{"id":4717,"date":"2017-06-07T18:55:55","date_gmt":"2017-06-07T18:55:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiply-polynomials\/"},"modified":"2017-08-15T20:33:25","modified_gmt":"2017-08-15T20:33:25","slug":"read-multiply-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-multiply-polynomials\/","title":{"raw":"Multiply Polynomials","rendered":"Multiply Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Find the product of monomials<\/li>\r\n \t<li>Find the product of polynomials and monomials<\/li>\r\n \t<li>Find the product of two binomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Find the product of monomials<\/h2>\r\nMultiplying <strong>polynomials<\/strong> involves applying the rules of exponents and the distributive property to simplify the product. Polynomial\u00a0multiplication can be useful in modeling real world situations. Understanding polynomial products is an important step in learning to solve algebraic equations involving polynomials. There are many, varied uses for polynomials including the generation of 3D graphics for entertainment and industry, as in the image below.\r\n\r\n[caption id=\"attachment_4567\" align=\"aligncenter\" width=\"431\"]<img class=\"wp-image-4567\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185551\/Screen-Shot-2016-06-03-at-12.33.29-PM-300x154.png\" alt=\"Bumpy, irregular surfaces on a grid made with AutoCAD\" width=\"431\" height=\"221\" \/> Surfaces made from polynomials with AutoCAD[\/caption]\r\n\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>Find the product of polynomials and monomials<\/h2>\r\nThe 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. In 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. 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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185552\/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\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^{2}\\left(2x^{2}-5x+1\\right)=14x^{4}-35^{3}+7x^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p class=\"no-indent\" style=\"text-align: left\">The following video provides more examples of multiplying a monomial and a polynomial.<\/p>\r\nhttps:\/\/youtu.be\/bwTmApTV_8o\r\n<h2 class=\"no-indent\" style=\"text-align: left\">Find the product of two binomials<\/h2>\r\nNow let's explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You'll use each binomial as one of the dimensions of a rectangle, and their product as the area.\r\n\r\nThe model below shows [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]:\r\n\r\n[caption id=\"attachment_4607\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-4607\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185555\/Screen-Shot-2016-06-04-at-12.15.06-PM-300x290.png\" alt=\"Visual representation of multiplying two binomials.\" width=\"300\" height=\"290\" \/> Visual representation of multiplying two binomials.[\/caption]\r\n\r\nEach binomial is expanded into variable terms and constants, [latex]x+4[\/latex], along the top of the model and [latex]x+2[\/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[\/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[\/latex].\r\n\r\nYou can use the distributive property to determine the product of two binomials.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify. [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]\r\n[reveal-answer q=\"186797\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"186797\"]Distribute the [latex]x[\/latex] over [latex]x+2[\/latex], then distribute 4 over [latex]x+2[\/latex].\r\n<p style=\"text-align: center\">[latex]x\\left(x\\right)+x\\left(2\\right)+4\\left(x\\right)+4\\left(2\\right)[\/latex]<\/p>\r\nMultiply.\r\n<p style=\"text-align: center\">[latex]x^{2}+2x+4x+8[\/latex]<\/p>\r\nCombine like terms [latex]\\left(2x+4x\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]x^{2}+6x+8[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(x+4\\right)\\left(2x+2\\right)=x^{2}+6x+8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nLook back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[\/latex] comes from. Can you see where you multiply [latex]x[\/latex] by [latex]x + 2[\/latex], and where you get [latex]x^{2}[\/latex]\u00a0from [latex]x\\left(x\\right)[\/latex]?\r\n\r\nAnother way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[\/latex] in [latex]x+4[\/latex] gets multiplied by both the [latex]x[\/latex] and the 2 from [latex]x+2[\/latex], and the 4 gets multiplied by both the [latex]x[\/latex] and the 2.\r\n\r\nThe following video provides an example of multiplying two binomials using an area model as well as repeated distribution.\r\n\r\nhttps:\/\/youtu.be\/u4Hgl0BrUlo\r\n\r\nIn the next section we will explore other methods for multiplying two binomials, and\u00a0become aware of the different forms that binomials can have.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Find the product of monomials<\/li>\n<li>Find the product of polynomials and monomials<\/li>\n<li>Find the product of two binomials<\/li>\n<\/ul>\n<\/div>\n<h2>Find the product of monomials<\/h2>\n<p>Multiplying <strong>polynomials<\/strong> involves applying the rules of exponents and the distributive property to simplify the product. Polynomial\u00a0multiplication can be useful in modeling real world situations. Understanding polynomial products is an important step in learning to solve algebraic equations involving polynomials. There are many, varied uses for polynomials including the generation of 3D graphics for entertainment and industry, as in the image below.<\/p>\n<div id=\"attachment_4567\" style=\"width: 441px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4567\" class=\"wp-image-4567\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185551\/Screen-Shot-2016-06-03-at-12.33.29-PM-300x154.png\" alt=\"Bumpy, irregular surfaces on a grid made with AutoCAD\" width=\"431\" height=\"221\" \/><\/p>\n<p id=\"caption-attachment-4567\" class=\"wp-caption-text\">Surfaces made from polynomials with AutoCAD<\/p>\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>Find the product of polynomials and monomials<\/h2>\n<p>The 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. 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. 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\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185552\/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>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^{2}\\left(2x^{2}-5x+1\\right)=14x^{4}-35^{3}+7x^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p class=\"no-indent\" style=\"text-align: left\">The following video provides more examples of multiplying a monomial and a polynomial.<\/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 class=\"no-indent\" style=\"text-align: left\">Find the product of two binomials<\/h2>\n<p>Now let&#8217;s explore multiplying two binomials. For those of you that use pictures to learn, you can draw an area model to help make sense of the process. You&#8217;ll use each binomial as one of the dimensions of a rectangle, and their product as the area.<\/p>\n<p>The model below shows [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]:<\/p>\n<div id=\"attachment_4607\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4607\" class=\"size-medium wp-image-4607\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185555\/Screen-Shot-2016-06-04-at-12.15.06-PM-300x290.png\" alt=\"Visual representation of multiplying two binomials.\" width=\"300\" height=\"290\" \/><\/p>\n<p id=\"caption-attachment-4607\" class=\"wp-caption-text\">Visual representation of multiplying two binomials.<\/p>\n<\/div>\n<p>Each binomial is expanded into variable terms and constants, [latex]x+4[\/latex], along the top of the model and [latex]x+2[\/latex] along the left side. The product of each pair of terms is a colored rectangle. The total area is the sum of all of these small rectangles, [latex]x^{2}+2x+4x+8[\/latex], If you combine all the like terms, you can write the product, or area, as [latex]x^{2}+6x+8[\/latex].<\/p>\n<p>You can use the distributive property to determine the product of two binomials.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify. [latex]\\left(x+4\\right)\\left(x+2\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186797\">Show Solution<\/span><\/p>\n<div id=\"q186797\" class=\"hidden-answer\" style=\"display: none\">Distribute the [latex]x[\/latex] over [latex]x+2[\/latex], then distribute 4 over [latex]x+2[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]x\\left(x\\right)+x\\left(2\\right)+4\\left(x\\right)+4\\left(2\\right)[\/latex]<\/p>\n<p>Multiply.<\/p>\n<p style=\"text-align: center\">[latex]x^{2}+2x+4x+8[\/latex]<\/p>\n<p>Combine like terms [latex]\\left(2x+4x\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]x^{2}+6x+8[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(x+4\\right)\\left(2x+2\\right)=x^{2}+6x+8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Look back at the model above to see where each piece of [latex]x^{2}+2x+4x+8[\/latex] comes from. Can you see where you multiply [latex]x[\/latex] by [latex]x + 2[\/latex], and where you get [latex]x^{2}[\/latex]\u00a0from [latex]x\\left(x\\right)[\/latex]?<\/p>\n<p>Another way to look at multiplying binomials is to see that each term in one binomial is multiplied by each term in the other binomial. Look at the example above: the [latex]x[\/latex] in [latex]x+4[\/latex] gets multiplied by both the [latex]x[\/latex] and the 2 from [latex]x+2[\/latex], and the 4 gets multiplied by both the [latex]x[\/latex] and the 2.<\/p>\n<p>The following video provides an example of multiplying two binomials using an area model as well as repeated distribution.<\/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<p>In the next section we will explore other methods for multiplying two binomials, and\u00a0become aware of the different forms that binomials can have.<\/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-4717\">\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>Ex: Multiplying Monomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/30x8hY32B0o\">https:\/\/youtu.be\/30x8hY32B0o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Multiplying Using the Distributive Property. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bwTmApTV_8o\">https:\/\/youtu.be\/bwTmApTV_8o<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Multiply Binomials Using An Area Model and Using Repeated Distribution. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/u4Hgl0BrUlo\">https:\/\/youtu.be\/u4Hgl0BrUlo<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. 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