{"id":10836,"date":"2017-06-05T21:25:11","date_gmt":"2017-06-05T21:25:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10836"},"modified":"2020-10-22T09:28:41","modified_gmt":"2020-10-22T09:28:41","slug":"multiplying-a-polynomial-by-a-monomial","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/multiplying-a-polynomial-by-a-monomial\/","title":{"raw":"12.2.c - Multiplying a Polynomial by a Monomial","rendered":"12.2.c &#8211; Multiplying a Polynomial by a Monomial"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply a polynomial by a monomial<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the exponents section, we\u00a0practiced\u00a0multiplying\u00a0monomials together, like we did with this expression: [latex]24{x}^{8}2{x}^{5}[\/latex]. The only thing different between that section and this one is that we called it simplifying, and now we are calling it polynomial multiplication. \u00a0Remember that simplifying a mathematical expression means performing as many operations as we can until there are no more to perform, including multiplication. \u00a0In this section we will show examples of how to multiply more than just monomials. \u00a0We will multiply monomials with\u00a0binomials and trinomials. We will also learn some techniques for multiplying two binomials together.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nMultiply. [latex]-9x^{3}\\cdot 3x^{2}[\/latex]\r\n\r\n[reveal-answer q=\"322242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"322242\"]\r\n\r\nRearrange the factors.\r\n<p style=\"text-align: center\">[latex]-9\\cdot3\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\r\nMultiply constants. Remember that a positive number times a negative number yields a negative number.\r\n<p style=\"text-align: center\">[latex]-27\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\r\nMultiply variable terms. Remember to add the exponents when multiplying exponents with the same base.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}-27\\cdot x^{3+2}\\\\-27\\cdot x^{5}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]-9x^{3}\\cdot 3x^{2}=-27x^{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThat\u2019s it! When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents, like this:\r\n<p style=\"text-align: center\">[latex] \\displaystyle 5{{a}^{4}}\\cdot 7{{a}^{6}}=35{{a}^{10}}[\/latex]<\/p>\r\nThe following video provides more examples of multiplying monomials with different exponents.\r\n\r\nhttps:\/\/youtu.be\/30x8hY32B0o\r\n<h2 id=\"title2\">Find the product of polynomials and monomials<\/h2>\r\nPreviously, 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!\u00a0\u00a0The distributive property can be used to multiply a monomial and a binomial. Just remember that the monomial must be multiplied by each term in the binomial.\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<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<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\nIn the next example, you will see how to multiply a second degree monomial with a binomial. \u00a0Note the use of exponent rules.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]5x^2\\left(4x^{2}+3x\\right)[\/latex]\r\n[reveal-answer q=\"176215\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"176215\"]Distribute the monomial to each term of the binomial. Multiply coefficients and variables separately.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}5x^2\\left(4x^{2}\\right)+5x^2\\left(3x\\right)\\\\\\text{ }\\\\=20x^{2+2}+15x^{2+1}\\\\\\text{ }\\\\=20x^{4}+15x^{3}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]5x^2\\left(4x^{2}+3x\\right)=20x^{4}+15x^{3}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNow let's add another layer by multiplying a monomial by a trinomial. Multiplying a monomial by a trinomial works in much the same way as multiplying a monomial by a binomial.\u00a0 Consider the expression [latex]2x\\left(2x^{2}+5x+10\\right)[\/latex].\r\n\r\nThis expression can be modeled with a sketch like the one below.\r\n\r\n<img class=\"aligncenter wp-image-2204 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/24212311\/Screen-Shot-2016-03-24-at-2.22.48-PM.png\" alt=\"2x times 2x squared equals 4x cubed. 2x times 5x equals 10x squared. 2x times 10 equals 20x.\" width=\"508\" height=\"79\" \/>\r\n<p style=\"text-align: left\">The only difference between this example and the previous one is there is one more term to distribute the monomial to.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}2x\\left(2x^{2}+5x+10\\right)=2x\\left(2x^{2}\\right)+2x\\left(5x\\right)=2x\\left(10\\right)\\\\=4x^{3}+10x^{2}+20x\\end{array}[\/latex]<\/p>\r\nYou will always need to pay attention to negative signs when you are multiplying. Watch\u00a0what happens to the sign on the terms in the trinomial when it is multiplied by a negative monomial in the next example.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]-7x\\left(2x^{2}-5x+1\\right)[\/latex]\r\n[reveal-answer q=\"590272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"590272\"]\r\n\r\nDistribute the monomial to each term in the trinomial.\r\n<p style=\"text-align: center\">[latex]-7x\\left(2x^{2}\\right)-7x\\left(-5x\\right)-7x\\left(1\\right)[\/latex]<\/p>\r\n&nbsp;\r\n\r\nMultiply.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}-14x^{1+2}+35x^{1+1}-7x\\\\\\text{ }\\\\-14x^{3}+35x^{2}-7x\\end{array}[\/latex]<\/p>\r\nRewrite addition of terms with negative coefficients as subtraction.\r\n<h4>Answer<\/h4>\r\n[latex]-7x\\left(2x^{2}-5x+1\\right)=-14x^{3}+35x^{2}-7x[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\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<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\nNow we will have the monomial as the second factor.\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","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply a polynomial by a monomial<\/li>\n<\/ul>\n<\/div>\n<p>In the exponents section, we\u00a0practiced\u00a0multiplying\u00a0monomials together, like we did with this expression: [latex]24{x}^{8}2{x}^{5}[\/latex]. The only thing different between that section and this one is that we called it simplifying, and now we are calling it polynomial multiplication. \u00a0Remember that simplifying a mathematical expression means performing as many operations as we can until there are no more to perform, including multiplication. \u00a0In this section we will show examples of how to multiply more than just monomials. \u00a0We will multiply monomials with\u00a0binomials and trinomials. We will also learn some techniques for multiplying two binomials together.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Multiply. [latex]-9x^{3}\\cdot 3x^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q322242\">Show Solution<\/span><\/p>\n<div id=\"q322242\" class=\"hidden-answer\" style=\"display: none\">\n<p>Rearrange the factors.<\/p>\n<p style=\"text-align: center\">[latex]-9\\cdot3\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\n<p>Multiply constants. Remember that a positive number times a negative number yields a negative number.<\/p>\n<p style=\"text-align: center\">[latex]-27\\cdot x^{3}\\cdot x^{2}[\/latex]<\/p>\n<p>Multiply variable terms. Remember to add the exponents when multiplying exponents with the same base.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}-27\\cdot x^{3+2}\\\\-27\\cdot x^{5}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]-9x^{3}\\cdot 3x^{2}=-27x^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>That\u2019s it! When multiplying monomials, multiply the coefficients together, and then multiply the variables together. Remember, if two variables have the same base, follow the rules of exponents, like this:<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle 5{{a}^{4}}\\cdot 7{{a}^{6}}=35{{a}^{10}}[\/latex]<\/p>\n<p>The following video provides more examples of multiplying monomials with different exponents.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Multiplying Monomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/30x8hY32B0o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">Find the product of polynomials and monomials<\/h2>\n<p>Previously, 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!\u00a0\u00a0The distributive property can be used to multiply a monomial and a binomial. Just remember that the monomial must be multiplied by each term in the binomial.<\/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<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<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>In the next example, you will see how to multiply a second degree monomial with a binomial. \u00a0Note the use of exponent rules.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]5x^2\\left(4x^{2}+3x\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q176215\">Show Solution<\/span><\/p>\n<div id=\"q176215\" class=\"hidden-answer\" style=\"display: none\">Distribute the monomial to each term of the binomial. Multiply coefficients and variables separately.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}5x^2\\left(4x^{2}\\right)+5x^2\\left(3x\\right)\\\\\\text{ }\\\\=20x^{2+2}+15x^{2+1}\\\\\\text{ }\\\\=20x^{4}+15x^{3}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]5x^2\\left(4x^{2}+3x\\right)=20x^{4}+15x^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Now let&#8217;s add another layer by multiplying a monomial by a trinomial. Multiplying a monomial by a trinomial works in much the same way as multiplying a monomial by a binomial.\u00a0 Consider the expression [latex]2x\\left(2x^{2}+5x+10\\right)[\/latex].<\/p>\n<p>This expression can be modeled with a sketch like the one below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2204 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/24212311\/Screen-Shot-2016-03-24-at-2.22.48-PM.png\" alt=\"2x times 2x squared equals 4x cubed. 2x times 5x equals 10x squared. 2x times 10 equals 20x.\" width=\"508\" height=\"79\" \/><\/p>\n<p style=\"text-align: left\">The only difference between this example and the previous one is there is one more term to distribute the monomial to.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}2x\\left(2x^{2}+5x+10\\right)=2x\\left(2x^{2}\\right)+2x\\left(5x\\right)=2x\\left(10\\right)\\\\=4x^{3}+10x^{2}+20x\\end{array}[\/latex]<\/p>\n<p>You will always need to pay attention to negative signs when you are multiplying. Watch\u00a0what happens to the sign on the terms in the trinomial when it is multiplied by a negative monomial in the next example.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]-7x\\left(2x^{2}-5x+1\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q590272\">Show Solution<\/span><\/p>\n<div id=\"q590272\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute the monomial to each term in the trinomial.<\/p>\n<p style=\"text-align: center\">[latex]-7x\\left(2x^{2}\\right)-7x\\left(-5x\\right)-7x\\left(1\\right)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}-14x^{1+2}+35x^{1+1}-7x\\\\\\text{ }\\\\-14x^{3}+35x^{2}-7x\\end{array}[\/latex]<\/p>\n<p>Rewrite addition of terms with negative coefficients as subtraction.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]-7x\\left(2x^{2}-5x+1\\right)=-14x^{3}+35x^{2}-7x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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<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>Now we will have the monomial as the second factor.<\/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\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-10836\">\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>Question ID 146206, 146204, 146203, 146201, 146198, 146197. <strong>Authored 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>Ex: Multiplying Using the Distributive Property. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <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><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/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":21046,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146206, 146204, 146203, 146201, 146198, 146197\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Multiplying Using the Distributive Property\",\"author\":\"James Sousa (mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/bwTmApTV_8o\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"900491ffb15747d7abbbe31f06329d3c, 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