{"id":340,"date":"2021-06-04T00:05:45","date_gmt":"2021-06-04T00:05:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/multiplying-a-polynomial-by-a-monomial\/"},"modified":"2022-01-01T22:33:01","modified_gmt":"2022-01-01T22:33:01","slug":"multiplying-a-polynomial-by-a-monomial","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/multiplying-a-polynomial-by-a-monomial\/","title":{"raw":"8.4.2: The Distributive Property","rendered":"8.4.2: The Distributive Property"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Multiply polynomials using the distributive property.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Distributive property<\/strong>: [latex]a(b+c)=ab+ac[\/latex] for all terms [latex]a,b,c[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Multiplying Monomials<\/h2>\r\nIn the last section, we\u00a0practiced\u00a0multiplying\u00a0monomials together. In 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<h4>Solution<\/h4>\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<\/div>\r\nWhen 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.\r\n<h2 id=\"title2\">The Distributive Property<\/h2>\r\nPreviously, we used the <em><strong>Distributive Property<\/strong><\/em> to simplify expressions such as [latex]2\\left(x - 3\\right)[\/latex]. We 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, we can say we 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 onto 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<h4>Solution<\/h4>\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><span style=\"font-size: 11.52px;\">Simplify.<\/span><\/td>\r\n<td>[latex]3\\cdot x+3\\cdot 7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]3x+21[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\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<h4>Solution<\/h4>\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>Simplify.<\/td>\r\n<td>[latex]x\\cdot x -8\\cdot x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td><span style=\"font-size: 11.52px;\">[latex]x^2-8x[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\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<h4>Solution<\/h4>\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><span style=\"font-size: 11.52px;\">Simplify.<\/span><\/td>\r\n<td>[latex]10x\\cdot{4x}+10x\\cdot{y}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]40x^2+10xy[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\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\nIn the next example, we 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<h4>Solution<\/h4>\r\nDistribute 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}+3x\\right)\\\\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<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify:\r\n<ol>\r\n \t<li>[latex]-3x^2\\left (5x^2-8x\\right )[\/latex]<\/li>\r\n \t<li>[latex]2x^4\\left (9x^3-7x^2\\right )[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm719\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm719\"]\r\n<ol>\r\n \t<li>[latex]-15x^4+24x^3[\/latex]<\/li>\r\n \t<li>[latex]18x^7-14x^6[\/latex][\/hidden-answer]<\/li>\r\n<\/ol>\r\n<\/div>\r\n&nbsp;\r\n\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{equation}\\begin{aligned}&amp;\\;\\;\\;\\;\\,2x\\left(2x^{2}+5x+10\\right)\\\\&amp;=2x\\left(2x^{2}\\right)+2x\\left(5x\\right)+2x\\left(10\\right)\\\\&amp;=4x^{3}+10x^{2}+20x\\end{aligned}\\end{equation}[\/latex]<\/p>\r\nWe always need to pay attention to negative signs when we 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<h4>Solution<\/h4>\r\n<p style=\"text-align: center;\">[latex]-7x\\left(2x^{2}-5x+1\\right)[\/latex]<\/p>\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<\/div>\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<h4>Solution<\/h4>\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<\/div>\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<h4>Solution<\/h4>\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<\/div>\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\nIn the next example, the monomial is 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<h4>Solution<\/h4>\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><span style=\"font-size: 11.52px;\">Simplify.<\/span><\/td>\r\n<td>[latex]x\\cdot p+3\\cdot p[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]xp+3p[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\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\n&nbsp;\r\n\r\nThe following video shows more examples of how to multiply monomials with other polynomials.\r\n\r\nhttps:\/\/youtu.be\/bwTmApTV_8o\r\n\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Multiply polynomials using the distributive property.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Distributive property<\/strong>: [latex]a(b+c)=ab+ac[\/latex] for all terms [latex]a,b,c[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h2>Multiplying Monomials<\/h2>\n<p>In the last section, we\u00a0practiced\u00a0multiplying\u00a0monomials together. In 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<h4>Solution<\/h4>\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<p>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.<\/p>\n<h2 id=\"title2\">The Distributive Property<\/h2>\n<p>Previously, we used the <em><strong>Distributive Property<\/strong><\/em> to simplify expressions such as [latex]2\\left(x - 3\\right)[\/latex]. We 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, we can say we 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 onto 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<h4>Solution<\/h4>\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><span style=\"font-size: 11.52px;\">Simplify.<\/span><\/td>\n<td>[latex]3\\cdot x+3\\cdot 7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]3x+21[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\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<h4>Solution<\/h4>\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>Simplify.<\/td>\n<td>[latex]x\\cdot x -8\\cdot x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td><span style=\"font-size: 11.52px;\">[latex]x^2-8x[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\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<h4>Solution<\/h4>\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><span style=\"font-size: 11.52px;\">Simplify.<\/span><\/td>\n<td>[latex]10x\\cdot{4x}+10x\\cdot{y}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]40x^2+10xy[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\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>In the next example, we 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<h4>Solution<\/h4>\n<p>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}+3x\\right)\\\\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 class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify:<\/p>\n<ol>\n<li>[latex]-3x^2\\left (5x^2-8x\\right )[\/latex]<\/li>\n<li>[latex]2x^4\\left (9x^3-7x^2\\right )[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm719\">Show Answer<\/span><\/p>\n<div id=\"qhjm719\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]-15x^4+24x^3[\/latex]<\/li>\n<li>[latex]18x^7-14x^6[\/latex]<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<p>&nbsp;<\/p>\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{equation}\\begin{aligned}&\\;\\;\\;\\;\\,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{aligned}\\end{equation}[\/latex]<\/p>\n<p>We always need to pay attention to negative signs when we 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<h4>Solution<\/h4>\n<p style=\"text-align: center;\">[latex]-7x\\left(2x^{2}-5x+1\\right)[\/latex]<\/p>\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 class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Multiply: [latex]-2x\\left(5{x}^{2}+7x - 3\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\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 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<h4>Solution<\/h4>\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 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>In the next example, the monomial is 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<h4>Solution<\/h4>\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><span style=\"font-size: 11.52px;\">Simplify.<\/span><\/td>\n<td>[latex]x\\cdot p+3\\cdot p[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td>[latex]xp+3p[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\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>&nbsp;<\/p>\n<p>The following video shows more examples of how to multiply monomials with other polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Multiplying Using the Distributive Property\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/bwTmApTV_8o?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-340\">\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><li>Try It hjm719. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <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>Revised and adapted: 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":23485,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Revised and adapted: 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 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