{"id":1193,"date":"2013-06-06T08:41:57","date_gmt":"2013-06-06T08:41:57","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=1193"},"modified":"2026-03-25T12:48:22","modified_gmt":"2026-03-25T17:48:22","slug":"polynomials","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/polynomials\/","title":{"rendered":"Polynomials"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_n3hurGDMHbk\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_n3hurGDMHbk\" data-source-videoID=\"n3hurGDMHbk\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Polynomials Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Polynomials\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_n3hurGDMHbk:hover {cursor:pointer;} img#videoThumbnailImage_n3hurGDMHbk {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1137-polynomials-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_n3hurGDMHbk\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_n3hurGDMHbk\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Polynomials\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_n3hurGDMHbk\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_n3hurGDMHbk\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_n3hurGDMHbk\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction shL_Function() {\n  var x = document.getElementById(\"shL\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"shL_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"shL\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#A_Review_of_Polynomials\" class=\"smooth-scroll\">A Review of Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Types_of_Polynomial_Expressions\" class=\"smooth-scroll\">Types of Polynomial Expressions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Multiplying_Polynomials\" class=\"smooth-scroll\">Multiplying Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Dividing_Polynomials\" class=\"smooth-scroll\">Dividing Polynomials<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Rational_Expressions\" class=\"smooth-scroll\">Rational Expressions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Polynomial_Practice_Questions\" class=\"smooth-scroll\">Polynomial Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about the algebra of polynomial and <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/rational-expressions\/\">rational expressions<\/a>!  In this video we will explore:<\/p>\n<ul>\n<li>Multiplying polynomials<\/li>\n<li>Factoring polynomials<\/li>\n<li>Dividing polynomials <\/li>\n<li>Rational expressions<\/li>\n<li>Operations with rational expressions<\/li>\n<\/ul>\n<h2><span id=\"A_Review_of_Polynomials\" class=\"m-toc-anchor\"><\/span>A Review of Polynomials<\/h2>\n<p>\nBefore we get heavily into polynomial algebra, let\u2019s recap a few important terms.<\/p>\n<p>A <strong>polynomial<\/strong> is a mathematical expression of one or more algebraic terms, each of which consists of a constant multiplied by one or more variables raised to a non-negative integral power (such as \\(a+bx+cx^2\\)).<\/p>\n<p>The key pieces to remember are:<\/p>\n<div class=\"transcriptcallout\" style=\"text-align: left;\">\n<ol style=\"margin-left: 20px; margin-bottom: 0em;\">\n<li style=\"margin-bottom: 10px;\">Polynomials consist of one or more terms (of the general form \\(ax^n\\)) that are added or subtracted. The sample in the definition has 3 terms.<\/li>\n<li style=\"margin-bottom: 10px;\">The coefficients of the terms can be any type of number.<\/li>\n<li>The powers of the terms can only be non-negative integers (including 0).<\/li>\n<\/ol>\n<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s look at some quick examples:<\/p>\n<ul>\n<li style=\"margin-bottom: 10px;\">\\(2x^2-4x+5\\) is a polynomial expression, but \\(2x^{2.5}-4x+5\\) is NOT a polynomial expression. Why? Because of the decimal exponent. Remember, the exponents in a polynomial expression have to be integers.<\/li>\n<li>\\(0.4x+1\\) is a polynomial expression, but \\(x^{-3}+1\\) is NOT, because of the negative exponent.<\/li>\n<\/ul>\n<h2><span id=\"Types_of_Polynomial_Expressions\" class=\"m-toc-anchor\"><\/span>Types of Polynomial Expressions<\/h2>\n<p>\nNow that we\u2019ve looked at some polynomial expressions, let\u2019s look at the different types:<\/p>\n<div class=\"transcriptcallout\" style=\"text-align: left;\">\n<ul style=\"list-style-type: none; margin-left: 20px; margin-bottom: 0em;\">\n<li style=\"margin-bottom: 10px;\">A <strong>monomial<\/strong> is a polynomial consisting of 1 term. For example, \\(x\\) and 2 are monomials.<\/li>\n<li style=\"margin-bottom: 10px;\">A <strong>binomial<\/strong> is a polynomial consisting of 2 terms. \\(x + 1\\) and \\(x^3+4\\) are binomials.<\/li>\n<li style=\"margin-bottom: 10px;\">A <strong>trinomial<\/strong> is a polynomial consisting of 3 terms. \\(12x^4+3x^3+1\\) is a trinomial.<\/li>\n<li>A <strong>degree<\/strong> is the highest exponent in a polynomial.\n<ul style=\"list-style-type: circle; margin-left: 40px; margin-top: 0.5em;\">\n<li style=\"margin-bottom: 7px;\">2 is a <span style=\"font-weight: 550;\">degree-0<\/span> polynomial<\/span>\n<li style=\"margin-bottom: 7px;\">\\(x+2\\) is a <span style=\"font-weight: 550;\">degree-1<\/span> polynomial<\/li>\n<li style=\"margin-bottom: 7px;\">\\(x^2+3\\) is a <span style=\"font-weight: 550;\">degree-2<\/span> polynomial<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<p>\n&nbsp;<br \/>\nNow let\u2019s move on to multiplying!<\/p>\n<h2><span id=\"Multiplying_Polynomials\" class=\"m-toc-anchor\"><\/span>Multiplying Polynomials<\/h2>\n<p>\nOne of the traditional methods taught for multiplying polynomials is FOIL. This method highlights the need for organization.<\/p>\n<p>Multiplying polynomials is an application of the <strong>distributive property<\/strong>. The key idea is that every term of each polynomial needs to be multiplied by every term of every other polynomial.<\/p>\n<p>One way to help visualize and organize polynomial multiplication is with an area model, which looks like this. Let\u2019s multiply \\(3x+4\\) by \\(-2x+-5\\):<\/p>\n<table class=\"ATable\" style=\"width: 90%; border: none; margin: auto;\">\n<thead>\n<tr>\n<th style=\"border-left:0px; border-top:0px; border-bottom:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; border-bottom:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-right:0px; border-top:0px; border-bottom:0px; border-left:0px; background-color:white\"><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-right:0px; border-bottom:0px\"><\/td>\n<td style=\"border-left:0px; border-top:0px\"><\/td>\n<td style=\"background-color:#E1E1E1\"><strong>\\(3x\\)<\/strong><\/td>\n<td style=\"background-color:#E1E1E1\"><strong>\\(4\\)<\/strong><\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-bottom:0px\"><\/td>\n<td style=\"background-color:#E1E1E1\">\\(-2x\\)<\/td>\n<td>\\((3x)(2x)\\)<br \/>\n<span style=\"color:green\">\\(-6x^2\\)<\/td>\n<td>\\((4)(-2x)\\)<br \/>\n<span style=\"color:green\">\\(-8x\\)<\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-bottom:0px\"><\/td>\n<td style=\"background-color:#E1E1E1\">\\(-5\\)<\/td>\n<td>\\((3x)(-5)\\)<br \/>\n<span style=\"color:green\">\\(15x\\)<\/td>\n<td>\\((4)(-5)\\)<br \/>\n<span style=\"color:green\">\\(-20\\)<\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px; border-left:0px\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nSo the product is \\(-6x^2-8x-15x-20\\), which then simplifies to \\(-6x^2-23x-20\\).<\/p>\n<p>The area model demonstrates how to multiply every term of one polynomial by every term of the other and it is expandable for polynomials of any size. Let\u2019s try another example. <\/p>\n<div class=\"table-container\">\n<table class=\"ATable\" style=\"width: 90%; border: none; margin: auto;\">\n<thead>\n<tr>\n<th style=\"border-left:0px; border-top:0px; border-bottom:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; border-bottom:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-left:0px; border-top:0px; border-right:0px; background-color:white\"><\/th>\n<th style=\"border-right:0px; border-top:0px; border-bottom:0px; border-left:0px; background-color:white\"><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-right:0px; border-bottom:0px\"><\/td>\n<td style=\"border-left:0px; border-top:0px\"><\/td>\n<td style=\"background-color:#E1E1E1\"><strong>\\(2a^3\\)<\/strong><\/td>\n<td style=\"background-color:#E1E1E1\"><strong>\\(-4a^2\\)<\/strong><\/td>\n<td style=\"background-color:#E1E1E1\"><strong>\\(+a\\)<\/strong><\/td>\n<td style=\"background-color:#E1E1E1\"><strong>\\(-12\\)<\/strong><\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-bottom:0px\"><\/td>\n<td style=\"background-color:#E1E1E1\">\\(7b^5\\)<\/td>\n<td>\\(14a^3b^5\\)<\/td>\n<td>\\(-28a^2b^5\\)<\/td>\n<td>\\(7ab^5\\)<\/td>\n<td>\\(-84b^5\\)<\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-bottom:0px\"><\/td>\n<td style=\"background-color:#E1E1E1\">\\(-2b^2\\)<\/td>\n<td>\\(-4a^3b^2\\)<\/td>\n<td>\\(8a^2b^2\\)<\/td>\n<td>\\(-2ab^2\\)<\/td>\n<td>\\(24b^2\\)<\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px\"><\/td>\n<\/tr>\n<tr>\n<td style=\"border-left:0px; border-top:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-left:0px; border-bottom:0px; border-right:0px\"><\/td>\n<td style=\"border-right:0px; border-top:0px; border-bottom:0px; border-left:0px\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"scroll-message\">(Scroll to see more &rarr;)<\/div>\n<p>\n&nbsp;<br \/>\nOur terms on the left are \\(7b^5\\) and \\(-2b^2\\), and our terms on top are \\(2a^3\\), \\(-4a^2\\), \\(a\\), and \\(-12\\). We need to simply multiply all the terms, then add the resulting terms together. <\/p>\n<p>None of these are like terms, so they can\u2019t be combined. So our answer is:<\/p>\n<div class=\"examplesentence\">\\(14a^3b^5 \u2013 4a^3b^2 \u2013 28a^2b^5\\)\\( + 8a^2b^2 + 7ab^5 \u2013 2ab^2\\)\\( \u2013 84b^5 + 24b^2\\)<\/div>\n<p>\n&nbsp;<br \/>\nSo this is our final answer.<\/p>\n<h3><span id=\"Factoring\" class=\"m-toc-anchor\"><\/span>Factoring<\/h3>\n<p>\nAs we\u2019ll see throughout this video, working with polynomials is almost identical to working with simple numbers. When we think of factors, for example, we may think of numbers that are multiplied to create another number. For example, 2 and 3 are factors of 6. 20 and 5 are factors of 100. <\/p>\n<p>Often, we look for numbers\u2019 prime factorizations. The prime factorization of 30 is \\(2(3)(5)\\) because all the factors are prime numbers. Think of factoring as undoing the area model we just explored.<\/p>\n<p>Often, the terms of polynomials have <strong>common factors<\/strong>.  <\/p>\n<p>Consider this polynomial:<\/p>\n<div class=\"examplesentence\">\\(20a^2b + 10a^3 + 15a^2b^3 + 5a^2\\)<\/div>\n<p>\n&nbsp;<br \/>\nEach term has a common factor of \\(5a^2\\): \\((5a^2)(4b) + (5a^2)(2a)\\) \\(+ (5a^2)(3b^3) + (5a^2)(1)\\). So this can be rewritten as \\(5a^2(4b + 2a + 3b^3 + 1)\\).  This is the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/prime-factorization\/\">prime factorization<\/a> of the polynomial. It cannot be factored any further.<\/p>\n<p>Sometimes, factoring happens in groups. It appears as if this polynomial cannot be factored:<\/p>\n<div class=\"examplesentence\">\\(2b^2x + 2ax + a + b^2\\)<\/div>\n<p>\n&nbsp;<br \/>\nRearranging to group the \\(a\\) terms and the \\(b\\) terms, we have<\/p>\n<div class=\"examplesentence\">\\((2ax + a) + (2b^2x + b2)\\)<\/div>\n<p>\n&nbsp;<br \/>\nPulling out common factors, we have<\/p>\n<div class=\"examplesentence\">\\(a(2x + 1) + b^2(2x + 1)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow we can see the common factor of \\(2x + 1\\), so we have this:<\/p>\n<div class=\"examplesentence\">\\((2x + 1)(a + b^2)\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can always check our factoring work by multiplying the factors to be sure we end up with the original polynomial.<\/p>\n<h2><span id=\"Dividing_Polynomials\" class=\"m-toc-anchor\"><\/span>Dividing Polynomials<\/h2>\n<p>\nPolynomial division is closely related to multiplication and factoring. Consider the number 6.  Its prime factors are 2 and 3. \\(2 \\times 3 = 6\\), so 6 divided by 2 is 3 and 6 divided by 3 is 2.<\/p>\n<p>This polynomial \\(-6x^2 \u2013 23x \u2013 20\\) has prime factors \\((3x+4)\\) and \\((-2x-5)\\). <\/p>\n<div class=\"examplesentence\">\\(\\frac{(-6x^2 \u2013 23x \u2013 20)}{(3x+4)}=(-2x-5)\\)<br \/>\nand<br \/>\n\\(\\frac{(-6x^2 \u2013 23x \u2013 20)}{(-2x-5)} = (3x+4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nOne polynomial can be divided by another as long as the degree of the dividend is greater than or equal to the degree of the divisor. Here\u2019s how it works:<\/p>\n<p>Make sure the exponents of both polynomials are in descending order and put a 0 in for any missing in the sequence (for example, \\(2x^2 + 1\\) would become \\(2x^2 + 0x + 1\\)).<\/p>\n<p>Divide the first terms. Here, the question is \u201chow many times does \\(3x\\) go into \\(-6x^2\\)?\u201d<\/p>\n<p>Multiply \\(3x+4\\) by \\(-2x^2\\).<\/p>\n<p>Subtract the product.<\/p>\n<p>Bring down the next term in the divisor.<\/p>\n<p>Repeat. How many times does \\(3x\\) go into \\(-15x\\)?<\/p>\n<p>Multiply.<\/p>\n<p>Subtract.<\/p>\n<p>There\u2019s a remainder of 0, so \\(-2x \u2013 5\\) and \\(3x + 4\\) are factors of \\(-6x^2-23x-20\\), as we already knew.<\/p>\n<h2><span id=\"Rational_Expressions\" class=\"m-toc-anchor\"><\/span>Rational Expressions<\/h2>\n<p>\n<strong>Rational expressions<\/strong> are closely related to rational numbers. The word rational comes from the word ratio. Just as rational numbers are defined as having the form integer\/integer, rational expressions are defined by the form polynomial\/polynomial.<\/p>\n<p>Here are some examples of rational expressions: \\(\\frac{x^2+2}{3x}, \\frac{1}{4a^2}, \\frac{4z^4-z^3-10z+2}{11z^5-22z^3}\\)<\/p>\n<p>When common factors occur in the numerator and denominator of a rational function, they can be divided to make 1. This is often taught as \u201ccanceling out\u201d, but it actually stems from the fact that, for example, \\(\\frac{2}{2} = 1\\).<\/p>\n<p>We can remove the common factors from these rational expressions:<\/p>\n<div class=\"examplesentence\">\\(\\frac{2x^2-6x}{2x^3+8x}=\\frac{2x(x-3)}{2x(x^2+4)}=\\frac{x-3}{x^2+4}\\)<br \/>\n\\(\\frac{x^2+2x+1}{x^2+3x+2}=\\frac{(x+1)(x+1)}{(x+1)(x+2)}=\\frac{x+1}{x+2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nRational expressions behave like fractions. To add or subtract them, we need to get a common denominator. For example:<\/p>\n<div class=\"examplesentence\">\\(\\frac{x^2+2}{3x}+\\frac{2x}{x^2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nSince one expression has \\(3x\\) as a denominator and the other has \\(x^2\\), a common denominator of \\((3x)(x2)\\) can be created.<\/p>\n<div class=\"examplesentence\">\\(\\frac{x^2+2}{3x} \\times \\frac{x^2}{x^2}+\\frac{2x}{x^2} \\times \\frac{3x}{3x}\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember to multiply each numerator by the same factor so that we are essentially multiplying each expression by 1.<\/p>\n<div class=\"examplesentence\">\\(\\frac{(x^2+2)(x^2)}{(3x)(x^2)}+\\frac{(2x)(3x)}{(x^2)(3x)}\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow multiply through and simplify.<\/p>\n<div class=\"examplesentence\">\\(\\frac{x^4+2x^2}{3x^3}+\\frac{6x^2}{3x^3}=\\frac{x^4+8x^2}{3x^3}\\)\\(=\\frac{x^2+8}{3x}\\)<\/div>\n<p>\n&nbsp;<br \/>\nTo multiply them, we multiply straight across. For example:<\/p>\n<div class=\"examplesentence\">\\(\\frac{x^2+2}{3x} \\times \\frac{2x}{x^2}=\\frac{(x^2+2)(2x)}{(3x)(x^2)}\\)\\(=\\frac{2x^3+4x}{3x^3}=\\frac{2x^2+4}{3x^2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd to divide them, we multiply by the reciprocal:<\/p>\n<div class=\"examplesentence\">\\(\\frac{x^2+2}{3x} \\div \\frac{2x}{x^2}\\)\\(=\\frac{x^2+2}{3x} \\times \\frac{x^2}{2x}\\)\\(=\\frac{(x^2+2)(x^2)}{(3x)(2x)}=\\frac{x^4+2x^2}{6x^2}\\)\\(=\\frac{x^2+2}{6}\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd there you have it! I hope this video has increased your understanding of polynomial expressions.<\/p>\n<p>Thanks for watching, and happy studying!<\/p>\n<div style=\"text-align: center;\"><a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/factoring-polynomials\/\">Factoring Polynomials<\/a><\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Polynomial_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Polynomial Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nMultiply the following polynomials using the FOIL method.<\/p>\n<div class=\"yellow-math-quote\">\\((3x+2)(x\u22121)\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">\\(3x^2-2x-1\\)<\/div><div class=\"PQ\"  id=\"PQ-1-2\">\\(4x^2-3x-2\\)<\/div><div class=\"PQ\"  id=\"PQ-1-3\">\\(6x^2-x-4\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-4\">\\(3x^2-x-2\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The FOIL method reminds us to multiply polynomials in the order of \u201cFirst, Outer, Inner, Last\u201d. This ensures that we do not miss any terms when distributing.<\/p>\n<ul style=\"list-style-type: none; padding-left: 0; margin-left: 1.5em\">\n<li style=\"display: flex; align-items: flex-start\"><strong style=\"display: inline-block; width: 1em; text-align: right; font-weight: 600; margin-right: 0.35em; flex-shrink: 0;\">F:<\/strong> First term times first term<\/li>\n<li style=\"display: flex; align-items: flex-start\"><strong style=\"display: inline-block; width: 1em; text-align: right; font-weight: 600; margin-right: 0.35em; flex-shrink: 0;\">O:<\/strong> Outer term times outer term<\/li>\n<li style=\"display: flex; align-items: flex-start\"><strong style=\"display: inline-block; width: 1em; text-align: right; font-weight: 600; margin-right: 0.35em; flex-shrink: 0;\">I:<\/strong> Inner term times inner term<\/li>\n<li style=\"display: flex; align-items: flex-start\"><strong style=\"display: inline-block; width: 1em; text-align: right; font-weight: 600; margin-right: 0.35em; flex-shrink: 0;\">L:<\/strong> Last term times last term<\/li>\n<\/ul>\n<p style=\"text-align: center; line-height: 35px\">\n\\(3x\\times x=3x^2\\)<br \/>\n\\(3x\\times \u22121=\u22123x\\)<br \/>\n\\(2\\times x=2x\\)<br \/>\n\\(2\\times (-1)=\u22122\\)\n<\/p>\n<p>Simplifying \\(3x^2-3x+2x-2\\) give us \\(3x^2-x-2\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nFactor the following polynomial:<\/p>\n<div class=\"yellow-math-quote\">\\(6x^2+13x+6\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\((2x+1)(3x+6)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\((3x+2)(3x+2)\\) <\/div><div class=\"PQ\"  id=\"PQ-2-3\">\\((2x+2)(13x+2)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-4\">\\((2x+3)(3x+2)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When factoring this polynomial, first set up two empty sets of parentheses that are multiplied by each other.<\/p>\n<p style=\"text-align: center\">\\((\\:\\:)(\\:\\:)\\)<\/p>\n<p>Now look at the first term \\(6x^2\\). This can be factored into  \\(3x\\) times \\(2x\\) because the product is still \\(6x\\). Now we have:<\/p>\n<p style=\"text-align: center\">\\((2x)(3x)\\)<\/p>\n<p>Now look at the other two terms from the original polynomial: \\(13x + 6\\). What two numbers will multiply to 6 but combine to \\(13x\\) when distributed?<\/p>\n<p>The numbers 2 and 3 will multiply to 6, and when distributed using FOIL, they will combine to \\(13x\\).<\/p>\n<p>So our factors now become \\((2x+3)(3x+2)\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nDivide the following polynomials:<\/p>\n<div class=\"yellow-math-quote\">\\((x^2+11x+10)\\div(x+1)\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">\\((x+20)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">\\((x+10)\\)<\/div><div class=\"PQ\"  id=\"PQ-3-3\">\\((x-5)\\)<\/div><div class=\"PQ\"  id=\"PQ-3-4\">\\((x-17)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The process follows the same procedures as long division. The answer can be checked by multiplying \\((x+1)(x+10)\\). The product will be \\((x^2+11x+10)\\), so we know that the two factors are correct. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Long-division-polynomials-example.svg\" alt=\"Long division of (x^2 + 11x + 10) by (x + 1), showing steps and a remainder of zero.\" width=\"214\" height=\"160\" class=\"aligncenter size-full wp-image-274159\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>Simplify the following expression by removing the common factors. <\/p>\n<div class=\"yellow-math-quote\">\\(\\dfrac{(2x^2+5x-3)}{(6x^2+18x)}\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\"><span style=\"font-size: 120%\">\\(\\frac{(3x-1)}{2x}\\)<\/span><\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\"><span style=\"font-size: 120%\">\\(\\frac{(2x-1)}{6x}\\)<\/span><\/div><div class=\"PQ\"  id=\"PQ-4-3\"><span style=\"font-size: 120%\">\\(\\frac{(4x-3)}{6x}\\)<\/span><\/div><div class=\"PQ\"  id=\"PQ-4-4\"><span style=\"font-size: 120%\">\\(\\frac{(2x-2)}{3x}\\)<\/span><\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The expression \\((2x^2+5x-3)\\) can be factored into \\((2x\u22121)(x+3)\\).<\/p>\n<p>The expression \\((6x^2+18x)\\) can be factored into \\(6x(x+3)\\). <\/p>\n<p>Both have the common factor of \\((x+3)\\), so when divided, this portion will cancel out.<\/p>\n<p>When \\(\\frac{(x+3)}{(x+3)}\\) cancels out, \\(\\frac{(2x-1)}{6x}\\) is all that remains. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nSimplify the following polynomial by multiplying:<\/p>\n<div class=\"yellow-math-quote\">\\(4x(x^2+3)\\)<\/div>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">\\(3x+12x\\)<\/div><div class=\"PQ\"  id=\"PQ-5-2\">\\(4x^4+12x\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">\\(4x^3+12x\\)<\/div><div class=\"PQ\"  id=\"PQ-5-4\">\\(4x^3+2x\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>To simplify the polynomial we can distribute the term \\(4x\\) into each term within the parenthesis.<\/p>\n<p style=\"text-align:center; line-height: 35px;\">\n\\(4x\\times x^2=4x^3\\)<br \/>\n\\(4x \\times 3=12x\\)<\/p>\n<p>The product of \\(4x(x^2+3)\\) is \\(4x^3+12x\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/algebra-i\/\">Return to Algebra I Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Algebra I Videos<\/p>\n","protected":false},"author":1,"featured_media":100249,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-1193","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-algebra-basics-videos","7":"page_category-math-advertising-group","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/1193","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=1193"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/1193\/revisions"}],"predecessor-version":[{"id":279430,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/1193\/revisions\/279430"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100249"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=1193"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}