{"id":4710,"date":"2017-06-07T18:55:45","date_gmt":"2017-06-07T18:55:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-add-polynomials\/"},"modified":"2017-08-15T20:30:14","modified_gmt":"2017-08-15T20:30:14","slug":"read-add-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-add-polynomials\/","title":{"raw":"Add Polynomials","rendered":"Add Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Add polynomials<\/li>\r\n \t<li>Use horizontal and vertical organization to add polynomials<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Add polynomials<\/h2>\r\nAdding and subtracting <b>polynomials<\/b> may sound complicated, but it\u2019s really not much different from the addition and subtraction that you do every day. The main thing to remember is to look for and combine <b>like terms<\/b>.\r\n\r\nYou can add two (or more) polynomials as you have added algebraic expressions. You can remove the parentheses and combine like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(3b+5\\right)+\\left(2b+4\\right)[\/latex]\r\n\r\n[reveal-answer q=\"379821\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"379821\"]Regroup\r\n<p style=\"text-align: center\">[latex]\\left(3b+2b\\right)+\\left(5+4\\right)[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center\">[latex]5b + 9[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3b+5\\right)+\\left(2b+4\\right)=5b+9[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you are adding polynomials that have subtraction,\u00a0it is important to remember to keep the sign on each term as you are collecting like terms. \u00a0 The next example will show you how to regroup terms that are subtracted when you are collecting like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(-5x^{2}\u201310x+2\\right)+\\left(3x^{2}+7x\u20134\\right)[\/latex]\r\n\r\n[reveal-answer q=\"486380\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"486380\"]\r\n\r\nCollect like terms, making sure you keep the sign on each term. For example, when you collect\u00a0the [latex]x^2[\/latex] terms, make sure to keep the negative sign on [latex]-5x^2[\/latex].\r\n\r\nHelpful Hint: We find that it is easier to put the terms with a negative sign on the right of the terms that are positive. This would mean\u00a0that the\u00a0[latex]x^2[\/latex] terms would be grouped as\u00a0[latex]\\left(3x^{2}-5x^{2}\\right)[\/latex]. If both terms are negative, then it doesn't matter which is on the left or right.\r\n\r\nThe polynomial now looks like this, with like terms collected:\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\underbrace{\\left(3x^{2}-5x^{2}\\right)}+\\underbrace{\\left(7x-10x\\right)}+\\underbrace{\\left(2-4\\right)}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x^2\\text{ terms }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\text{ terms}\\,\\,\\,\\,\\,\\,\\,\\,\\text{ constants }\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left\">The [latex]x^2[\/latex] terms will simplify to [latex]-2x^{2}[\/latex]<\/p>\r\n<p style=\"text-align: left\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex]<\/p>\r\n<p style=\"text-align: left\">The constant terms will simplify to [latex]-2[\/latex]<\/p>\r\n<p style=\"text-align: left\">\u00a0Rewrite the polynomial with it's simplified terms, keeping the sign on each term.<\/p>\r\n<p style=\"text-align: center\">[latex]-2x^{2}-3x-2[\/latex]<\/p>\r\n<p style=\"text-align: left\">As a matter of convention, we write polynomials in descending order based on degree. \u00a0Notice how we put the\u00a0[latex]x^2[\/latex] term first, the\u00a0[latex]x[\/latex] term second and the constant term last.<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(-5x^{2}-10x+2\\right)+\\left(3x^{2}+7x-4\\right)=-2x^{2}-3x-2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this \u201cvertical\u201d method of adding polynomials:\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(3x^{2}+2x-7\\right)+\\left(7x^{2}-4x+8\\right)[\/latex]\r\n\r\n[reveal-answer q=\"425224\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"425224\"]Write one polynomial below the other, making sure to line up like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\+7x^{2}-4x+8\\end{array}[\/latex]<\/p>\r\nCombine like terms, paying close attention to the signs.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\\\underline{+7x^{2}-4x+8}\\\\10x^{2}-2x+1\\end{array}[\/latex]<b>\u00a0<\/b><\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(3x^{2}+3x-7\\right)+\\left(7x^{2}-4x+8\\right)=10x^{2}-2x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn't so tidy. When there isn't a matching like term for every term, there will be empty places in the vertical arrangement.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nAdd. [latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)[\/latex]\r\n\r\n[reveal-answer q=\"232680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"232680\"]Write one polynomial below the other, lining up like terms vertically.\r\n\r\nTo keep track of like terms, you can insert zeros where there aren't any shared like terms. This is optional, but some find it helpful.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+2\\\\+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10\\end{array}[\/latex]<\/p>\r\nCombine like terms, paying close attention to the signs.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+\\,\\,\\,2\\\\\\underline{+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10}\\\\4x^{3}\\,+\\,\\,x^{2}-6x+12\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)=4x^{3}+x^{2}-6x+12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may be thinking, how is this different than combining like terms, which we did in the last section? The answer is, it's not really. We just added a layer to combining like terms by adding more terms to combine. :) Polynomials are a useful tool for describing the behavior of anything that isn't linear, and sometimes you may need to add them.\r\n\r\nIn the following video, you will see more examples of combining like terms by adding polynomials.\r\n\r\nhttps:\/\/youtu.be\/KYZR7g7QcF4\r\n\r\nIn the next section we will subtract polynomials.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Add polynomials<\/li>\n<li>Use horizontal and vertical organization to add polynomials<\/li>\n<\/ul>\n<\/div>\n<h2>Add polynomials<\/h2>\n<p>Adding and subtracting <b>polynomials<\/b> may sound complicated, but it\u2019s really not much different from the addition and subtraction that you do every day. The main thing to remember is to look for and combine <b>like terms<\/b>.<\/p>\n<p>You can add two (or more) polynomials as you have added algebraic expressions. You can remove the parentheses and combine like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(3b+5\\right)+\\left(2b+4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q379821\">Show Solution<\/span><\/p>\n<div id=\"q379821\" class=\"hidden-answer\" style=\"display: none\">Regroup<\/p>\n<p style=\"text-align: center\">[latex]\\left(3b+2b\\right)+\\left(5+4\\right)[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center\">[latex]5b + 9[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3b+5\\right)+\\left(2b+4\\right)=5b+9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you are adding polynomials that have subtraction,\u00a0it is important to remember to keep the sign on each term as you are collecting like terms. \u00a0 The next example will show you how to regroup terms that are subtracted when you are collecting like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(-5x^{2}\u201310x+2\\right)+\\left(3x^{2}+7x\u20134\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q486380\">Show Solution<\/span><\/p>\n<div id=\"q486380\" class=\"hidden-answer\" style=\"display: none\">\n<p>Collect like terms, making sure you keep the sign on each term. For example, when you collect\u00a0the [latex]x^2[\/latex] terms, make sure to keep the negative sign on [latex]-5x^2[\/latex].<\/p>\n<p>Helpful Hint: We find that it is easier to put the terms with a negative sign on the right of the terms that are positive. This would mean\u00a0that the\u00a0[latex]x^2[\/latex] terms would be grouped as\u00a0[latex]\\left(3x^{2}-5x^{2}\\right)[\/latex]. If both terms are negative, then it doesn&#8217;t matter which is on the left or right.<\/p>\n<p>The polynomial now looks like this, with like terms collected:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}\\underbrace{\\left(3x^{2}-5x^{2}\\right)}+\\underbrace{\\left(7x-10x\\right)}+\\underbrace{\\left(2-4\\right)}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x^2\\text{ terms }\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\text{ terms}\\,\\,\\,\\,\\,\\,\\,\\,\\text{ constants }\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left\">The [latex]x^2[\/latex] terms will simplify to [latex]-2x^{2}[\/latex]<\/p>\n<p style=\"text-align: left\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex]<\/p>\n<p style=\"text-align: left\">The constant terms will simplify to [latex]-2[\/latex]<\/p>\n<p style=\"text-align: left\">\u00a0Rewrite the polynomial with it&#8217;s simplified terms, keeping the sign on each term.<\/p>\n<p style=\"text-align: center\">[latex]-2x^{2}-3x-2[\/latex]<\/p>\n<p style=\"text-align: left\">As a matter of convention, we write polynomials in descending order based on degree. \u00a0Notice how we put the\u00a0[latex]x^2[\/latex] term first, the\u00a0[latex]x[\/latex] term second and the constant term last.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(-5x^{2}-10x+2\\right)+\\left(3x^{2}+7x-4\\right)=-2x^{2}-3x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this \u201cvertical\u201d method of adding polynomials:<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(3x^{2}+2x-7\\right)+\\left(7x^{2}-4x+8\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q425224\">Show Solution<\/span><\/p>\n<div id=\"q425224\" class=\"hidden-answer\" style=\"display: none\">Write one polynomial below the other, making sure to line up like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\+7x^{2}-4x+8\\end{array}[\/latex]<\/p>\n<p>Combine like terms, paying close attention to the signs.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}3x^{2}+2x-7\\\\\\underline{+7x^{2}-4x+8}\\\\10x^{2}-2x+1\\end{array}[\/latex]<b>\u00a0<\/b><\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(3x^{2}+3x-7\\right)+\\left(7x^{2}-4x+8\\right)=10x^{2}-2x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Sometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn&#8217;t so tidy. When there isn&#8217;t a matching like term for every term, there will be empty places in the vertical arrangement.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Add. [latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q232680\">Show Solution<\/span><\/p>\n<div id=\"q232680\" class=\"hidden-answer\" style=\"display: none\">Write one polynomial below the other, lining up like terms vertically.<\/p>\n<p>To keep track of like terms, you can insert zeros where there aren&#8217;t any shared like terms. This is optional, but some find it helpful.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+2\\\\+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10\\end{array}[\/latex]<\/p>\n<p>Combine like terms, paying close attention to the signs.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{r}4x^{3}+5x^{2}-6x+\\,\\,\\,2\\\\\\underline{+0\\,\\,-4x^{2}\\,\\,+0\\,\\,+10}\\\\4x^{3}\\,+\\,\\,x^{2}-6x+12\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(4x^{3}+5x^{2}-6x+2\\right)+\\left(-4x^{2}+10\\right)=4x^{3}+x^{2}-6x+12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may be thinking, how is this different than combining like terms, which we did in the last section? The answer is, it&#8217;s not really. We just added a layer to combining like terms by adding more terms to combine. :) Polynomials are a useful tool for describing the behavior of anything that isn&#8217;t linear, and sometimes you may need to add them.<\/p>\n<p>In the following video, you will see more examples of combining like terms by adding polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Adding Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/KYZR7g7QcF4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the next section we will subtract polynomials.<\/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-4710\">\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: Adding Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/KYZR7g7QcF4\">https:\/\/youtu.be\/KYZR7g7QcF4<\/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>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 and Education. <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><\/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":17533,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Ex: Adding Polynomials\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KYZR7g7QcF4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and 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