{"id":756,"date":"2021-09-15T19:28:43","date_gmt":"2021-09-15T19:28:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=756"},"modified":"2022-01-01T21:39:02","modified_gmt":"2022-01-01T21:39:02","slug":"8-3-adding-and-subtracting-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/8-3-adding-and-subtracting-polynomials\/","title":{"raw":"8.3: Addition and Subtraction of Polynomials","rendered":"8.3: Addition and Subtraction of Polynomials"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Combine like terms through addition and subtraction<\/li>\r\n \t<li>Add and subtract monomials<\/li>\r\n \t<li>Add and subtract polynomials<\/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>Like terms<\/strong>: Terms that have identical variables with identical exponents<\/li>\r\n \t<li><strong>Sum<\/strong>: the answer when two or more terms are added<\/li>\r\n \t<li><strong>Difference<\/strong>: the answer when two or more terms are subtracted<\/li>\r\n \t<li><strong>Commutative property of addition<\/strong>: reordering terms in an expression results in an equivalent expression [latex]a+b=b+a[\/latex]<\/li>\r\n \t<li><strong>Associative property of addition<\/strong>:\u00a0regrouping terms in an expression results in an equivalent expression [latex](a+b)+c=a+(b+c)[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Combining Like Terms<\/h3>\r\nA polynomial may need to be simplified. One way to simplify a polynomial is to combine the <em><b>like terms,<\/b><\/em> if there are any. Two or more terms in a polynomial are like terms if they have identical variables with the identical exponents. For example, [latex]3x^{2}[\/latex] and [latex]-5x^{2}[\/latex] are like terms: They both have [latex]x[\/latex] as the variable, and the exponent is [latex]2[\/latex] 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 like terms and unlike terms.\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}, 7x, 7y, -8x^{3}, 9y, -3x^{2}, 8y^{2}[\/latex]<\/p>\r\n&nbsp;\r\n<h4>Solution<\/h4>\r\nLike 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:7y, 9y, 8y^{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 [latex]<em>x[\/latex]-<\/em>terms [latex]7x^{3}[\/latex]\u00a0and [latex]-8x^{3}[\/latex]\u00a0have the same exponent.\r\n\r\nThe [latex]<em>y[\/latex]-<\/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<\/div>\r\n&nbsp;\r\n\r\nWe can use the <em><strong>distributive property<\/strong><\/em> to simplify the sum of like terms. Recall that the distributive property of addition over multiplication 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 [latex]18[\/latex]. So we 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<h4>Solution<\/h4>\r\n[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\nRewrite 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 sum as the new coefficient:\r\n<p style=\"text-align: center;\">[latex]-2x^2[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}-5x^{2}=-2x^{2}[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nNotice that combining like terms through addition or subtraction involves adding or subtracting the coefficients to find the new coefficient of the like term.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSimplify [latex]6a^{4}+4a^{4}[\/latex].\r\n<h4>Solution<\/h4>\r\nNotice 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, [latex]6[\/latex] and [latex]4[\/latex].\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<p style=\"text-align: center;\">[latex]10a^4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]6a^{4}+4a^{4}=10a^{4}[\/latex]\r\n\r\n<\/div>\r\nWhen we have a polynomial with more terms, we have to be careful that we combine <i>only<\/i> like terms<i>.<\/i> If two terms are not like terms, we 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<h4>Solution<\/h4>\r\nFirst 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&nbsp;\r\n\r\nWrite the polynomial in standard (descending) form and group the like terms:\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 [latex]x[\/latex] 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(3\\right)x+1\\end{array}[\/latex]<\/p>\r\nWrite the sum as the new coefficient:\r\n<p style=\"text-align: center;\">[latex]3x^2+3x+1[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]3x^{2}+3x+x+1+5x=3x^{2}+3x+1[\/latex]\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/1epjbVO_qU4\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]-5x^{2}+3x-x-1+5x[\/latex] by combining like terms.\r\n\r\n[reveal-answer q=\"hjm913\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm913\"][latex]-5x^2+7x-1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nSimplify [latex]7x^{2}+3x^2-4x-7+5x-3[\/latex] by combining like terms.\r\n\r\n[reveal-answer q=\"hjm698\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm698\"][latex]10x^2+x-10[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Adding and Subtracting Monomials<\/h2>\r\nAdding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent.\r\n<p style=\"text-align: left;\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"52\" height=\"47\" \/><\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<p style=\"text-align: left;\">Recall that when combining like terms only the coefficients are combined, never the exponents.<\/p>\r\n\r\n<div><\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\nHere is a brief summary of the steps we will follow to add or subtract monomials.\r\n<div class=\"textbox shaded\">\r\n<h3>ADDING AND SUBTRACTING MONOMIALS<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.Pay attention to signs when adding or subtracting monomials.\u00a0 In the example below, we are subtracting a monomial with a negative coefficient.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]11n-\\left(-8n\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-id1168468510729\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]11n-\\left(-8n\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write subtraction of a negative as addition of a positive<\/td>\r\n<td>[latex]11n+(+8n)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]19n[\/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]146315[\/ohm_question]\r\n\r\n[ohm_question]146077[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nIn order to add monomials, they must be like terms. If the variables are not the same, even if their exponents have the same value, they are not like terms and therefore cannot be added together.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nSimplify: [latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-id1168468675660\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>[latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Reorder the terms.<\/td>\r\n<td>[latex]{a}^{2}-7{a}^{2}+4{b}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Combine like terms.<\/td>\r\n<td>[latex]-6{a}^{{}^{2}}+4{b}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember, [latex]-6{a}^{2}[\/latex] and [latex]4{b}^{2}[\/latex] are not like terms. The variables are not the same.\r\n\r\n<\/div>\r\nIn this example, we switched the order of two of the monomials so that like terms were written next to each other. This is an example of the <em><strong>commutative property of addition<\/strong><\/em>.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146078[\/ohm_question]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Adding Polynomials<\/h2>\r\nAdding and subtracting <em><b>polynomials<\/b><\/em>\u00a0can be thought of as just adding and subtracting multiple monomials i.e., combining like terms.\u00a0We use both the Commutative and Associative properties to add and subtract polynomials. Using these two properties we can group like terms that can then be added or subtracted.\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<h4>Solution<\/h4>\r\nReorder and 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<\/div>\r\nIn this example, we reordered the terms using the <em><strong>commutative property of addition<\/strong><\/em>. We also regrouped the terms using the <em><strong>associative property of addition<\/strong><\/em>.\r\n\r\nWhen we add polynomials that include negative coefficients,\u00a0it is important to remember to keep the negative sign with the term it belongs to.\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<h4>Solution<\/h4>\r\nCollect like terms, making sure to keep the sign of each term.\r\n\r\n&nbsp;\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(-5x^{2}+3x^{2}\\right)}+\\underbrace{\\left(-10x+7x\\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] since [latex]-5+3=-2[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex] since [latex]-10+7=-3[\/latex]<\/p>\r\n<p style=\"text-align: left;\">The constant terms will simplify to [latex]-2[\/latex] since [latex]2-4=-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 standard form (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<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the sum: [latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex].\r\n<h4>Solution<\/h4>\r\n<p style=\"text-align: left;\">Collect like terms:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(4{x}^{2}+3x^2\\right )+ \\left (-5x-8x\\right )+\\left (1-9\\right)[\/latex]<\/p>\r\nCombine like terms:\r\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{2}\\right )+ \\left (-13x\\right )+\\left (-8\\right)[\/latex]<\/p>\r\nSimply and remove parentheses:\r\n<p style=\"text-align: center;\">[latex]7{x}^{2}-13x-8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div><\/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<h4>Solution<\/h4>\r\nWrite 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<\/div>\r\nSometimes in a vertical arrangement, we 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<h4>Solution<\/h4>\r\nWrite one polynomial below the other, lining up like terms vertically.\r\n\r\nTo keep track of like terms, insert zeros where there aren't any shared like terms. This is optional, but it may be 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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146084[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe following video shows more examples of combining like terms by adding polynomials.\r\n\r\nhttps:\/\/youtu.be\/KYZR7g7QcF4\r\n<h2><\/h2>\r\n<h2>Subtracting Polynomials<\/h2>\r\nWhen we subtract one polynomial from another, we must distribute the subtraction sign by multiplying every term in the polynomial being subtracted by [latex]-1[\/latex].\u00a0 This is equivalent to finding the opposite of the polynomial being subtracted. We then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract. [latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nMultiply the polynomial being subtracted by [latex]-1[\/latex]. i.e., change\u00a0the sign of <i>each<\/i> term in the polynomial [latex]9x^{2}+10x+5[\/latex]! All the terms are positive, so they will all become negative.\r\n<p style=\"text-align: center;\">[latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(15x^{2}+12x+20\\right)-9x^{2}-10x-5[\/latex]<\/p>\r\nReorder to match like terms, remember to check\u00a0the sign of each term.\r\n<p style=\"text-align: center;\">[latex]15x^{2}-9x^{2}+12x-10x+20-5[\/latex]<\/p>\r\nCombine like terms.\r\n<p style=\"text-align: center;\">[latex]6x^{2}+2x+15[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(15x^{2}+12x+20\\right)-\\left(9x^{2}+10x+5\\right)=6x^{2}+2x+15[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\"><img class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"52\" height=\"47\" \/>When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.<\/div>\r\n<div><\/div>\r\nIn the following example we 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 of\u00a0[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)\\text{ and }\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]\r\n<h4 style=\"text-align: left;\">Solution<\/h4>\r\n[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]\r\n<p style=\"text-align: left;\">[latex]\\begin{array}{ccc}\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)\\text{ }\\hfill &amp; \\text{Distribute -1 to each term in the second polynomial}.\\hfill \\\\ 7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill &amp; \\text{Reorder to collect like terms}.\\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\r\n<\/div>\r\n&nbsp;\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract: [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nDistribute -1 to each term by changing the sign of each term in the polynomial [latex]7x^{3}+5x^{2}\u20138x+10[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-7x^{3}-5x^{2}+8x-10[\/latex]<\/p>\r\nReorder to put like terms together and combine like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\underbrace{14x^{3}-7x^{3}}+\\underbrace{3x^{2}-5x^{2}}-\\underbrace{5x+8x}+\\underbrace{14-10}\\\\=7x^{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Write the resulting polynomial with each term's sign in front.<\/p>\r\n<p style=\"text-align: center;\">[latex]7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nWe can also use the vertical approach to keep our terms organized.\u00a0 But we must distribute the subtraction by multiplying the polynomial being subtracted by\u00a0 [latex]-1[\/latex] before we add like terms.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSubtract:\u00a0 [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]\r\n<h4>Solution<\/h4>\r\nReorganizing using the vertical approach.\r\n<p style=\"text-align: center;\">[latex]14x^{3}+3x^{2}-5x+14-\\left(7x^{3}+5x^{2}-8x+10\\right)[\/latex]<\/p>\r\nChange the signs, and combine like terms.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}14x^{3}+3x^{2}-5x+14\\,\\,\\,\\,\\\\\\underline{-7x^{3}-5x^{2}+8x-10}\\\\=7x^{3}-2x^{2}+3x+4\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]\r\n\r\n<\/div>\r\nHowever you choose to combine polynomials is up to you\u2014the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.\r\n\r\nWhen we add polynomials as we did in the last example, we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146079[\/ohm_question]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSubtract [latex]\\left({m}^{2}-3m+8\\right)[\/latex] from [latex]\\left(9{m}^{2}-7m+4\\right)[\/latex].\r\n<h4>Solution<\/h4>\r\nWhen we are asked to subtract A from B, we start with B and subtract A: B \u2013 A\r\n\r\n[latex]\\left(9{m}^{2}-7m+4\\right)-\\left({m}^{2}-3m+8\\right)[\/latex]\u00a0 \u00a0 \u00a0 Remove parentheses: Distribute the \u2013 sign.\r\n\r\n[latex]9{m}^{2}-7m+4-{m}^{2}+3m-8[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Reorder to collect like terms.\r\n\r\n[latex]9{m}^{2}-{m}^{2}<span style=\"font-size: 1rem; text-align: initial;\">-7m+3m<\/span>+4<span style=\"font-size: 1rem; text-align: initial;\">-8[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Simplify by combining like terms.<\/span>\r\n\r\n[latex]8{m}^{2}<span style=\"font-size: 1rem; text-align: initial;\">-4m-<\/span>4<span style=\"font-size: 1rem; text-align: initial;\">[\/latex]\u00a0<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>TRY\u00a0IT<\/h3>\r\n[ohm_question]146085[\/ohm_question]\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nThe following video shows more examples of subtracting polynomials.\r\n\r\nhttps:\/\/youtu.be\/xq-zVm25VC0\r\n\r\n&nbsp;\r\n\r\nThe next video shows more examples of adding and subtracting polynomials.\r\n\r\nhttps:\/\/youtu.be\/jiq3toC7wGM\r\n<h2>Summary<\/h2>\r\nWe have seen that subtracting a polynomial is equivalent to adding the opposite of the polynomial being subtracted. This means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are like terms.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Combine like terms through addition and subtraction<\/li>\n<li>Add and subtract monomials<\/li>\n<li>Add and subtract polynomials<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Like terms<\/strong>: Terms that have identical variables with identical exponents<\/li>\n<li><strong>Sum<\/strong>: the answer when two or more terms are added<\/li>\n<li><strong>Difference<\/strong>: the answer when two or more terms are subtracted<\/li>\n<li><strong>Commutative property of addition<\/strong>: reordering terms in an expression results in an equivalent expression [latex]a+b=b+a[\/latex]<\/li>\n<li><strong>Associative property of addition<\/strong>:\u00a0regrouping terms in an expression results in an equivalent expression [latex](a+b)+c=a+(b+c)[\/latex]<\/li>\n<\/ul>\n<\/div>\n<h3>Combining Like Terms<\/h3>\n<p>A polynomial may need to be simplified. One way to simplify a polynomial is to combine the <em><b>like terms,<\/b><\/em> if there are any. Two or more terms in a polynomial are like terms if they have identical variables with the identical exponents. For example, [latex]3x^{2}[\/latex] and [latex]-5x^{2}[\/latex] are like terms: They both have [latex]x[\/latex] as the variable, and the exponent is [latex]2[\/latex] 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 like terms and unlike terms.<\/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}, 7x, 7y, -8x^{3}, 9y, -3x^{2}, 8y^{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<h4>Solution<\/h4>\n<p>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:7y, 9y, 8y^{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 [latex]<em>x[\/latex]&#8211;<\/em>terms [latex]7x^{3}[\/latex]\u00a0and [latex]-8x^{3}[\/latex]\u00a0have the same exponent.<\/p>\n<p>The [latex]<em>y[\/latex]&#8211;<\/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<p>&nbsp;<\/p>\n<p>We can use the <em><strong>distributive property<\/strong><\/em> to simplify the sum of like terms. Recall that the distributive property of addition over multiplication 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 [latex]18[\/latex]. So we 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<h4>Solution<\/h4>\n<p>[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>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 sum as the new coefficient:<\/p>\n<p style=\"text-align: center;\">[latex]-2x^2[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}-5x^{2}=-2x^{2}[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Notice that combining like terms through addition or subtraction involves adding or subtracting the coefficients to find the new coefficient of the like term.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Simplify [latex]6a^{4}+4a^{4}[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>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, [latex]6[\/latex] and [latex]4[\/latex].<\/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<p style=\"text-align: center;\">[latex]10a^4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]6a^{4}+4a^{4}=10a^{4}[\/latex]<\/p>\n<\/div>\n<p>When we have a polynomial with more terms, we have to be careful that we combine <i>only<\/i> like terms<i>.<\/i> If two terms are not like terms, we 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<h4>Solution<\/h4>\n<p>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>&nbsp;<\/p>\n<p>Write the polynomial in standard (descending) form and group the like terms:<\/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 [latex]x[\/latex] 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(3\\right)x+1\\end{array}[\/latex]<\/p>\n<p>Write the sum as the new coefficient:<\/p>\n<p style=\"text-align: center;\">[latex]3x^2+3x+1[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]3x^{2}+3x+x+1+5x=3x^{2}+3x+1[\/latex]<\/p>\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<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]-5x^{2}+3x-x-1+5x[\/latex] by combining like terms.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm913\">Show Answer<\/span><\/p>\n<div id=\"qhjm913\" class=\"hidden-answer\" style=\"display: none\">[latex]-5x^2+7x-1[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Simplify [latex]7x^{2}+3x^2-4x-7+5x-3[\/latex] by combining like terms.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm698\">Show Answer<\/span><\/p>\n<div id=\"qhjm698\" class=\"hidden-answer\" style=\"display: none\">[latex]10x^2+x-10[\/latex]<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<h2>Adding and Subtracting Monomials<\/h2>\n<p>Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent.<\/p>\n<p style=\"text-align: left;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"52\" height=\"47\" \/><\/p>\n<div class=\"textbox shaded\">\n<p style=\"text-align: left;\">Recall that when combining like terms only the coefficients are combined, never the exponents.<\/p>\n<div><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Here is a brief summary of the steps we will follow to add or subtract monomials.<\/p>\n<div class=\"textbox shaded\">\n<h3>ADDING AND SUBTRACTING MONOMIALS<strong><br \/>\n<\/strong><\/h3>\n<ol>\n<li>Combine like terms.<\/li>\n<li>Simplify and write in standard form.Pay attention to signs when adding or subtracting monomials.\u00a0 In the example below, we are subtracting a monomial with a negative coefficient.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]11n-\\left(-8n\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-id1168468510729\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]11n-\\left(-8n\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Write subtraction of a negative as addition of a positive<\/td>\n<td>[latex]11n+(+8n)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]19n[\/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=\"ohm146315\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146315&theme=oea&iframe_resize_id=ohm146315&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146077\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146077&theme=oea&iframe_resize_id=ohm146077&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>In order to add monomials, they must be like terms. If the variables are not the same, even if their exponents have the same value, they are not like terms and therefore cannot be added together.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Simplify: [latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-id1168468675660\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td><\/td>\n<td>[latex]{a}^{2}+4{b}^{2}-7{a}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reorder the terms.<\/td>\n<td>[latex]{a}^{2}-7{a}^{2}+4{b}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Combine like terms.<\/td>\n<td>[latex]-6{a}^{{}^{2}}+4{b}^{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember, [latex]-6{a}^{2}[\/latex] and [latex]4{b}^{2}[\/latex] are not like terms. The variables are not the same.<\/p>\n<\/div>\n<p>In this example, we switched the order of two of the monomials so that like terms were written next to each other. This is an example of the <em><strong>commutative property of addition<\/strong><\/em>.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146078\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146078&theme=oea&iframe_resize_id=ohm146078&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2><\/h2>\n<h2>Adding Polynomials<\/h2>\n<p>Adding and subtracting <em><b>polynomials<\/b><\/em>\u00a0can be thought of as just adding and subtracting multiple monomials i.e., combining like terms.\u00a0We use both the Commutative and Associative properties to add and subtract polynomials. Using these two properties we can group like terms that can then be added or subtracted.<\/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<h4>Solution<\/h4>\n<p>Reorder and 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<p>In this example, we reordered the terms using the <em><strong>commutative property of addition<\/strong><\/em>. We also regrouped the terms using the <em><strong>associative property of addition<\/strong><\/em>.<\/p>\n<p>When we add polynomials that include negative coefficients,\u00a0it is important to remember to keep the negative sign with the term it belongs to.<\/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<h4>Solution<\/h4>\n<p>Collect like terms, making sure to keep the sign of each term.<\/p>\n<p>&nbsp;<\/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(-5x^{2}+3x^{2}\\right)}+\\underbrace{\\left(-10x+7x\\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] since [latex]-5+3=-2[\/latex]<\/p>\n<p style=\"text-align: left;\">The\u00a0[latex]x[\/latex] will simplify to [latex]-3x[\/latex] since [latex]-10+7=-3[\/latex]<\/p>\n<p style=\"text-align: left;\">The constant terms will simplify to [latex]-2[\/latex] since [latex]2-4=-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 standard form (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 class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the sum: [latex]\\left(4{x}^{2}-5x+1\\right)+\\left(3{x}^{2}-8x - 9\\right)[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p style=\"text-align: left;\">Collect like terms:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(4{x}^{2}+3x^2\\right )+ \\left (-5x-8x\\right )+\\left (1-9\\right)[\/latex]<\/p>\n<p>Combine like terms:<\/p>\n<p style=\"text-align: center;\">[latex]\\left(7{x}^{2}\\right )+ \\left (-13x\\right )+\\left (-8\\right)[\/latex]<\/p>\n<p>Simply and remove parentheses:<\/p>\n<p style=\"text-align: center;\">[latex]7{x}^{2}-13x-8[\/latex]<\/p>\n<\/div>\n<div><\/div>\n<div><\/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<h4>Solution<\/h4>\n<p>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<p>Sometimes in a vertical arrangement, we 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<h4>Solution<\/h4>\n<p>Write one polynomial below the other, lining up like terms vertically.<\/p>\n<p>To keep track of like terms, insert zeros where there aren&#8217;t any shared like terms. This is optional, but it may be 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 class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146084\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146084&theme=oea&iframe_resize_id=ohm146084&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The following video shows more examples of combining like terms by adding polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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<h2><\/h2>\n<h2>Subtracting Polynomials<\/h2>\n<p>When we subtract one polynomial from another, we must distribute the subtraction sign by multiplying every term in the polynomial being subtracted by [latex]-1[\/latex].\u00a0 This is equivalent to finding the opposite of the polynomial being subtracted. We then combine like terms. The easiest mistake to make when subtracting one polynomial from another is to forget to change the sign of EVERY term in the polynomial being subtracted.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract. [latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Multiply the polynomial being subtracted by [latex]-1[\/latex]. i.e., change\u00a0the sign of <i>each<\/i> term in the polynomial [latex]9x^{2}+10x+5[\/latex]! All the terms are positive, so they will all become negative.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(15x^{2}+12x+20\\right)\u2013\\left(9x^{2}+10x+5\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left(15x^{2}+12x+20\\right)-9x^{2}-10x-5[\/latex]<\/p>\n<p>Reorder to match like terms, remember to check\u00a0the sign of each term.<\/p>\n<p style=\"text-align: center;\">[latex]15x^{2}-9x^{2}+12x-10x+20-5[\/latex]<\/p>\n<p>Combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]6x^{2}+2x+15[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(15x^{2}+12x+20\\right)-\\left(9x^{2}+10x+5\\right)=6x^{2}+2x+15[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"52\" height=\"47\" \/>When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.<\/div>\n<div><\/div>\n<p>In the following example we 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 of\u00a0[latex]\\left(7{x}^{4}-{x}^{2}+6x+1\\right)\\text{ and }\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)[\/latex]<\/p>\n<h4 style=\"text-align: left;\">Solution<\/h4>\n<p>[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;\">[latex]\\begin{array}{ccc}\\left(7{x}^{4}-{x}^{2}+6x+1\\right)-\\left(5{x}^{3}-2{x}^{2}+3x+2\\right)\\text{ }\\hfill & \\text{Distribute -1 to each term in the second polynomial}.\\hfill \\\\ 7{x}^4-{x}^2+6x+1-5{x}^3+2{x}^{2}-3x-2\\text{ }\\hfill & \\text{Reorder to collect like terms}.\\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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract: [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Distribute -1 to each term by changing the sign of each term in the polynomial [latex]7x^{3}+5x^{2}\u20138x+10[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-7x^{3}-5x^{2}+8x-10[\/latex]<\/p>\n<p>Reorder to put like terms together and combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\underbrace{14x^{3}-7x^{3}}+\\underbrace{3x^{2}-5x^{2}}-\\underbrace{5x+8x}+\\underbrace{14-10}\\\\=7x^{3}\\,\\,\\,\\,\\,\\,\\,\\,\\,=-2x^{2}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=3x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=4\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Write the resulting polynomial with each term&#8217;s sign in front.<\/p>\n<p style=\"text-align: center;\">[latex]7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>We can also use the vertical approach to keep our terms organized.\u00a0 But we must distribute the subtraction by multiplying the polynomial being subtracted by\u00a0 [latex]-1[\/latex] before we add like terms.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Subtract:\u00a0 [latex]\\left(14x^{3}+3x^{2}\u20135x+14\\right)\u2013\\left(7x^{3}+5x^{2}\u20138x+10\\right)[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>Reorganizing using the vertical approach.<\/p>\n<p style=\"text-align: center;\">[latex]14x^{3}+3x^{2}-5x+14-\\left(7x^{3}+5x^{2}-8x+10\\right)[\/latex]<\/p>\n<p>Change the signs, and combine like terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}14x^{3}+3x^{2}-5x+14\\,\\,\\,\\,\\\\\\underline{-7x^{3}-5x^{2}+8x-10}\\\\=7x^{3}-2x^{2}+3x+4\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>[latex]\\left(14x^{3}+3x^{2}-5x+14\\right)-\\left(7x^{3}+5x^{2}-8x+10\\right)=7x^{3}-2x^{2}+3x+4[\/latex]<\/p>\n<\/div>\n<p>However you choose to combine polynomials is up to you\u2014the key point is to identify like terms, keep track of their signs, and be able to organize them accurately.<\/p>\n<p>When we add polynomials as we did in the last example, we can rewrite the expression without parentheses and then combine like terms. But when we subtract polynomials, we must be very careful with the signs.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146079\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146079&theme=oea&iframe_resize_id=ohm146079&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Subtract [latex]\\left({m}^{2}-3m+8\\right)[\/latex] from [latex]\\left(9{m}^{2}-7m+4\\right)[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>When we are asked to subtract A from B, we start with B and subtract A: B \u2013 A<\/p>\n<p>[latex]\\left(9{m}^{2}-7m+4\\right)-\\left({m}^{2}-3m+8\\right)[\/latex]\u00a0 \u00a0 \u00a0 Remove parentheses: Distribute the \u2013 sign.<\/p>\n<p>[latex]9{m}^{2}-7m+4-{m}^{2}+3m-8[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Reorder to collect like terms.<\/p>\n<p>[latex]9{m}^{2}-{m}^{2}<span style=\"font-size: 1rem; text-align: initial;\">-7m+3m<\/span>+4<span style=\"font-size: 1rem; text-align: initial;\">-8[\/latex]\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0Simplify by combining like terms.<\/span><\/p>\n<p>[latex]8{m}^{2}<span style=\"font-size: 1rem; text-align: initial;\">-4m-<\/span>4<span style=\"font-size: 1rem; text-align: initial;\">[\/latex]\u00a0<\/span><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>TRY\u00a0IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146085\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146085&theme=oea&iframe_resize_id=ohm146085&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The following video shows more examples of subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Subtracting Polynomials\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/xq-zVm25VC0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<p>The next video shows more examples of adding and subtracting polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-4\" 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>Summary<\/h2>\n<p>We have seen that subtracting a polynomial is equivalent to adding the opposite of the polynomial being subtracted. This means changing the sign of each term in the polynomial and then reorganizing all the terms to make it easier to combine those that are like terms.<\/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-756\">\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 146085, 146084, 146078, 146070, 146073. <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>Identify Like Terms and Combine Like Terms. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <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><li>Ex: Subtracting Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/xq-zVm25VC0\">https:\/\/youtu.be\/xq-zVm25VC0<\/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>Try It hjm913; hjm698. <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: Adding Polynomials. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com). <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>Ex: Adding and Subtracting Polynomials. <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><\/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":422605,"menu_order":4,"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 146085, 146084, 146078, 146070, 146073\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Adding Polynomials\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/KYZR7g7QcF4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Adding and Subtracting Polynomials\",\"author\":\"\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/jiq3toC7wGM\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Identify Like Terms and Combine Like Terms\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/1epjbVO_qU4\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex: Subtracting Polynomials\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/xq-zVm25VC0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Try It hjm913; 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