{"id":747,"date":"2021-09-15T17:35:09","date_gmt":"2021-09-15T17:35:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=747"},"modified":"2021-12-30T21:50:33","modified_gmt":"2021-12-30T21:50:33","slug":"8-1-definition-of-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/8-1-definition-of-polynomials\/","title":{"raw":"8.1: Polynomials in One Variable","rendered":"8.1: Polynomials in One Variable"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Define a polynomial<\/li>\r\n \t<li>Determine the degree of a polynomial<\/li>\r\n \t<li>Classify a polynomial as a monomial, binomial or trinomial<\/li>\r\n \t<li>Determine the leading term and leading coefficient of a polynomial<\/li>\r\n \t<li>Write a polynomial in standard form (descending order)<\/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>Term<\/strong>: a number, variable, or product or quotient of numbers and variables<\/li>\r\n \t<li><strong>Coefficient<\/strong>: a number multiplied onto a variable; the number in a term<\/li>\r\n \t<li><strong>Algebraic expression<\/strong>: terms that are combined using operations like [latex]+,\\;-,\\;\\times,\\;\\div[\/latex]. etc.<\/li>\r\n \t<li><strong>Polynomial<\/strong>: an algebraic expression of the form [latex]a_{n}x^n+a_{n-1}x^{n-1}+ ... + a_{1}x+a_{0}[\/latex]<\/li>\r\n \t<li><strong>Monomial<\/strong>:a polynomial with one terms<\/li>\r\n \t<li><strong>Binomial<\/strong>: a polynomial with two terms<\/li>\r\n \t<li><strong>Trinomial<\/strong>:\u00a0a polynomial with three terms<\/li>\r\n \t<li><strong>Degree<\/strong>: the largest exponent<\/li>\r\n \t<li><strong>Descending order<\/strong>:\u00a0written from highest to lowest terms based on degree<\/li>\r\n \t<li><strong>Standard Form<\/strong>: a polynomial written in descending order<\/li>\r\n \t<li><strong>Leading term<\/strong>: the monomial with the highest degree<\/li>\r\n \t<li><strong>Leading coefficient<\/strong>: the coefficient of the leading term<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title1\">Polynomials in One Variable<\/h2>\r\nWe previously defined a <em><strong>term<\/strong><\/em> to be a number, a variable, or the product or quotient of numbers and variables. For example, [latex]6x,\\; \\frac{4x^2}{5y^3},\\; -4x^3y^2[\/latex] are all terms.\u00a0 The <em><strong>coefficient<\/strong><\/em> of a term is the number in the term. So, for the list of examples just given, the coefficients are, [latex]6,\\;\\frac{4}{5},\\;-4[\/latex]. When algebraic terms are combined using <em><strong>operations<\/strong><\/em> like [latex]+,\\;-,\\;\\times,\\;\\div[\/latex]. etc., they form an <em><strong>algebraic expression<\/strong><\/em>.\u00a0 For example, [latex]5x^3-\\frac{3}{4x^2}+6[\/latex] is an algebraic expression containing three terms.\r\n\r\nA <em><strong>polynomial<\/strong><\/em>\u00a0in one variable is a special algebraic expression that consists of a term or a sum (or difference) of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and a variable with a whole number exponent.\r\n\r\nFor example,\r\n\r\n[latex]5x^2+7x-3[\/latex] is a polynomial with three terms: [latex]5x^2,\\;7x,\\;-3[\/latex]. The coefficient of [latex]x^2[\/latex] is [latex]5[\/latex]; the coefficient of [latex]x[\/latex] is [latex]7[\/latex]; and [latex]-3[\/latex] is the constant term.\r\n\r\n[latex]-\\frac{4}{3}x^9+8x^4-\\frac{3}{5}x^2-9[\/latex] is a polynomial with four terms: [latex]-\\frac{4}{3}x^9,\\;8x^4,\\;-\\frac{3}{5}x^2,\\;-9[\/latex]. The coefficient of [latex]x^9[\/latex] is [latex]-\\frac{4}{3}[\/latex]; the coefficient of [latex]x^4[\/latex] is [latex]8[\/latex]; the coefficient of [latex]x^2[\/latex] is [latex]\\frac{3}{5}[\/latex]; the constant term [latex]-9[\/latex].\r\n\r\n[latex]\\sqrt{5}x^3[\/latex] and [latex]4.563[\/latex] are polynomials with just one term.\r\n\r\nHowever, [latex]\\frac{3}{x},\\;\\;x^{\\frac{1}{2}},\\;\\;\\frac{4}{y^2}-4x^3[\/latex] are NOT polynomials because the exponents on the variables are not whole numbers. Whole number exponents exclude radicals (fractional exponents) and variables on the denominator of a fraction (negative exponents).\r\n\r\nThe following table is intended to help us tell the difference between what is a polynomial and what is not.\r\n<table>\r\n<thead>\r\n<tr>\r\n<td>IS a Polynomial<\/td>\r\n<td>Is NOT a Polynomial<\/td>\r\n<td>Because<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\r\n<td>[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\r\n<td>Polynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\frac{y}{4}-y^3[\/latex]<\/td>\r\n<td>[latex]\\frac{2}{y}+4[\/latex]<\/td>\r\n<td>Polynomials only have variables in the numerator<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\r\n<td>\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\r\n<td>Variables under a root are not allowed in polynomials<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe basic building block of a polynomial is a <em><b>monomial<\/b><\/em>. A monomial is a term of the form [latex]a{x}^{n}[\/latex], where [latex]a[\/latex] is a constant and [latex]n[\/latex] is a whole number. A monomial can be a number, a variable, or the product of a number and variable with an exponent. The number part of the term is called the <em><b>coefficient<\/b><\/em>.\r\n\r\nExamples of monomials:\r\n<ul>\r\n \t<li>constant: [latex]{6}[\/latex]<\/li>\r\n \t<li>variable with a coefficient of [latex]1[\/latex]: [latex]{x}[\/latex]<\/li>\r\n \t<li>product of a coefficient and a variable: [latex]{6x}[\/latex]<\/li>\r\n \t<li>product of a coefficient and a variable with a whole number exponent: [latex]6{x}^{3}[\/latex]<\/li>\r\n<\/ul>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064120\/image003.jpg\" alt=\"The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent.\" width=\"183\" height=\"82\" \/>\r\n\r\nThe coefficient can be any real number, including [latex]0[\/latex]. The exponent of the variable must be a whole number (i.e., [latex]0, 1, 2, 3,...[\/latex]).\u00a0\u00a0The value of the exponent is the <em><b>degree<\/b><\/em> of the monomial. Remember that a variable, such as [latex]x[\/latex], that appears to have no exponent really has an exponent of [latex]1[\/latex]: [latex]x=x^1[\/latex]. A monomial with no variable has a degree of \u00a0[latex]0[\/latex].\u00a0A number such as [latex]3[\/latex] can be written as [latex]3x^{0}[\/latex], s<span style=\"font-size: 1em;\">ince\u00a0[latex]x^{0}=1[\/latex] if [latex]x\\neq0<\/span><span style=\"font-size: 1em;\">[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">.\u00a0 A monomial cannot have a variable in the denominator as that would have a negative exponent, nor can it have a variable under a radical as that would have a fractional exponent.<\/span>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nIdentify the coefficient, variable, and degree\u00a0of the monomial:\r\n1) [latex]9[\/latex]\r\n2) [latex]x[\/latex]\r\n3) [latex]\\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex]\r\n<h4>Solution<\/h4>\r\n1) [latex]9[\/latex] is a constant so there is no variable. Since [latex]9=9x^0[\/latex], the degree is [latex]0[\/latex] and the coefficient is [latex]9[\/latex].\u00a0[latex]9[\/latex] is called a constant term.\r\n\r\n2) The variable is [latex]x[\/latex].\r\n\r\nThe degree of [latex]x[\/latex] is [latex]1[\/latex], because [latex]x=x^{1}[\/latex].\r\n\r\nThe coefficient of [latex]x[\/latex] is [latex]1[\/latex], because [latex]x=1x[\/latex].\r\n\r\n3) The variable is [latex]k[\/latex].\r\n\r\nThe degree of[latex]\\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex] is [latex]8[\/latex].\r\n\r\nThe coefficient of [latex]k^{8}[\/latex]\u00a0is [latex] \\displaystyle \\frac{3}{5}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nIdentify the coefficient, variable, and degree\u00a0of the monomial:\r\n\r\n1. [latex]6x^7[\/latex]\r\n\r\n2. [latex]-9x[\/latex]\r\n\r\n3. [latex]\\frac{4}{7}[\/latex]\r\n\r\n[reveal-answer q=\"hjm334\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm334\"]\r\n\r\n1. Coefficient = [latex]6[\/latex]; variable = [latex]x[\/latex]; degree = [latex]7[\/latex]\r\n\r\n2.\u00a0Coefficient = [latex]-9[\/latex]; variable = [latex]x[\/latex]; degree = [latex]1[\/latex]\r\n\r\n3.\u00a0Coefficient = [latex]\\frac{4}{7}[\/latex]; no variable; degree = [latex]0[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Classifying Polynomials<\/h2>\r\nThe word\u00a0<strong><i>polynomial<\/i><\/strong>\u00a0joins two diverse roots: the Greek\u00a0<i>poly<\/i>, meaning \"many\", and the Latin\u00a0<i>nomen<\/i>, meaning \"name.\" Consequently, polynomial means many names, or in math, many terms. In this section, we will look at different ways that we classify polynomials. First, we will classify polynomials by the number of terms in the polynomial and then we will classify them by the largest exponent.\r\n<h3>By Number of Terms<\/h3>\r\nA polynomial is defined as a monomial or the sum (or difference) of monomials. This means that a polynomial that contains just one term is called a <em><strong>monomial<\/strong><\/em>.\u00a0 A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <em><strong>binomial<\/strong><\/em>.\u00a0 A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <em><strong>trinomial<\/strong><\/em>. After three terms we tend to simply refer to them as polynomials.\r\n<div class=\"textbox shaded\">\r\n<h3>Polynomials<\/h3>\r\npolynomial\u2014A monomial; or two or more monomials, combined by addition (or subtraction)\r\nmonomial\u2014A polynomial with exactly one term (\"mono\" means one)\r\nbinomial\u2014 A polynomial with exactly two terms (\"bi\" means two)\r\ntrinomial\u2014A polynomial with exactly three terms (\"tri\" means three)\r\n\r\n<\/div>\r\nHere are some examples of polynomials classified by the number of terms.\r\n<table id=\"fs-id1171105397687\" class=\"unnumbered column-header\" summary=\"The table has four rows and four columns. The first column lists Polynomial, Monomial, Binomial, and Trinomial. The columns list examples of each. The first row lists b plu 1, 4y squared minus 7y plus 2, and 5x to the fifth minus 4x to the fourth plus x cubed minus 9x plus 1. The second row lists 5, 4b squared, and negative9x cubed. The third row lists 3a minus 7, y squared minus 9, and 17x cubed plus 14x squared. The fourth row lists x squared minus 5x plus 6, 4y squared minus 7y plus 2, and 5a to the fourth minus 3a cubed plus a.\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>Polynomial<\/strong><\/td>\r\n<td>[latex]b+1[\/latex]<\/td>\r\n<td>[latex]4{y}^{2}-7y+2[\/latex]<\/td>\r\n<td>[latex]5{x}^{5}-4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>Monomial<\/strong><\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]4{b}^{2}[\/latex]<\/td>\r\n<td>[latex]-9{x}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>Binomial<\/strong><\/td>\r\n<td>[latex]3a - 7[\/latex]<\/td>\r\n<td>[latex]{y}^{2}-9[\/latex]<\/td>\r\n<td>[latex]17{x}^{3}+14{x}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>Trinomial<\/strong><\/td>\r\n<td>[latex]{x}^{2}-5x+6[\/latex]<\/td>\r\n<td>[latex]4{y}^{2}-7y+2[\/latex]<\/td>\r\n<td>[latex]5{a}^{4}-3{a}^{3}+a[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNotice that every monomial, binomial, and trinomial is also a polynomial. in other words:\r\n<p style=\"text-align: center;\">{monomials} [latex]\\subset[\/latex] {binomials} [latex]\\subset[\/latex] {trinomials}\u00a0[latex]\\subset[\/latex] {polynomials}<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nDetermine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:\r\n\r\n1. [latex]8{x}^{2}-7x - 9[\/latex]\r\n2. [latex]-5{a}^{4}[\/latex]\r\n3. [latex]{x}^{4}-7{x}^{3}-6{x}^{2}+5x+2[\/latex]\r\n4. [latex]11 - 4{y}^{3}[\/latex]\r\n5. [latex]n[\/latex]\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-118\" class=\"unnumbered unstyled\" summary=\".\">\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th>Polynomial<\/th>\r\n<th>Number of terms<\/th>\r\n<th>Type<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td>[latex]8{x}^{2}-7x - 9[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>Trinomial<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>[latex]-5{a}^{4}[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>Monomial<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.<\/td>\r\n<td>[latex]{x}^{4}-7{x}^{3}-6{x}^{2}+5x+2[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>Polynomial<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4.<\/td>\r\n<td>[latex]11 - 4{y}^{3}[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>Binomial<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5.<\/td>\r\n<td>[latex]n[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>Monomial<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFor the following expressions, determine whether they are a polynomial. If so, categorize them as a monomial, binomial, or trinomial.\r\n<ol>\r\n \t<li>[latex]\\frac{x-3}{1-x}+x^2[\/latex]<\/li>\r\n \t<li>[latex]t^2+2t-3[\/latex]<\/li>\r\n \t<li>[latex]x^3+\\frac{x}{8}[\/latex]<\/li>\r\n \t<li>[latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>[latex]\\frac{x-3}{1-x}+x^2[\/latex] is not a polynomial because it violates the rule that polynomials cannot have variables in the denominator of a fraction (negative exponent).<\/li>\r\n \t<li>[latex]t^2+2t-3[\/latex] is a polynomial because it is an expression whose monomial terms are connected by addition and subtraction. \u00a0There are three terms in this polynomial so it is a trinomial.<\/li>\r\n \t<li>[latex]x^3+\\frac{x}{8}[\/latex]is a polynomial because it is an expression whose monomial terms are connected by addition and subtraction. \u00a0There are two terms in this polynomial so it is a binomial.<\/li>\r\n \t<li>[latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]\u00a0is not a polynomial because it violates the rule that polynomials cannot have variables\u00a0under a root (fractional exponent).<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146073[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video shows more examples of how to identify and categorize polynomials.\r\n\r\nhttps:\/\/youtu.be\/nPAqfuoSbPI\r\n<h3>By Degree<\/h3>\r\nWe can find the <em><strong>degree<\/strong><\/em> of a one variable polynomial by identifying the highest power of the variable that occurs in the polynomial. Polynomials can be classified by the degree of the polynomial. The degree of a polynomial is the degree of its highest degree monomial term. So the degree of [latex]2x^{3}+3x^{2}+8x+5[\/latex] is [latex]3[\/latex].\r\n\r\nA one variable polynomial is said to be written in <em><strong>standard form<\/strong><\/em> when the terms are arranged from the highest degree to the lowest degree. This is referred to as <em><strong>descending order<\/strong><\/em>. When it is written in standard form it is easy to determine the degree of the polynomial.\u00a0 The term with the highest degree is called the<em> <strong>leading term<\/strong><\/em> because it is written first in standard form. The coefficient of the leading term is called the <em><strong>leading coefficient<\/strong><\/em>.\r\n\r\n<img class=\"wp-image-2550 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15150341\/Screen-Shot-2016-07-15-at-8.03.13-AM-300x150.png\" alt=\"4x^3 - 9x^2 + 6x, with the text &quot;degree = 3&quot; and an arrow pointing at the exponent on x^3, and the text &quot;leading term =4&quot; with an arrow pointing at the 4. \" width=\"504\" height=\"252\" \/>\r\n<div class=\"textbox shaded\">\r\n<h3>Degree of a ONE VARIABLE Polynomial<\/h3>\r\nThe degree of a term is the exponent of its variable.\r\nThe degree of a constant is [latex]0[\/latex].\r\nThe degree of a polynomial is the highest degree of all its terms.\r\n\r\n<\/div>\r\nWhen the coefficient of a polynomial term is\u00a0[latex]0[\/latex], e.g., [latex]0x^2[\/latex], we usually do not write the term at all because [latex]0x^n=0[\/latex], and adding\u00a0[latex]0[\/latex] does not change the value of the polynomial. However, we will see later, that occasionally, it is necessary to include all terms of the polynomial, including terms with zero coefficients.\r\n\r\nA term without a variable is called a <em><b>constant <\/b><\/em>term, and the degree of that term is\u00a0[latex]0[\/latex].\u00a0 This is because any constant [latex]a[\/latex] can be written as [latex]ax^0[\/latex], since [latex]x^0=1[\/latex]. For example, we can write the polynomial [latex]3x+13[\/latex] as\u00a0[latex]3x^{1}+13x^{0}[\/latex].\u00a0 Although this is not how we would normally write this, it allows us to see that [latex]13[\/latex] is the constant term because its degree is 0, and the degree of\u00a0[latex]3x[\/latex] is 1.\u00a0 Consequently, the degree of the binomial[latex]3x+13[\/latex] is 1.\r\n\r\nIf a polynomial does not have a constant term, like the polynomial [latex]14x^{3}+3x[\/latex] we say that the constant term is\u00a0[latex]0[\/latex].\r\n\r\nLet's see how this works by looking at several polynomials. We'll take it step by step, starting with monomials, and then progressing to polynomials with more terms.\r\n\r\nRemember: Any variable base written without an exponent has an implied exponent of [latex]1[\/latex], and any constant term has an implied exponent of zero.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 20%;\"><strong>Monomials<\/strong><\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]5[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]4b^2[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]-9x^3[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]-18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\">Degree<\/td>\r\n<td style=\"width: 20%; text-align: center;\">0<\/td>\r\n<td style=\"width: 20%; text-align: center;\">2<\/td>\r\n<td style=\"width: 20%; text-align: center;\">3<\/td>\r\n<td style=\"width: 20%; text-align: center;\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\"><strong>Binomials<\/strong><\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]b+\\frac{1}{7}[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]3a-7[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]y^2-9[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]17x^3+14x^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\">Degree of each term<\/td>\r\n<td style=\"width: 20%; text-align: center;\">1\u00a0 \u00a00<\/td>\r\n<td style=\"width: 20%; text-align: center;\">1\u00a0 0<\/td>\r\n<td style=\"width: 20%; text-align: center;\">2\u00a0 \u00a00<\/td>\r\n<td style=\"width: 20%; text-align: center;\">3\u00a0 \u00a02<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\">Degree of polynomial<\/td>\r\n<td style=\"width: 20%; text-align: center;\">1<\/td>\r\n<td style=\"width: 20%; text-align: center;\">1<\/td>\r\n<td style=\"width: 20%; text-align: center;\">2<\/td>\r\n<td style=\"width: 20%; text-align: center;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%;\"><strong>Trinomial<\/strong><\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]x^2-5x+6[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]4y^2-7y+2[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]5a^12-3a^9+a[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]\\frac{1}{2}x^4+2x^2-5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\">Degree of each term<\/td>\r\n<td style=\"width: 20%; text-align: center;\">2\u00a0 \u00a0 1\u00a0 \u00a0 0<\/td>\r\n<td style=\"width: 20%; text-align: center;\">2\u00a0 \u00a0 1\u00a0 \u00a0 0<\/td>\r\n<td style=\"width: 20%; text-align: center;\">12\u00a0 \u00a0 9\u00a0 \u00a0 1<\/td>\r\n<td style=\"width: 20%; text-align: center;\">4\u00a0 \u00a0 2\u00a0 \u00a0 0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\">Degree of polynomial<\/td>\r\n<td style=\"width: 20%; text-align: center;\">2<\/td>\r\n<td style=\"width: 20%; text-align: center;\">2<\/td>\r\n<td style=\"width: 20%; text-align: center;\">12<\/td>\r\n<td style=\"width: 20%; text-align: center;\">4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: left;\"><strong>Polynomial<\/strong><\/td>\r\n<td style=\"width: 20%; text-align: center;\"><span style=\"font-size: 12.8px;\">[latex]-7x^8+9x^6-7x^4+x^2-8[\/latex]<\/span><\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]\\frac{3}{5}x^7-8x^6+\\frac{7}{8}x+9[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]5.4x^{17}-8.1x^{13}+2.8x^{10}-6.5x^5[\/latex]<\/td>\r\n<td style=\"width: 20%; text-align: center;\">[latex]7+6y-9y^2+6y^3-3y^4+y^5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\">Degree of each term<\/td>\r\n<td style=\"width: 20%; text-align: center;\"><span style=\"font-size: 12.8px;\">8\u00a0 \u00a0 6\u00a0 \u00a0 4\u00a0 \u00a0 2\u00a0 \u00a0 0<\/span><\/td>\r\n<td style=\"width: 20%; text-align: center;\">7\u00a0 \u00a0 6\u00a0 \u00a0 1\u00a0 \u00a0 0<\/td>\r\n<td style=\"width: 20%; text-align: center;\">17\u00a0 \u00a0 13\u00a0 \u00a0 10\u00a0 \u00a0 5<\/td>\r\n<td style=\"width: 20%; text-align: center;\">0\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a03\u00a0 \u00a04\u00a0 \u00a05<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 20%; text-align: right;\">Degree of polynomial<\/td>\r\n<td style=\"width: 20%; text-align: center;\">8<\/td>\r\n<td style=\"width: 20%; text-align: center;\">7<\/td>\r\n<td style=\"width: 20%; text-align: center;\">17<\/td>\r\n<td style=\"width: 20%; text-align: center;\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the degree of the following polynomials:\r\n\r\n1. [latex]4x[\/latex]\r\n2. [latex]3{x}^{3}-5x+7[\/latex]\r\n3. [latex]-11[\/latex]\r\n4. [latex]-6{x}^{2}+9x - 3[\/latex]\r\n5. [latex]8x+2[\/latex]\r\n<h4>Solution<\/h4>\r\n<table id=\"eip-id1168468469502\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.<\/td>\r\n<td>[latex]4x[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The exponent of [latex]x[\/latex] is one. [latex]x={x}^{1}[\/latex]<\/td>\r\n<td>The degree is [latex]1[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2.<\/td>\r\n<td>[latex]3{x}^{3}-5x+7[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The highest degree of all the terms is [latex]3[\/latex].<\/td>\r\n<td>The degree is [latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>3.<\/td>\r\n<td>[latex]-11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The degree of a constant is [latex]0[\/latex].<\/td>\r\n<td>The degree is [latex]0[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>4.<\/td>\r\n<td>[latex]-6{x}^{2}+9x - 3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The highest degree of all the terms is [latex]2[\/latex].<\/td>\r\n<td>The degree is [latex]2[\/latex].<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>5.<\/td>\r\n<td>[latex]8x+2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>The highest degree of all the terms is [latex]1[\/latex].<\/td>\r\n<td>The degree is [latex]1[\/latex].<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nWorking with polynomials is easier when we list the terms in <em><strong>descending order<\/strong><\/em> of degrees. When a polynomial is written this way, it is said to be in <em><strong>standard form<\/strong><\/em>. Look back at the polynomials in the previous example. Notice that they are all written in standard form. Get in the habit of writing polynomials in standard form\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146070[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFor the following polynomials, write them in standard form (descending order), identify the degree, the leading term, and the leading coefficient.\r\n<ol>\r\n \t<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\r\n \t<li>[latex]5{t}^{5}+7t-2{t}^{3}[\/latex]<\/li>\r\n \t<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\r\n<\/ol>\r\n<h4>Solution<\/h4>\r\n<ol>\r\n \t<li>Standard form: [latex]-4x^32x^2+3[\/latex]. The highest power of [latex]x[\/latex]\u00a0is\u00a0[latex]3[\/latex], so the degree is\u00a0[latex]3[\/latex]. The leading term is [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\r\n \t<li>Standard form: [latex]5t^5-2t^3+7t[\/latex].\u00a0 The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\r\n \t<li>Standard form:[latex]-p^3+6p-2[\/latex].\u00a0 The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\r\n<\/ol>\r\n&nbsp;\r\n\r\n<\/div>\r\nThe following video shows how to identify the terms, leading coefficient, and degree of a polynomial.\r\n\r\nhttps:\/\/youtu.be\/3u16B2PN9zk\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFor the following polynomials, write them in standard form (descending order), identify the degree, the leading term, and the leading coefficient.\r\n<ol>\r\n \t<li>[latex]9+2{x}^{2}-5{x}^{5}[\/latex]<\/li>\r\n \t<li>[latex]-5{t}^{5}+7+8t-4{t}^{3}[\/latex]<\/li>\r\n \t<li>[latex]4p^4-{p}^{7}-2p[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm847\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm847\"]\r\n<ol>\r\n \t<li>Standard form: [latex]-5x^5+2x^2+9[\/latex]; Degree = 5; Leading term = [latex]-5x^5[\/latex]; Leading coefficient =\u00a0[latex]-5[\/latex]<\/li>\r\n \t<li>Standard form: [latex]-5t^5-4t^3+8t+7[\/latex]; Degree = 5; Leading term = [latex]-5t^5[\/latex]; Leading coefficient =\u00a0[latex]-5[\/latex]<\/li>\r\n \t<li>Standard form: [latex]-p^7+4p^4-2p[\/latex]; Degree = 7; Leading term = [latex]-p^7[\/latex]; Leading coefficient =\u00a0[latex]-1[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Define a polynomial<\/li>\n<li>Determine the degree of a polynomial<\/li>\n<li>Classify a polynomial as a monomial, binomial or trinomial<\/li>\n<li>Determine the leading term and leading coefficient of a polynomial<\/li>\n<li>Write a polynomial in standard form (descending order)<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key WORDS<\/h3>\n<ul>\n<li><strong>Term<\/strong>: a number, variable, or product or quotient of numbers and variables<\/li>\n<li><strong>Coefficient<\/strong>: a number multiplied onto a variable; the number in a term<\/li>\n<li><strong>Algebraic expression<\/strong>: terms that are combined using operations like [latex]+,\\;-,\\;\\times,\\;\\div[\/latex]. etc.<\/li>\n<li><strong>Polynomial<\/strong>: an algebraic expression of the form [latex]a_{n}x^n+a_{n-1}x^{n-1}+ ... + a_{1}x+a_{0}[\/latex]<\/li>\n<li><strong>Monomial<\/strong>:a polynomial with one terms<\/li>\n<li><strong>Binomial<\/strong>: a polynomial with two terms<\/li>\n<li><strong>Trinomial<\/strong>:\u00a0a polynomial with three terms<\/li>\n<li><strong>Degree<\/strong>: the largest exponent<\/li>\n<li><strong>Descending order<\/strong>:\u00a0written from highest to lowest terms based on degree<\/li>\n<li><strong>Standard Form<\/strong>: a polynomial written in descending order<\/li>\n<li><strong>Leading term<\/strong>: the monomial with the highest degree<\/li>\n<li><strong>Leading coefficient<\/strong>: the coefficient of the leading term<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title1\">Polynomials in One Variable<\/h2>\n<p>We previously defined a <em><strong>term<\/strong><\/em> to be a number, a variable, or the product or quotient of numbers and variables. For example, [latex]6x,\\; \\frac{4x^2}{5y^3},\\; -4x^3y^2[\/latex] are all terms.\u00a0 The <em><strong>coefficient<\/strong><\/em> of a term is the number in the term. So, for the list of examples just given, the coefficients are, [latex]6,\\;\\frac{4}{5},\\;-4[\/latex]. When algebraic terms are combined using <em><strong>operations<\/strong><\/em> like [latex]+,\\;-,\\;\\times,\\;\\div[\/latex]. etc., they form an <em><strong>algebraic expression<\/strong><\/em>.\u00a0 For example, [latex]5x^3-\\frac{3}{4x^2}+6[\/latex] is an algebraic expression containing three terms.<\/p>\n<p>A <em><strong>polynomial<\/strong><\/em>\u00a0in one variable is a special algebraic expression that consists of a term or a sum (or difference) of terms in which each term is a\u00a0real\u00a0number, a variable, or the product of a\u00a0real\u00a0number and a variable with a whole number exponent.<\/p>\n<p>For example,<\/p>\n<p>[latex]5x^2+7x-3[\/latex] is a polynomial with three terms: [latex]5x^2,\\;7x,\\;-3[\/latex]. The coefficient of [latex]x^2[\/latex] is [latex]5[\/latex]; the coefficient of [latex]x[\/latex] is [latex]7[\/latex]; and [latex]-3[\/latex] is the constant term.<\/p>\n<p>[latex]-\\frac{4}{3}x^9+8x^4-\\frac{3}{5}x^2-9[\/latex] is a polynomial with four terms: [latex]-\\frac{4}{3}x^9,\\;8x^4,\\;-\\frac{3}{5}x^2,\\;-9[\/latex]. The coefficient of [latex]x^9[\/latex] is [latex]-\\frac{4}{3}[\/latex]; the coefficient of [latex]x^4[\/latex] is [latex]8[\/latex]; the coefficient of [latex]x^2[\/latex] is [latex]\\frac{3}{5}[\/latex]; the constant term [latex]-9[\/latex].<\/p>\n<p>[latex]\\sqrt{5}x^3[\/latex] and [latex]4.563[\/latex] are polynomials with just one term.<\/p>\n<p>However, [latex]\\frac{3}{x},\\;\\;x^{\\frac{1}{2}},\\;\\;\\frac{4}{y^2}-4x^3[\/latex] are NOT polynomials because the exponents on the variables are not whole numbers. Whole number exponents exclude radicals (fractional exponents) and variables on the denominator of a fraction (negative exponents).<\/p>\n<p>The following table is intended to help us tell the difference between what is a polynomial and what is not.<\/p>\n<table>\n<thead>\n<tr>\n<td>IS a Polynomial<\/td>\n<td>Is NOT a Polynomial<\/td>\n<td>Because<\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]2x^2-\\frac{1}{2}x -9[\/latex]<\/td>\n<td>[latex]\\frac{2}{x^{2}}+x[\/latex]<\/td>\n<td>Polynomials only have variables in the numerator<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\frac{y}{4}-y^3[\/latex]<\/td>\n<td>[latex]\\frac{2}{y}+4[\/latex]<\/td>\n<td>Polynomials only have variables in the numerator<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\sqrt{12}\\left(a\\right)+9[\/latex]<\/td>\n<td>\u00a0[latex]\\sqrt{a}+7[\/latex]<\/td>\n<td>Variables under a root are not allowed in polynomials<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The basic building block of a polynomial is a <em><b>monomial<\/b><\/em>. A monomial is a term of the form [latex]a{x}^{n}[\/latex], where [latex]a[\/latex] is a constant and [latex]n[\/latex] is a whole number. A monomial can be a number, a variable, or the product of a number and variable with an exponent. The number part of the term is called the <em><b>coefficient<\/b><\/em>.<\/p>\n<p>Examples of monomials:<\/p>\n<ul>\n<li>constant: [latex]{6}[\/latex]<\/li>\n<li>variable with a coefficient of [latex]1[\/latex]: [latex]{x}[\/latex]<\/li>\n<li>product of a coefficient and a variable: [latex]{6x}[\/latex]<\/li>\n<li>product of a coefficient and a variable with a whole number exponent: [latex]6{x}^{3}[\/latex]<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04064120\/image003.jpg\" alt=\"The expression 6x to the power of 3. 6 is the coefficient, x is the variable, and the power of 3 is the exponent.\" width=\"183\" height=\"82\" \/><\/p>\n<p>The coefficient can be any real number, including [latex]0[\/latex]. The exponent of the variable must be a whole number (i.e., [latex]0, 1, 2, 3,...[\/latex]).\u00a0\u00a0The value of the exponent is the <em><b>degree<\/b><\/em> of the monomial. Remember that a variable, such as [latex]x[\/latex], that appears to have no exponent really has an exponent of [latex]1[\/latex]: [latex]x=x^1[\/latex]. A monomial with no variable has a degree of \u00a0[latex]0[\/latex].\u00a0A number such as [latex]3[\/latex] can be written as [latex]3x^{0}[\/latex], s<span style=\"font-size: 1em;\">ince\u00a0[latex]x^{0}=1[\/latex] if [latex]x\\neq0<\/span><span style=\"font-size: 1em;\">[\/latex]<\/span><span style=\"font-size: 1rem; text-align: initial;\">.\u00a0 A monomial cannot have a variable in the denominator as that would have a negative exponent, nor can it have a variable under a radical as that would have a fractional exponent.<\/span><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Identify the coefficient, variable, and degree\u00a0of the monomial:<br \/>\n1) [latex]9[\/latex]<br \/>\n2) [latex]x[\/latex]<br \/>\n3) [latex]\\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex]<\/p>\n<h4>Solution<\/h4>\n<p>1) [latex]9[\/latex] is a constant so there is no variable. Since [latex]9=9x^0[\/latex], the degree is [latex]0[\/latex] and the coefficient is [latex]9[\/latex].\u00a0[latex]9[\/latex] is called a constant term.<\/p>\n<p>2) The variable is [latex]x[\/latex].<\/p>\n<p>The degree of [latex]x[\/latex] is [latex]1[\/latex], because [latex]x=x^{1}[\/latex].<\/p>\n<p>The coefficient of [latex]x[\/latex] is [latex]1[\/latex], because [latex]x=1x[\/latex].<\/p>\n<p>3) The variable is [latex]k[\/latex].<\/p>\n<p>The degree of[latex]\\displaystyle \\frac{3}{5}{{k}^{8}}[\/latex] is [latex]8[\/latex].<\/p>\n<p>The coefficient of [latex]k^{8}[\/latex]\u00a0is [latex]\\displaystyle \\frac{3}{5}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Identify the coefficient, variable, and degree\u00a0of the monomial:<\/p>\n<p>1. [latex]6x^7[\/latex]<\/p>\n<p>2. [latex]-9x[\/latex]<\/p>\n<p>3. [latex]\\frac{4}{7}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm334\">Show Answer<\/span><\/p>\n<div id=\"qhjm334\" class=\"hidden-answer\" style=\"display: none\">\n<p>1. Coefficient = [latex]6[\/latex]; variable = [latex]x[\/latex]; degree = [latex]7[\/latex]<\/p>\n<p>2.\u00a0Coefficient = [latex]-9[\/latex]; variable = [latex]x[\/latex]; degree = [latex]1[\/latex]<\/p>\n<p>3.\u00a0Coefficient = [latex]\\frac{4}{7}[\/latex]; no variable; degree = [latex]0[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Classifying Polynomials<\/h2>\n<p>The word\u00a0<strong><i>polynomial<\/i><\/strong>\u00a0joins two diverse roots: the Greek\u00a0<i>poly<\/i>, meaning &#8220;many&#8221;, and the Latin\u00a0<i>nomen<\/i>, meaning &#8220;name.&#8221; Consequently, polynomial means many names, or in math, many terms. In this section, we will look at different ways that we classify polynomials. First, we will classify polynomials by the number of terms in the polynomial and then we will classify them by the largest exponent.<\/p>\n<h3>By Number of Terms<\/h3>\n<p>A polynomial is defined as a monomial or the sum (or difference) of monomials. This means that a polynomial that contains just one term is called a <em><strong>monomial<\/strong><\/em>.\u00a0 A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <em><strong>binomial<\/strong><\/em>.\u00a0 A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <em><strong>trinomial<\/strong><\/em>. After three terms we tend to simply refer to them as polynomials.<\/p>\n<div class=\"textbox shaded\">\n<h3>Polynomials<\/h3>\n<p>polynomial\u2014A monomial; or two or more monomials, combined by addition (or subtraction)<br \/>\nmonomial\u2014A polynomial with exactly one term (&#8220;mono&#8221; means one)<br \/>\nbinomial\u2014 A polynomial with exactly two terms (&#8220;bi&#8221; means two)<br \/>\ntrinomial\u2014A polynomial with exactly three terms (&#8220;tri&#8221; means three)<\/p>\n<\/div>\n<p>Here are some examples of polynomials classified by the number of terms.<\/p>\n<table id=\"fs-id1171105397687\" class=\"unnumbered column-header\" summary=\"The table has four rows and four columns. The first column lists Polynomial, Monomial, Binomial, and Trinomial. The columns list examples of each. The first row lists b plu 1, 4y squared minus 7y plus 2, and 5x to the fifth minus 4x to the fourth plus x cubed minus 9x plus 1. The second row lists 5, 4b squared, and negative9x cubed. The third row lists 3a minus 7, y squared minus 9, and 17x cubed plus 14x squared. The fourth row lists x squared minus 5x plus 6, 4y squared minus 7y plus 2, and 5a to the fourth minus 3a cubed plus a.\">\n<tbody>\n<tr valign=\"top\">\n<td><strong>Polynomial<\/strong><\/td>\n<td>[latex]b+1[\/latex]<\/td>\n<td>[latex]4{y}^{2}-7y+2[\/latex]<\/td>\n<td>[latex]5{x}^{5}-4{x}^{4}+{x}^{3}+8{x}^{2}-9x+1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>Monomial<\/strong><\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]4{b}^{2}[\/latex]<\/td>\n<td>[latex]-9{x}^{3}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>Binomial<\/strong><\/td>\n<td>[latex]3a - 7[\/latex]<\/td>\n<td>[latex]{y}^{2}-9[\/latex]<\/td>\n<td>[latex]17{x}^{3}+14{x}^{2}[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>Trinomial<\/strong><\/td>\n<td>[latex]{x}^{2}-5x+6[\/latex]<\/td>\n<td>[latex]4{y}^{2}-7y+2[\/latex]<\/td>\n<td>[latex]5{a}^{4}-3{a}^{3}+a[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Notice that every monomial, binomial, and trinomial is also a polynomial. in other words:<\/p>\n<p style=\"text-align: center;\">{monomials} [latex]\\subset[\/latex] {binomials} [latex]\\subset[\/latex] {trinomials}\u00a0[latex]\\subset[\/latex] {polynomials}<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:<\/p>\n<p>1. [latex]8{x}^{2}-7x - 9[\/latex]<br \/>\n2. [latex]-5{a}^{4}[\/latex]<br \/>\n3. [latex]{x}^{4}-7{x}^{3}-6{x}^{2}+5x+2[\/latex]<br \/>\n4. [latex]11 - 4{y}^{3}[\/latex]<br \/>\n5. [latex]n[\/latex]<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-118\" class=\"unnumbered unstyled\" summary=\".\">\n<thead>\n<tr>\n<th><\/th>\n<th>Polynomial<\/th>\n<th>Number of terms<\/th>\n<th>Type<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1.<\/td>\n<td>[latex]8{x}^{2}-7x - 9[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>Trinomial<\/td>\n<\/tr>\n<tr>\n<td>2.<\/td>\n<td>[latex]-5{a}^{4}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>Monomial<\/td>\n<\/tr>\n<tr>\n<td>3.<\/td>\n<td>[latex]{x}^{4}-7{x}^{3}-6{x}^{2}+5x+2[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>Polynomial<\/td>\n<\/tr>\n<tr>\n<td>4.<\/td>\n<td>[latex]11 - 4{y}^{3}[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>Binomial<\/td>\n<\/tr>\n<tr>\n<td>5.<\/td>\n<td>[latex]n[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>Monomial<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>For the following expressions, determine whether they are a polynomial. If so, categorize them as a monomial, binomial, or trinomial.<\/p>\n<ol>\n<li>[latex]\\frac{x-3}{1-x}+x^2[\/latex]<\/li>\n<li>[latex]t^2+2t-3[\/latex]<\/li>\n<li>[latex]x^3+\\frac{x}{8}[\/latex]<\/li>\n<li>[latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>[latex]\\frac{x-3}{1-x}+x^2[\/latex] is not a polynomial because it violates the rule that polynomials cannot have variables in the denominator of a fraction (negative exponent).<\/li>\n<li>[latex]t^2+2t-3[\/latex] is a polynomial because it is an expression whose monomial terms are connected by addition and subtraction. \u00a0There are three terms in this polynomial so it is a trinomial.<\/li>\n<li>[latex]x^3+\\frac{x}{8}[\/latex]is a polynomial because it is an expression whose monomial terms are connected by addition and subtraction. \u00a0There are two terms in this polynomial so it is a binomial.<\/li>\n<li>[latex]\\frac{\\sqrt{y}}{2}-y-1[\/latex]\u00a0is not a polynomial because it violates the rule that polynomials cannot have variables\u00a0under a root (fractional exponent).<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146073\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146073&theme=oea&iframe_resize_id=ohm146073&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows more examples of how to identify and categorize polynomials.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine if an Expression is a Polynomial\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/nPAqfuoSbPI?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3>By Degree<\/h3>\n<p>We can find the <em><strong>degree<\/strong><\/em> of a one variable polynomial by identifying the highest power of the variable that occurs in the polynomial. Polynomials can be classified by the degree of the polynomial. The degree of a polynomial is the degree of its highest degree monomial term. So the degree of [latex]2x^{3}+3x^{2}+8x+5[\/latex] is [latex]3[\/latex].<\/p>\n<p>A one variable polynomial is said to be written in <em><strong>standard form<\/strong><\/em> when the terms are arranged from the highest degree to the lowest degree. This is referred to as <em><strong>descending order<\/strong><\/em>. When it is written in standard form it is easy to determine the degree of the polynomial.\u00a0 The term with the highest degree is called the<em> <strong>leading term<\/strong><\/em> because it is written first in standard form. The coefficient of the leading term is called the <em><strong>leading coefficient<\/strong><\/em>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2550 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15150341\/Screen-Shot-2016-07-15-at-8.03.13-AM-300x150.png\" alt=\"4x^3 - 9x^2 + 6x, with the text &quot;degree = 3&quot; and an arrow pointing at the exponent on x^3, and the text &quot;leading term =4&quot; with an arrow pointing at the 4.\" width=\"504\" height=\"252\" \/><\/p>\n<div class=\"textbox shaded\">\n<h3>Degree of a ONE VARIABLE Polynomial<\/h3>\n<p>The degree of a term is the exponent of its variable.<br \/>\nThe degree of a constant is [latex]0[\/latex].<br \/>\nThe degree of a polynomial is the highest degree of all its terms.<\/p>\n<\/div>\n<p>When the coefficient of a polynomial term is\u00a0[latex]0[\/latex], e.g., [latex]0x^2[\/latex], we usually do not write the term at all because [latex]0x^n=0[\/latex], and adding\u00a0[latex]0[\/latex] does not change the value of the polynomial. However, we will see later, that occasionally, it is necessary to include all terms of the polynomial, including terms with zero coefficients.<\/p>\n<p>A term without a variable is called a <em><b>constant <\/b><\/em>term, and the degree of that term is\u00a0[latex]0[\/latex].\u00a0 This is because any constant [latex]a[\/latex] can be written as [latex]ax^0[\/latex], since [latex]x^0=1[\/latex]. For example, we can write the polynomial [latex]3x+13[\/latex] as\u00a0[latex]3x^{1}+13x^{0}[\/latex].\u00a0 Although this is not how we would normally write this, it allows us to see that [latex]13[\/latex] is the constant term because its degree is 0, and the degree of\u00a0[latex]3x[\/latex] is 1.\u00a0 Consequently, the degree of the binomial[latex]3x+13[\/latex] is 1.<\/p>\n<p>If a polynomial does not have a constant term, like the polynomial [latex]14x^{3}+3x[\/latex] we say that the constant term is\u00a0[latex]0[\/latex].<\/p>\n<p>Let&#8217;s see how this works by looking at several polynomials. We&#8217;ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.<\/p>\n<p>Remember: Any variable base written without an exponent has an implied exponent of [latex]1[\/latex], and any constant term has an implied exponent of zero.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 20%;\"><strong>Monomials<\/strong><\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]5[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]4b^2[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]-9x^3[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]-18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\">Degree<\/td>\n<td style=\"width: 20%; text-align: center;\">0<\/td>\n<td style=\"width: 20%; text-align: center;\">2<\/td>\n<td style=\"width: 20%; text-align: center;\">3<\/td>\n<td style=\"width: 20%; text-align: center;\">0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\"><strong>Binomials<\/strong><\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]b+\\frac{1}{7}[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]3a-7[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]y^2-9[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]17x^3+14x^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\">Degree of each term<\/td>\n<td style=\"width: 20%; text-align: center;\">1\u00a0 \u00a00<\/td>\n<td style=\"width: 20%; text-align: center;\">1\u00a0 0<\/td>\n<td style=\"width: 20%; text-align: center;\">2\u00a0 \u00a00<\/td>\n<td style=\"width: 20%; text-align: center;\">3\u00a0 \u00a02<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\">Degree of polynomial<\/td>\n<td style=\"width: 20%; text-align: center;\">1<\/td>\n<td style=\"width: 20%; text-align: center;\">1<\/td>\n<td style=\"width: 20%; text-align: center;\">2<\/td>\n<td style=\"width: 20%; text-align: center;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%;\"><strong>Trinomial<\/strong><\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]x^2-5x+6[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]4y^2-7y+2[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]5a^12-3a^9+a[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]\\frac{1}{2}x^4+2x^2-5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\">Degree of each term<\/td>\n<td style=\"width: 20%; text-align: center;\">2\u00a0 \u00a0 1\u00a0 \u00a0 0<\/td>\n<td style=\"width: 20%; text-align: center;\">2\u00a0 \u00a0 1\u00a0 \u00a0 0<\/td>\n<td style=\"width: 20%; text-align: center;\">12\u00a0 \u00a0 9\u00a0 \u00a0 1<\/td>\n<td style=\"width: 20%; text-align: center;\">4\u00a0 \u00a0 2\u00a0 \u00a0 0<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\">Degree of polynomial<\/td>\n<td style=\"width: 20%; text-align: center;\">2<\/td>\n<td style=\"width: 20%; text-align: center;\">2<\/td>\n<td style=\"width: 20%; text-align: center;\">12<\/td>\n<td style=\"width: 20%; text-align: center;\">4<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<td style=\"width: 20%; text-align: center;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: left;\"><strong>Polynomial<\/strong><\/td>\n<td style=\"width: 20%; text-align: center;\"><span style=\"font-size: 12.8px;\">[latex]-7x^8+9x^6-7x^4+x^2-8[\/latex]<\/span><\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]\\frac{3}{5}x^7-8x^6+\\frac{7}{8}x+9[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]5.4x^{17}-8.1x^{13}+2.8x^{10}-6.5x^5[\/latex]<\/td>\n<td style=\"width: 20%; text-align: center;\">[latex]7+6y-9y^2+6y^3-3y^4+y^5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\">Degree of each term<\/td>\n<td style=\"width: 20%; text-align: center;\"><span style=\"font-size: 12.8px;\">8\u00a0 \u00a0 6\u00a0 \u00a0 4\u00a0 \u00a0 2\u00a0 \u00a0 0<\/span><\/td>\n<td style=\"width: 20%; text-align: center;\">7\u00a0 \u00a0 6\u00a0 \u00a0 1\u00a0 \u00a0 0<\/td>\n<td style=\"width: 20%; text-align: center;\">17\u00a0 \u00a0 13\u00a0 \u00a0 10\u00a0 \u00a0 5<\/td>\n<td style=\"width: 20%; text-align: center;\">0\u00a0 \u00a01\u00a0 \u00a02\u00a0 \u00a03\u00a0 \u00a04\u00a0 \u00a05<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 20%; text-align: right;\">Degree of polynomial<\/td>\n<td style=\"width: 20%; text-align: center;\">8<\/td>\n<td style=\"width: 20%; text-align: center;\">7<\/td>\n<td style=\"width: 20%; text-align: center;\">17<\/td>\n<td style=\"width: 20%; text-align: center;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the degree of the following polynomials:<\/p>\n<p>1. [latex]4x[\/latex]<br \/>\n2. [latex]3{x}^{3}-5x+7[\/latex]<br \/>\n3. [latex]-11[\/latex]<br \/>\n4. [latex]-6{x}^{2}+9x - 3[\/latex]<br \/>\n5. [latex]8x+2[\/latex]<\/p>\n<h4>Solution<\/h4>\n<table id=\"eip-id1168468469502\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.<\/td>\n<td>[latex]4x[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The exponent of [latex]x[\/latex] is one. [latex]x={x}^{1}[\/latex]<\/td>\n<td>The degree is [latex]1[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>2.<\/td>\n<td>[latex]3{x}^{3}-5x+7[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The highest degree of all the terms is [latex]3[\/latex].<\/td>\n<td>The degree is [latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>3.<\/td>\n<td>[latex]-11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The degree of a constant is [latex]0[\/latex].<\/td>\n<td>The degree is [latex]0[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>4.<\/td>\n<td>[latex]-6{x}^{2}+9x - 3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The highest degree of all the terms is [latex]2[\/latex].<\/td>\n<td>The degree is [latex]2[\/latex].<\/td>\n<\/tr>\n<tr>\n<td>5.<\/td>\n<td>[latex]8x+2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>The highest degree of all the terms is [latex]1[\/latex].<\/td>\n<td>The degree is [latex]1[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Working with polynomials is easier when we list the terms in <em><strong>descending order<\/strong><\/em> of degrees. When a polynomial is written this way, it is said to be in <em><strong>standard form<\/strong><\/em>. Look back at the polynomials in the previous example. Notice that they are all written in standard form. Get in the habit of writing polynomials in standard form<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146070\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146070&theme=oea&iframe_resize_id=ohm146070&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>For the following polynomials, write them in standard form (descending order), identify the degree, the leading term, and the leading coefficient.<\/p>\n<ol>\n<li>[latex]3+2{x}^{2}-4{x}^{3}[\/latex]<\/li>\n<li>[latex]5{t}^{5}+7t-2{t}^{3}[\/latex]<\/li>\n<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n<h4>Solution<\/h4>\n<ol>\n<li>Standard form: [latex]-4x^32x^2+3[\/latex]. The highest power of [latex]x[\/latex]\u00a0is\u00a0[latex]3[\/latex], so the degree is\u00a0[latex]3[\/latex]. The leading term is [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\n<li>Standard form: [latex]5t^5-2t^3+7t[\/latex].\u00a0 The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\n<li>Standard form:[latex]-p^3+6p-2[\/latex].\u00a0 The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<\/div>\n<p>The following video shows how to identify the terms, leading coefficient, and degree of a polynomial.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex:  Intro to Polynomials in One Variable\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3u16B2PN9zk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>For the following polynomials, write them in standard form (descending order), identify the degree, the leading term, and the leading coefficient.<\/p>\n<ol>\n<li>[latex]9+2{x}^{2}-5{x}^{5}[\/latex]<\/li>\n<li>[latex]-5{t}^{5}+7+8t-4{t}^{3}[\/latex]<\/li>\n<li>[latex]4p^4-{p}^{7}-2p[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm847\">Show Answer<\/span><\/p>\n<div id=\"qhjm847\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Standard form: [latex]-5x^5+2x^2+9[\/latex]; Degree = 5; Leading term = [latex]-5x^5[\/latex]; Leading coefficient =\u00a0[latex]-5[\/latex]<\/li>\n<li>Standard form: [latex]-5t^5-4t^3+8t+7[\/latex]; Degree = 5; Leading term = [latex]-5t^5[\/latex]; Leading coefficient =\u00a0[latex]-5[\/latex]<\/li>\n<li>Standard form: [latex]-p^7+4p^4-2p[\/latex]; Degree = 7; Leading term = [latex]-p^7[\/latex]; Leading coefficient =\u00a0[latex]-1[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/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-747\">\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>Determine if an Expression is a Polynomial. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning.. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nPAqfuoSbPI\">https:\/\/youtu.be\/nPAqfuoSbPI<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaption. <strong>Authored by<\/strong>: Hazel McKenna and Roxanne Brinkerhoff. <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><li>Ex: Intro to Polynomials in One Variable. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/3u16B2PN9zk\">https:\/\/youtu.be\/3u16B2PN9zk<\/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 hjm334; hjm847. <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>Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. <strong>Authored by<\/strong>:  Monterey Institute of Technology and Education. <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":422605,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Determine if an Expression is a Polynomial\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning.\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/nPAqfuoSbPI\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaption\",\"author\":\"Hazel McKenna and Roxanne Brinkerhoff\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program\",\"author\":\" Monterey Institute of Technology and Education\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Ex: Intro to Polynomials in One Variable\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/3u16B2PN9zk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Try It hjm334; hjm847\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-747","chapter","type-chapter","status-publish","hentry"],"part":663,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/747","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422605"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/747\/revisions"}],"predecessor-version":[{"id":2017,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/747\/revisions\/2017"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/663"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/747\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=747"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=747"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=747"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=747"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}