{"id":2562,"date":"2016-07-15T15:40:15","date_gmt":"2016-07-15T15:40:15","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=2562"},"modified":"2019-08-06T18:43:15","modified_gmt":"2019-08-06T18:43:15","slug":"read-multiply-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/chapter\/read-multiply-polynomials\/","title":{"raw":"Sums and Products of Polynomials","rendered":"Sums and Products of Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Add and subtract polynomials<\/li>\r\n \t<li>Multiply and divide polynomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Add and Subtract Polynomials<\/h2>\r\nWe can add and subtract polynomials by combining like terms which are terms that contain the same variables raised to the same exponents. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms; therefore, they cannot be added.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the sum.\r\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"222892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"222892\"]<\/p>\r\n<p style=\"text-align: left;\">[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nHere\u00a0is a summary of some helpful steps for adding and subtracting polynomials.\r\n<div class=\"textbox\">\r\n<h3>\u00a0How To: Given multiple polynomials, add or subtract them to simplify the expressions<strong>\r\n<\/strong><\/h3>\r\n<ol>\r\n \t<li>Combine like terms.<\/li>\r\n \t<li>Simplify and write in standard form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div>\r\n\r\nWhen you subtract polynomials, you will still be looking for like terms to combine, but you will need to pay attention to the sign of the terms you are combining. In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the difference.\r\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">[reveal-answer q=\"279648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"279648\"]<\/p>\r\n<p style=\"text-align: left;\">[latex]\\begin{array}{ccc}7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill &amp; \\text{Distribute}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\nNote that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.\r\n<p style=\"text-align: left;\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\nIn the following video, we show more examples of adding and subtracting polynomials.\r\n\r\nhttps:\/\/youtu.be\/jiq3toC7wGM\r\n<h2>Multiplying Polynomials<\/h2>\r\nMultiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms.\r\n\r\nYou may have used the distributive property to help you solve linear equations such as\u00a0[latex]2\\left(x+7\\right)=21[\/latex]. We can distribute the [latex]2[\/latex] in [latex]2\\left(x+7\\right)[\/latex] to obtain the equivalent expression [latex]2x+14[\/latex]. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.\r\n\r\nThe following video will provide you with examples of using the distributive property to find the product of\u00a0monomials and polynomials.\r\n\r\nhttps:\/\/youtu.be\/bwTmApTV_8o\r\n\r\nBelow is a summary of the steps we used to find the product of two polynomials using the distributive property.\r\n<div class=\"textbox\">\r\n<h3>How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression<\/h3>\r\n<ol>\r\n \t<li>Multiply each term of the first polynomial by each term of the second.<\/li>\r\n \t<li>Combine like terms.<\/li>\r\n \t<li>Simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<h2>Using FOIL to Multiply Binomials<\/h2>\r\nWe can also use a shortcut called the FOIL method when multiplying two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200224\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" \/>\r\n\r\nThe FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial and then combining like terms.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse FOIL to find the product. [latex](2x-18)(3x+3)[\/latex]\r\n[reveal-answer q=\"787670\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"787670\"]\r\n\r\nFind the product of the first terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200225\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" \/>\r\n\r\nFind the product of the outer terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200227\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" \/>\r\n\r\nFind the product of the inner terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200228\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" \/>\r\n\r\nFind the product of the last terms.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200229\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" \/>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill &amp; \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we show an example of how to use the FOIL method to multiply two binomials.\r\n\r\nhttps:\/\/youtu.be\/_MrdEFnXNGA\r\n\r\nThe following steps summarize the process for using FOIL to multiply two binomials. It is very important to note that this process only works for the product of two binomials. If you are multiplying a binomial\u00a0and a trinomial, it is better to use a table to keep track of your terms.\r\n<div class=\"textbox\">\r\n<h3>How To: Given two binomials, use FOIL to simplify the expression<\/h3>\r\n<ol>\r\n \t<li>Multiply the first terms of each binomial.<\/li>\r\n \t<li>Multiply the outer terms of the binomials.<\/li>\r\n \t<li>Multiply the inner terms of the binomials.<\/li>\r\n \t<li>Multiply the last terms of each binomial.<\/li>\r\n \t<li>Add the products.<\/li>\r\n \t<li>Combine like terms and simplify.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Add and subtract polynomials<\/li>\n<li>Multiply and divide polynomials<\/li>\n<\/ul>\n<\/div>\n<h2>Add and Subtract Polynomials<\/h2>\n<p>We can add and subtract polynomials by combining like terms which are terms that contain the same variables raised to the same exponents. For example, [latex]5{x}^{2}[\/latex] and [latex]-2{x}^{2}[\/latex] are like terms and can be added to get [latex]3{x}^{2}[\/latex], but [latex]3x[\/latex] and [latex]3{x}^{2}[\/latex] are not like terms; therefore, they cannot be added.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the sum.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(12{x}^{2}+9x - 21\\right)+\\left(4{x}^{3}+8{x}^{2}-5x+20\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q222892\">Show Solution<\/span><\/p>\n<div id=\"q222892\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\begin{array}{cc}4{x}^{3}+\\left(12{x}^{2}+8{x}^{2}\\right)+\\left(9x - 5x\\right)+\\left(-21+20\\right) \\hfill & \\text{Combine like terms}.\\hfill \\\\ 4{x}^{3}+20{x}^{2}+4x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<p>Here\u00a0is a summary of some helpful steps for adding and subtracting polynomials.<\/p>\n<div class=\"textbox\">\n<h3>\u00a0How To: Given multiple polynomials, add or subtract them to simplify the expressions<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Combine like terms.<\/li>\n<li>Simplify and write in standard form.<\/li>\n<\/ol>\n<\/div>\n<div>\n<p>When you subtract polynomials, you will still be looking for like terms to combine, but you will need to pay attention to the sign of the terms you are combining. In the following example we will show how to distribute the negative sign to each term of a polynomial that is being subtracted from another.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the difference.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q279648\">Show Solution<\/span><\/p>\n<div id=\"q279648\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: left;\">[latex]\\begin{array}{ccc}7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill & \\text{Distribute}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+\\left(-{x}^{2}+2{x}^{2}\\right)+\\left(6x - 3x\\right)+\\left(1 - 2\\right)\\text{ }\\hfill & \\text{Combine like terms}.\\hfill \\\\ 7{x}^{4}-5{x}^{3}+{x}^{2}+3x - 1\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<p>Note that finding the difference between two polynomials is the same as adding the opposite of the second polynomial to the first.<\/p>\n<p style=\"text-align: left;\"><\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show more examples of adding and subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Adding and Subtracting Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/jiq3toC7wGM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Multiplying Polynomials<\/h2>\n<p>Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms.<\/p>\n<p>You may have used the distributive property to help you solve linear equations such as\u00a0[latex]2\\left(x+7\\right)=21[\/latex]. We can distribute the [latex]2[\/latex] in [latex]2\\left(x+7\\right)[\/latex] to obtain the equivalent expression [latex]2x+14[\/latex]. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.<\/p>\n<p>The following video will provide you with examples of using the distributive property to find the product of\u00a0monomials and polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-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<p>Below is a summary of the steps we used to find the product of two polynomials using the distributive property.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the multiplication of two polynomials, use the distributive property to simplify the expression<\/h3>\n<ol>\n<li>Multiply each term of the first polynomial by each term of the second.<\/li>\n<li>Combine like terms.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<h2>Using FOIL to Multiply Binomials<\/h2>\n<p>We can also use a shortcut called the FOIL method when multiplying two binomials. It is called FOIL because we multiply the <strong>f<\/strong>irst terms, the <strong>o<\/strong>uter terms, the <strong>i<\/strong>nner terms, and then the <strong>l<\/strong>ast terms of each binomial.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200224\/CNX_CAT_Figure_01_04_003.jpg\" alt=\"Two quantities in parentheses are being multiplied, the first being: a times x plus b and the second being: c times x plus d. This expression equals ac times x squared plus ad times x plus bc times x plus bd. The terms ax and cx are labeled: First Terms. The terms ax and d are labeled: Outer Terms. The terms b and cx are labeled: Inner Terms. The terms b and d are labeled: Last Terms.\" \/><\/p>\n<p>The FOIL method arises out of the distributive property. We are simply multiplying each term of the first binomial by each term of the second binomial and then combining like terms.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use FOIL to find the product. [latex](2x-18)(3x+3)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q787670\">Show Solution<\/span><\/p>\n<div id=\"q787670\" class=\"hidden-answer\" style=\"display: none\">\n<p>Find the product of the first terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200225\/CNX_CAT_Figure_01_04_004.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the outer terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200227\/CNX_CAT_Figure_01_04_005.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the inner terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200228\/CNX_CAT_Figure_01_04_006.jpg\" alt=\"\" \/><\/p>\n<p>Find the product of the last terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200229\/CNX_CAT_Figure_01_04_007.jpg\" alt=\"\" \/><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}6{x}^{2}+6x - 54x - 54\\hfill & \\text{Add the products}.\\hfill \\\\ 6{x}^{2}+\\left(6x - 54x\\right)-54\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{2}-48x - 54\\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we show an example of how to use the FOIL method to multiply two binomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Multiply Binomials Using the FOIL Acronym\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/_MrdEFnXNGA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The following steps summarize the process for using FOIL to multiply two binomials. It is very important to note that this process only works for the product of two binomials. If you are multiplying a binomial\u00a0and a trinomial, it is better to use a table to keep track of your terms.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two binomials, use FOIL to simplify the expression<\/h3>\n<ol>\n<li>Multiply the first terms of each binomial.<\/li>\n<li>Multiply the outer terms of the binomials.<\/li>\n<li>Multiply the inner terms of the binomials.<\/li>\n<li>Multiply the last terms of each binomial.<\/li>\n<li>Add the products.<\/li>\n<li>Combine like terms and simplify.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n\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-2562\">\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 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><li>Multiply Binomials Using the FOIL Acronym. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/_MrdEFnXNGA\">https:\/\/youtu.be\/_MrdEFnXNGA<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay, et al.. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at: http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/li><li>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Adding and Subtracting Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/jiq3toC7wGM\">https:\/\/youtu.be\/jiq3toC7wGM<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Polynomial Multiplication Involving Binomials and Trinomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/bBKbldmlbqI\">https:\/\/youtu.be\/bBKbldmlbqI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay, et al.\",\"organization\":\"\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: 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