{"id":4708,"date":"2017-06-07T18:55:45","date_gmt":"2017-06-07T18:55:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-simplify-polynomials\/"},"modified":"2017-08-15T20:29:20","modified_gmt":"2017-08-15T20:29:20","slug":"read-simplify-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/aacc-collegealgebrafoundations\/chapter\/read-simplify-polynomials\/","title":{"raw":"Simplify Polynomials","rendered":"Simplify Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Simplify polynomials by collecting like terms<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_4439\" align=\"aligncenter\" width=\"488\"]<img class=\"wp-image-4439\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185544\/Apple_and_Orange_-_they_do_not_compare-300x206.jpg\" alt=\"Apple sitting next to an Orange\" width=\"488\" height=\"335\" \/> Apple and Orange[\/caption]\r\n<h2>Simplify polynomials by collecting like terms<\/h2>\r\nA polynomial may need to be simplified. One way to simplify a polynomial is to combine the <b>like terms<\/b> if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[\/latex] and [latex]-5x^{2}[\/latex] are like terms: They both have <i>x<\/i> as the variable, and the exponent is 2 for each. However, [latex]3x^{2}[\/latex]\u00a0and [latex]3x[\/latex]\u00a0are not like terms, because their exponents are different.\r\n\r\nHere are some examples of terms that are alike and some that are unlike.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Term<\/td>\r\n<td>Like Terms<\/td>\r\n<td>UNLike Terms<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a[\/latex]<\/td>\r\n<td>[latex]3a, \\,\\,\\,-2a,\\,\\,\\, \\frac{1}{2}a[\/latex]<\/td>\r\n<td>[latex]a^2,\\,\\,\\,\\frac{1}{a},\\,\\,\\, \\sqrt{a}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a^2[\/latex]<\/td>\r\n<td>[latex]-5a^2,\\,\\,\\,\\frac{1}{4}a^2,\\,\\,\\, 0.56a^2[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{a^2},\\,\\,\\,\\sqrt{a^2},\\,\\,\\, a^3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]ab[\/latex]<\/td>\r\n<td>[latex]7ab,\\,\\,\\,0.23ab,\\,\\,\\,\\frac{2}{3}ab,\\,\\,\\,-ab[\/latex]<\/td>\r\n<td>[latex]a^2b,\\,\\,\\,\\frac{1}{ab},\\,\\,\\,\\sqrt{ab} [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]ab^2[\/latex]<\/td>\r\n<td>\u00a0[latex]4ab^2,\\,\\,\\, \\frac{ab^2}{7},\\,\\,\\,0.4ab^2,\\,\\,\\, -a^2b[\/latex]<\/td>\r\n<td>\u00a0[latex]a^2b,\\,\\,\\, ab,\\,\\,\\,\\sqrt{ab^2},\\,\\,\\,\\frac{1}{ab^2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nWhich of these terms are like terms?\r\n<p style=\"text-align: center\">[latex]7x^{3}7x7y-8x^{3}9y-3x^{2}8y^{2}[\/latex]<\/p>\r\n[reveal-answer q=\"413363\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"413363\"]Like terms must have the same variables, so first identify which terms use the same variables.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}x:7x^{3}7x-8x^{3}-3x^{2}\\\\y:7y9y8y^{2}\\end{array}[\/latex]<\/p>\r\nLike terms must also have the same exponents. Identify which terms with the same variables also use the same exponents.\r\n\r\nThe <em>x<\/em> terms [latex]7x^{3}[\/latex]\u00a0and [latex]-8x^{3}[\/latex]\u00a0have the same exponent.\r\n\r\nThe <em>y<\/em> terms [latex]7y[\/latex] and [latex]9y[\/latex] have the same exponent.\r\n<h4>Answer<\/h4>\r\n[latex]7x^{3}[\/latex] and [latex]-8x^{3}[\/latex] are like terms.\r\n\r\n[latex]7y[\/latex] and [latex]9y[\/latex]\u00a0are like terms.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.\r\n<p style=\"text-align: center\">[latex]2\\left(3+6\\right)=2\\left(3\\right)+2\\left(6\\right)[\/latex]<\/p>\r\nBoth expressions equal 18. So you can write the expression in whichever form is the most useful.\r\n\r\nLet\u2019s see how we can use this property to combine like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]3x^{2}-5x^{2}[\/latex].\r\n\r\n[reveal-answer q=\"969840\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"969840\"][latex]3x^{2}[\/latex] and [latex]5x^{2}[\/latex]<sup>\u00a0<\/sup>are like terms.\r\n<p style=\"text-align: center\">[latex]3\\left(x^{2}\\right)-5\\left(x^{2}\\right)[\/latex]<\/p>\r\nWe can rewrite the expression as the product of the difference.\r\n<p style=\"text-align: center\">[latex]\\left(3-5\\right)\\left(x^{2}\\right)[\/latex]<\/p>\r\nCalculate [latex]3\u20135[\/latex].\r\n<p style=\"text-align: center\">[latex]\\left(-2\\right)\\left(x^{2}\\right)[\/latex]<\/p>\r\nWrite the difference of [latex]3 \u2013 5[\/latex] as the new coefficient.\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}-5x^{2}=-2x^{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nYou may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]6a^{4}+4a^{4}[\/latex].\r\n\r\n[reveal-answer q=\"840415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"840415\"]Notice that both terms have a number multiplied by [latex]a^{4}[\/latex]. This makes them like terms.\r\n<p style=\"text-align: center\">[latex]6a^{4}+4a^{4}[\/latex]<\/p>\r\nCombine the coefficients, 6 and 4.\r\n<p style=\"text-align: center\">[latex]\\left(6+4\\right)\\left(a^{4}\\right)[\/latex]<\/p>\r\nCalculate the sum.\r\n<p style=\"text-align: center\">[latex]\\left(10\\right)\\left(a^{4}\\right)[\/latex]<\/p>\r\nWrite the sum as the new coefficient.\r\n<h4>Answer<\/h4>\r\n[latex]6a^{4}+4a^{4}=10a^{4}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen you have a polynomial with more terms, you have to be careful that you combine <i>only<\/i> like terms<i>.<\/i> If two terms are not like terms, you can\u2019t combine them.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]3x^{2}+3x+x+1+5x[\/latex]\r\n\r\n[reveal-answer q=\"731804\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"731804\"]First identify which terms are <i>like terms<\/i>: only [latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex]\u00a0are like terms.\r\n\r\n[latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex] are like terms.\r\n\r\nUse the commutative and associative properties to group the like terms together.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+3x+x+1+5x\\\\3x^{2}+\\left(3x+x+5x\\right)+1\\end{array}[\/latex]<\/p>\r\nAdd the coefficients of the like terms. Remember that the coefficient of <em>x<\/em> is [latex]1\\left(x=1x\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+\\left(3+1+5\\right)x+1\\\\3x^{2}+\\left(9\\right)x+1\\end{array}[\/latex]<\/p>\r\nWrite the sum as the new coefficient.\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}+3x+x+1+5x=3x^{2}+9x+1[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/1epjbVO_qU4\r\n<h2>Summary<\/h2>\r\nPolynomials are algebraic expressions that contain any number of terms combined by using addition or subtraction. A term is a number, a variable, or a product of a number and one or more variables with exponents. Like terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials can be evaluated by substituting a given value of the variable into <i>each<\/i> instance of the variable, then using order of operations to complete the calculations.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Simplify polynomials by collecting like terms<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_4439\" style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4439\" class=\"wp-image-4439\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2011\/2017\/06\/07185544\/Apple_and_Orange_-_they_do_not_compare-300x206.jpg\" alt=\"Apple sitting next to an Orange\" width=\"488\" height=\"335\" \/><\/p>\n<p id=\"caption-attachment-4439\" class=\"wp-caption-text\">Apple and Orange<\/p>\n<\/div>\n<h2>Simplify polynomials by collecting like terms<\/h2>\n<p>A polynomial may need to be simplified. One way to simplify a polynomial is to combine the <b>like terms<\/b> if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[\/latex] and [latex]-5x^{2}[\/latex] are like terms: They both have <i>x<\/i> as the variable, and the exponent is 2 for each. However, [latex]3x^{2}[\/latex]\u00a0and [latex]3x[\/latex]\u00a0are not like terms, because their exponents are different.<\/p>\n<p>Here are some examples of terms that are alike and some that are unlike.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Term<\/td>\n<td>Like Terms<\/td>\n<td>UNLike Terms<\/td>\n<\/tr>\n<tr>\n<td>[latex]a[\/latex]<\/td>\n<td>[latex]3a, \\,\\,\\,-2a,\\,\\,\\, \\frac{1}{2}a[\/latex]<\/td>\n<td>[latex]a^2,\\,\\,\\,\\frac{1}{a},\\,\\,\\, \\sqrt{a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a^2[\/latex]<\/td>\n<td>[latex]-5a^2,\\,\\,\\,\\frac{1}{4}a^2,\\,\\,\\, 0.56a^2[\/latex]<\/td>\n<td>[latex]\\frac{1}{a^2},\\,\\,\\,\\sqrt{a^2},\\,\\,\\, a^3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]ab[\/latex]<\/td>\n<td>[latex]7ab,\\,\\,\\,0.23ab,\\,\\,\\,\\frac{2}{3}ab,\\,\\,\\,-ab[\/latex]<\/td>\n<td>[latex]a^2b,\\,\\,\\,\\frac{1}{ab},\\,\\,\\,\\sqrt{ab}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]ab^2[\/latex]<\/td>\n<td>\u00a0[latex]4ab^2,\\,\\,\\, \\frac{ab^2}{7},\\,\\,\\,0.4ab^2,\\,\\,\\, -a^2b[\/latex]<\/td>\n<td>\u00a0[latex]a^2b,\\,\\,\\, ab,\\,\\,\\,\\sqrt{ab^2},\\,\\,\\,\\frac{1}{ab^2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Which of these terms are like terms?<\/p>\n<p style=\"text-align: center\">[latex]7x^{3}7x7y-8x^{3}9y-3x^{2}8y^{2}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q413363\">Show Solution<\/span><\/p>\n<div id=\"q413363\" class=\"hidden-answer\" style=\"display: none\">Like terms must have the same variables, so first identify which terms use the same variables.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}x:7x^{3}7x-8x^{3}-3x^{2}\\\\y:7y9y8y^{2}\\end{array}[\/latex]<\/p>\n<p>Like terms must also have the same exponents. Identify which terms with the same variables also use the same exponents.<\/p>\n<p>The <em>x<\/em> terms [latex]7x^{3}[\/latex]\u00a0and [latex]-8x^{3}[\/latex]\u00a0have the same exponent.<\/p>\n<p>The <em>y<\/em> terms [latex]7y[\/latex] and [latex]9y[\/latex] have the same exponent.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]7x^{3}[\/latex] and [latex]-8x^{3}[\/latex] are like terms.<\/p>\n<p>[latex]7y[\/latex] and [latex]9y[\/latex]\u00a0are like terms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.<\/p>\n<p style=\"text-align: center\">[latex]2\\left(3+6\\right)=2\\left(3\\right)+2\\left(6\\right)[\/latex]<\/p>\n<p>Both expressions equal 18. So you can write the expression in whichever form is the most useful.<\/p>\n<p>Let\u2019s see how we can use this property to combine like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]3x^{2}-5x^{2}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q969840\">Show Solution<\/span><\/p>\n<div id=\"q969840\" class=\"hidden-answer\" style=\"display: none\">[latex]3x^{2}[\/latex] and [latex]5x^{2}[\/latex]<sup>\u00a0<\/sup>are like terms.<\/p>\n<p style=\"text-align: center\">[latex]3\\left(x^{2}\\right)-5\\left(x^{2}\\right)[\/latex]<\/p>\n<p>We can rewrite the expression as the product of the difference.<\/p>\n<p style=\"text-align: center\">[latex]\\left(3-5\\right)\\left(x^{2}\\right)[\/latex]<\/p>\n<p>Calculate [latex]3\u20135[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\left(-2\\right)\\left(x^{2}\\right)[\/latex]<\/p>\n<p>Write the difference of [latex]3 \u2013 5[\/latex] as the new coefficient.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}-5x^{2}=-2x^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>You may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]6a^{4}+4a^{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q840415\">Show Solution<\/span><\/p>\n<div id=\"q840415\" class=\"hidden-answer\" style=\"display: none\">Notice that both terms have a number multiplied by [latex]a^{4}[\/latex]. This makes them like terms.<\/p>\n<p style=\"text-align: center\">[latex]6a^{4}+4a^{4}[\/latex]<\/p>\n<p>Combine the coefficients, 6 and 4.<\/p>\n<p style=\"text-align: center\">[latex]\\left(6+4\\right)\\left(a^{4}\\right)[\/latex]<\/p>\n<p>Calculate the sum.<\/p>\n<p style=\"text-align: center\">[latex]\\left(10\\right)\\left(a^{4}\\right)[\/latex]<\/p>\n<p>Write the sum as the new coefficient.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]6a^{4}+4a^{4}=10a^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When you have a polynomial with more terms, you have to be careful that you combine <i>only<\/i> like terms<i>.<\/i> If two terms are not like terms, you can\u2019t combine them.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]3x^{2}+3x+x+1+5x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q731804\">Show Solution<\/span><\/p>\n<div id=\"q731804\" class=\"hidden-answer\" style=\"display: none\">First identify which terms are <i>like terms<\/i>: only [latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex]\u00a0are like terms.<\/p>\n<p>[latex]3x[\/latex], [latex]x[\/latex], and [latex]5x[\/latex] are like terms.<\/p>\n<p>Use the commutative and associative properties to group the like terms together.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+3x+x+1+5x\\\\3x^{2}+\\left(3x+x+5x\\right)+1\\end{array}[\/latex]<\/p>\n<p>Add the coefficients of the like terms. Remember that the coefficient of <em>x<\/em> is [latex]1\\left(x=1x\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{l}3x^{2}+\\left(3+1+5\\right)x+1\\\\3x^{2}+\\left(9\\right)x+1\\end{array}[\/latex]<\/p>\n<p>Write the sum as the new coefficient.<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}+3x+x+1+5x=3x^{2}+9x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Identify Like Terms and Combine Like\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/1epjbVO_qU4?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Polynomials are algebraic expressions that contain any number of terms combined by using addition or subtraction. A term is a number, a variable, or a product of a number and one or more variables with exponents. Like terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials can be evaluated by substituting a given value of the variable into <i>each<\/i> instance of the variable, then using order of operations to complete the calculations.<\/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-4708\">\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>Identify Like Terms and Combine Like. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/1epjbVO_qU4\">https:\/\/youtu.be\/1epjbVO_qU4<\/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>Apple and Orange - they do not compare. <strong>Authored by<\/strong>: By Michael Johnson . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/commons.wikimedia.org\/wiki\/File%3AApple_and_Orange_-_they_do_not_compare.jpg\">https:\/\/commons.wikimedia.org\/wiki\/File%3AApple_and_Orange_-_they_do_not_compare.jpg<\/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>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":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Apple and Orange - 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