{"id":3886,"date":"2021-05-20T18:29:55","date_gmt":"2021-05-20T18:29:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-differentiation-rules\/"},"modified":"2021-07-03T18:50:28","modified_gmt":"2021-07-03T18:50:28","slug":"review-for-differentiation-rules","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-differentiation-rules\/","title":{"raw":"Skills Review for Differentiation Rules","rendered":"Skills Review for Differentiation Rules"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Multiply a polynomial by a monomial&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Multiply a polynomial by a monomial<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Multiply binomials - FOIL&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Multiply binomials - FOIL<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Multiply polynomials - single variable&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Multiply polynomials - single variable<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Convert between radical and exponent notations&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Convert between radical and exponent notations<\/span><\/li>\r\n \t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the zero product principle to solve quadratic equations that can be factored&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Use the zero product principle to solve quadratic equations that can be factored<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Differentiation Rules section, you will learn methods that will allow you to find the derivative of a function quicker. Here we will review some basic algebraic principles including how to multiply polynomials and write radicals in exponential form. How to solve quadratic equations by factoring will also be discussed.\r\n<h2>Multiply Polynomials<\/h2>\r\nTo multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. In [latex]2\\left(x+7\\right)[\/latex] we can distribute [latex]2[\/latex] to obtain the 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<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<div class=\"textbox exercises\">\r\n<h3>Example: Multiplying Polynomials Using Distributive Property<\/h3>\r\nFind the product.\r\n<p style=\"text-align: center;\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"752165\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"752165\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill &amp; \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill &amp; \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill &amp; \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill &amp; \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can use a table to keep track of our work, as shown in the table below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.\r\n<table style=\"height: 45px;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 115px; height: 15px;\"><\/td>\r\n<td style=\"width: 159px; height: 15px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\r\n<td style=\"width: 154px; height: 15px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 116px; height: 15px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\r\n<td style=\"width: 159px; height: 15px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\r\n<td style=\"width: 154px; height: 15px;\">[latex]-2{x}^{2}[\/latex]<\/td>\r\n<td style=\"width: 116px; height: 15px;\">[latex]8x[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\r\n<td style=\"width: 159px; height: 15px;\">[latex]3{x}^{2}[\/latex]<\/td>\r\n<td style=\"width: 154px; height: 15px;\">[latex]-x[\/latex]<\/td>\r\n<td style=\"width: 116px; height: 15px;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]3864[\/ohm_question]\r\n\r\n<\/div>\r\nWatch this video to see more examples of how to use the distributive property to multiply polynomials.\r\n\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=6405041&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=bwTmApTV_8o&amp;video_target=tpm-plugin-r235hd14-bwTmApTV_8o\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExMultiplyingUsingTheDistributiveProperty_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \"Ex: Multiplying Using the Distributive Property\" here (opens in new window)<\/a>.\r\n\r\nA shortcut called FOIL is sometimes used to find the product of 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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205149\/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 is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.\r\n<div class=\"textbox\">\r\n<h3>How To: Given two binomials, Multiplying Using FOIL<\/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 class=\"textbox exercises\">\r\n<h3>Example: Multiplying Polynomials Using FOIL<\/h3>\r\nUse FOIL to find the product.\r\n\r\n[latex]\\left(2x-18\\right)\\left(3x + 3\\right)[\/latex]\r\n\r\n[reveal-answer q=\"698991\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698991\"]\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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205151\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205154\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205156\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205158\/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\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]93539[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Convert Between Radical and Exponential Form<\/h2>\r\nRadical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]a[\/latex] cannot be negative.\r\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/div>\r\nWe can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an <em>n<\/em>th root. The numerator tells us the power and the denominator tells us the root.\r\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\r\nAll of the properties of exponents that we learned for integer exponents also hold for rational exponents.\r\n<div class=\"textbox\">\r\n<h3>Rational Exponents<\/h3>\r\nRational exponents are another way to express principal <em>n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is\r\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}\\end{align}[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an expression with a rational exponent, write the expression as a radical.<\/h3>\r\n<ol>\r\n \t<li>Determine the power by looking at the numerator of the exponent.<\/li>\r\n \t<li>Determine the root by looking at the denominator of the exponent.<\/li>\r\n \t<li>Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Converting from radical to exponential form<\/h3>\r\nWrite [latex]\\sqrt[3]{{x}^{2}}[\/latex] with a rational exponent.\r\n\r\n[reveal-answer q=\"878113\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"878113\"]\r\n\r\nThe 2 tells us the power and the 3 tells us the root.\r\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{x}^{2}}={\\left(\\sqrt[3]{x}\\right)}^{2}={x}^{\\frac{2}{3}}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]\\sqrt{{y}^{5}}[\/latex] with a rational exponent.\r\n\r\n[reveal-answer q=\"937831\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"937831\"]\r\n\r\n[latex]{y}^{\\frac{5}{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Converting From Radical To Exponential Form In a Rational Expression<\/h3>\r\nWrite [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.\r\n\r\n[reveal-answer q=\"183909\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"183909\"]\r\n\r\nThe power is 2 and the root is 7, so the rational exponent will be [latex]\\dfrac{2}{7}[\/latex]. We get [latex]\\dfrac{4}{{a}^{\\frac{2}{7}}}[\/latex]. Using properties of exponents, we get [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}=4{a}^{\\frac{-2}{7}}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWrite [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.\r\n\r\n[reveal-answer q=\"522860\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"522860\"]\r\n\r\n[latex]x{\\left(5y\\right)}^{\\frac{9}{2}}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Solve Quadratic Equations by Factoring<\/h2>\r\nAn equation containing a second-degree polynomial is called a <strong>quadratic equation<\/strong>. For example, equations such as [latex]2{x}^{2}+3x - 1=0[\/latex] and [latex]{x}^{2}-4=0[\/latex] are quadratic equations. Often, the easiest method of solving a quadratic equation is by\u00a0<strong>factoring<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic equation, Solve it by factoring<\/h3>\r\n<ol>\r\n \t<li>Set the quadratic equation equal to 0.<\/li>\r\n \t<li>Factor.<\/li>\r\n \t<li>Set each factor equal to 0 and solve for the variable.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving A Quadratic Equation by Factoring<\/h3>\r\nFactor and solve the equation: [latex]{x}^{2}+x - 6=0[\/latex].\r\n\r\n[reveal-answer q=\"16111\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"16111\"]\r\n\r\nThe quadratic equation is already set equal to 0, so we can now factor it.\r\n<p style=\"text-align: center;\">[latex]\\left(x - 2\\right)\\left(x+3\\right)=0[\/latex]<\/p>\r\nTo solve this equation, now set each factor equal to zero and solve.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(x - 2\\right)\\left(x+3\\right)=0\\hfill \\\\ \\left(x - 2\\right)=0\\hfill \\\\ x=2\\hfill \\\\ \\left(x+3\\right)=0\\hfill \\\\ x=-3\\hfill \\end{array}[\/latex]<\/div>\r\nThe two solutions are [latex]x=2[\/latex] and [latex]x=-3[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFactor and solve the quadratic equation: [latex]{x}^{2}-5x - 6=0[\/latex].\r\n\r\n[reveal-answer q=\"220537\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"220537\"]\r\n\r\n[latex]\\left(x - 6\\right)\\left(x+1\\right)=0;x=6,x=-1[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]2029[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Multiply a polynomial by a monomial&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Multiply a polynomial by a monomial<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Multiply binomials - FOIL&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Multiply binomials &#8211; FOIL<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Multiply polynomials - single variable&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4611,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:[null,2,15389148],&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Multiply polynomials &#8211; single variable<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Convert between radical and exponent notations&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Convert between radical and exponent notations<\/span><\/li>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Use the zero product principle to solve quadratic equations that can be factored&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:4609,&quot;3&quot;:{&quot;1&quot;:0},&quot;12&quot;:0,&quot;15&quot;:&quot;Work Sans&quot;}\">Use the zero product principle to solve quadratic equations that can be factored<\/span><\/li>\n<\/ul>\n<\/div>\n<p>In the Differentiation Rules section, you will learn methods that will allow you to find the derivative of a function quicker. Here we will review some basic algebraic principles including how to multiply polynomials and write radicals in exponential form. How to solve quadratic equations by factoring will also be discussed.<\/p>\n<h2>Multiply Polynomials<\/h2>\n<p>To multiply a number by a polynomial, we use the distributive property. The number must be distributed to each term of the polynomial. In [latex]2\\left(x+7\\right)[\/latex] we can distribute [latex]2[\/latex] to obtain the 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<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<div class=\"textbox exercises\">\n<h3>Example: Multiplying Polynomials Using Distributive Property<\/h3>\n<p>Find the product.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(2x+1\\right)\\left(3{x}^{2}-x+4\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q752165\">Show Solution<\/span><\/p>\n<div id=\"q752165\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{array}{cc}2x\\left(3{x}^{2}-x+4\\right)+1\\left(3{x}^{2}-x+4\\right) \\hfill & \\text{Use the distributive property}.\\hfill \\\\ \\left(6{x}^{3}-2{x}^{2}+8x\\right)+\\left(3{x}^{2}-x+4\\right)\\hfill & \\text{Multiply}.\\hfill \\\\ 6{x}^{3}+\\left(-2{x}^{2}+3{x}^{2}\\right)+\\left(8x-x\\right)+4\\hfill & \\text{Combine like terms}.\\hfill \\\\ 6{x}^{3}+{x}^{2}+7x+4 \\hfill & \\text{Simplify}.\\hfill \\end{array}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can use a table to keep track of our work, as shown in the table below. Write one polynomial across the top and the other down the side. For each box in the table, multiply the term for that row by the term for that column. Then add all of the terms together, combine like terms, and simplify.<\/p>\n<table style=\"height: 45px;\" summary=\"A table with 3 rows and 4 columns. The first entry of the first row is empty, the others are labeled: three times x squared, negative x, and positive four. The first entry of the second row is labeled: two times x. The second entry reads: six times x cubed. The third entry reads: negative two times x squared. The fourth entry reads: eight times x. The first entry of the third row reads: positive one. The second entry reads: three times x squared. The third entry reads: negative x. The fourth entry reads: four.\">\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><\/td>\n<td style=\"width: 159px; height: 15px;\"><strong>[latex]3{x}^{2}[\/latex]<\/strong><\/td>\n<td style=\"width: 154px; height: 15px;\"><strong>[latex]-x[\/latex]<\/strong><\/td>\n<td style=\"width: 116px; height: 15px;\"><strong>[latex]+4[\/latex]<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]2x[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]6{x}^{3}\\\\[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-2{x}^{2}[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]8x[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 115px; height: 15px;\"><strong>[latex]+1[\/latex]<\/strong><\/td>\n<td style=\"width: 159px; height: 15px;\">[latex]3{x}^{2}[\/latex]<\/td>\n<td style=\"width: 154px; height: 15px;\">[latex]-x[\/latex]<\/td>\n<td style=\"width: 116px; height: 15px;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm3864\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=3864&theme=oea&iframe_resize_id=ohm3864&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Watch this video to see more examples of how to use the distributive property to multiply polynomials.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=6405041&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=bwTmApTV_8o&amp;video_target=tpm-plugin-r235hd14-bwTmApTV_8o\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/ExMultiplyingUsingTheDistributiveProperty_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for &#8220;Ex: Multiplying Using the Distributive Property&#8221; here (opens in new window)<\/a>.<\/p>\n<p>A shortcut called FOIL is sometimes used to find the product of 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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205149\/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 is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given two binomials, Multiplying Using FOIL<\/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 class=\"textbox exercises\">\n<h3>Example: Multiplying Polynomials Using FOIL<\/h3>\n<p>Use FOIL to find the product.<\/p>\n<p>[latex]\\left(2x-18\\right)\\left(3x + 3\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698991\">Show Solution<\/span><\/p>\n<div id=\"q698991\" class=\"hidden-answer\" style=\"display: none\">\nFind the product of the first terms.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21205151\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205154\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205156\/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\/wp-content\/uploads\/sites\/896\/2016\/10\/21205158\/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<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm93539\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=93539&theme=oea&iframe_resize_id=ohm93539&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Convert Between Radical and Exponential Form<\/h2>\n<p>Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index [latex]n[\/latex] is even, then [latex]a[\/latex] cannot be negative.<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{1}{n}}=\\sqrt[n]{a}[\/latex]<\/div>\n<p>We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an <em>n<\/em>th root. The numerator tells us the power and the denominator tells us the root.<\/p>\n<div style=\"text-align: center;\">[latex]{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}[\/latex]<\/div>\n<p>All of the properties of exponents that we learned for integer exponents also hold for rational exponents.<\/p>\n<div class=\"textbox\">\n<h3>Rational Exponents<\/h3>\n<p>Rational exponents are another way to express principal <em>n<\/em>th roots. The general form for converting between a radical expression with a radical symbol and one with a rational exponent is<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}{a}^{\\frac{m}{n}}={\\left(\\sqrt[n]{a}\\right)}^{m}=\\sqrt[n]{{a}^{m}}\\end{align}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given an expression with a rational exponent, write the expression as a radical.<\/h3>\n<ol>\n<li>Determine the power by looking at the numerator of the exponent.<\/li>\n<li>Determine the root by looking at the denominator of the exponent.<\/li>\n<li>Using the base as the radicand, raise the radicand to the power and use the root as the index.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Converting from radical to exponential form<\/h3>\n<p>Write [latex]\\sqrt[3]{{x}^{2}}[\/latex] with a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q878113\">Show Solution<\/span><\/p>\n<div id=\"q878113\" class=\"hidden-answer\" style=\"display: none\">\n<p>The 2 tells us the power and the 3 tells us the root.<\/p>\n<p style=\"text-align: center;\">[latex]\\sqrt[3]{{x}^{2}}={\\left(\\sqrt[3]{x}\\right)}^{2}={x}^{\\frac{2}{3}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]\\sqrt{{y}^{5}}[\/latex] with a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q937831\">Show Solution<\/span><\/p>\n<div id=\"q937831\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{y}^{\\frac{5}{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Converting From Radical To Exponential Form In a Rational Expression<\/h3>\n<p>Write [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}[\/latex] using a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q183909\">Show Solution<\/span><\/p>\n<div id=\"q183909\" class=\"hidden-answer\" style=\"display: none\">\n<p>The power is 2 and the root is 7, so the rational exponent will be [latex]\\dfrac{2}{7}[\/latex]. We get [latex]\\dfrac{4}{{a}^{\\frac{2}{7}}}[\/latex]. Using properties of exponents, we get [latex]\\dfrac{4}{\\sqrt[7]{{a}^{2}}}=4{a}^{\\frac{-2}{7}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Write [latex]x\\sqrt{{\\left(5y\\right)}^{9}}[\/latex] using a rational exponent.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q522860\">Show Solution<\/span><\/p>\n<div id=\"q522860\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x{\\left(5y\\right)}^{\\frac{9}{2}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Solve Quadratic Equations by Factoring<\/h2>\n<p>An equation containing a second-degree polynomial is called a <strong>quadratic equation<\/strong>. For example, equations such as [latex]2{x}^{2}+3x - 1=0[\/latex] and [latex]{x}^{2}-4=0[\/latex] are quadratic equations. Often, the easiest method of solving a quadratic equation is by\u00a0<strong>factoring<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic equation, Solve it by factoring<\/h3>\n<ol>\n<li>Set the quadratic equation equal to 0.<\/li>\n<li>Factor.<\/li>\n<li>Set each factor equal to 0 and solve for the variable.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving A Quadratic Equation by Factoring<\/h3>\n<p>Factor and solve the equation: [latex]{x}^{2}+x - 6=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q16111\">Show Solution<\/span><\/p>\n<div id=\"q16111\" class=\"hidden-answer\" style=\"display: none\">\n<p>The quadratic equation is already set equal to 0, so we can now factor it.<\/p>\n<p style=\"text-align: center;\">[latex]\\left(x - 2\\right)\\left(x+3\\right)=0[\/latex]<\/p>\n<p>To solve this equation, now set each factor equal to zero and solve.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}\\left(x - 2\\right)\\left(x+3\\right)=0\\hfill \\\\ \\left(x - 2\\right)=0\\hfill \\\\ x=2\\hfill \\\\ \\left(x+3\\right)=0\\hfill \\\\ x=-3\\hfill \\end{array}[\/latex]<\/div>\n<p>The two solutions are [latex]x=2[\/latex] and [latex]x=-3[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Factor and solve the quadratic equation: [latex]{x}^{2}-5x - 6=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q220537\">Show Solution<\/span><\/p>\n<div id=\"q220537\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(x - 6\\right)\\left(x+1\\right)=0;x=6,x=-1[\/latex]<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2029\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2029&theme=oea&iframe_resize_id=ohm2029&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-3886\">\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>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/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>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/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":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"\",\"organization\":\"Lumen 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