color{red}{(2x^2-5x+8)}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]2{x}^{3}-5{x}^{2}+8x+6{x}^{2}-15x+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]2{x}^{3}+{x}^{2}-7x+24[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146217\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146217&theme=oea&iframe_resize_id=ohm146217&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Now let&#8217;s do this same multiplication using the Vertical Method.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply using the Vertical Method: [latex]\\left(x+3\\right)\\left(2{x}^{2}-5x+8\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80068\">Show Solution<\/span><\/p>\n<div id=\"q80068\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<br \/>\nIt is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.<\/p>\n<table id=\"eip-id1168469786876\" class=\"unnumbered unstyled\" summary=\"A vertical multiplication problem is shown. 2 x squared minus 5x plus 8 times x plus 3 is written, with a line beneath it. The next line says,\">\n<tbody>\n<tr>\n<td><\/td>\n<td><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224533\/CNX_BMath_Figure_10_03_062_img-01.png\" alt=\".\" \/><\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]3[\/latex].<\/td>\n<td>[latex]6x^2-15x+24[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply [latex]\\left(2{x}^{2}-5x+8\\right)[\/latex] by [latex]x[\/latex] .<\/td>\n<td>[latex]2x^3-5x^2+8x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Add like terms.<\/td>\n<td>[latex]2x^3+x^2-7x+24[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146218\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146218&theme=oea&iframe_resize_id=ohm146218&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Watch the following video to see more examples of multiplying polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" 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","protected":false},"author":264444,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18265","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18265","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":8,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18265\/revisions"}],"predecessor-version":[{"id":18704,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18265\/revisions\/18704"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18265\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18265"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18265"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18265"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}