{"id":17796,"date":"2020-04-11T19:48:18","date_gmt":"2020-04-11T19:48:18","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/?post_type=chapter&#038;p=17796"},"modified":"2020-10-22T09:27:49","modified_gmt":"2020-10-22T09:27:49","slug":"classifying-polynomials","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/classifying-polynomials\/","title":{"raw":"12.1.b - Classifying Polynomials","rendered":"12.1.b &#8211; Classifying Polynomials"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify polynomials, monomials, binomials, and trinomials<\/li>\r\n \t<li>Determine the degree of polynomials<\/li>\r\n<\/ul>\r\n<\/div>\r\nPolynomials come in many forms. They can vary by how many terms, or monomials, make up the polynomial and they also can vary by the degrees of the monomials in the polynomial. 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 monomial with the largest exponent.\r\n<h2>Identify Polynomials, Monomials, Binomials, and Trinomials<\/h2>\r\nA monomial, or a sum and\/or difference of monomials, is called a polynomial.\u00a0\u00a0A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <strong>binomial<\/strong>.\u00a0 A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <strong>trinomial<\/strong>.\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 (\"poly\" means many)\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:\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. They are special members of the family of polynomials and so they have special names. We use the words \u2018monomial\u2019, \u2018binomial\u2019, and \u2018trinomial\u2019 when referring to these special polynomials and just call all the rest \u2018polynomials\u2019.\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\r\nSolution\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[reveal-answer q=\"239104\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"239104\"]\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.<\/li>\r\n \t<li>[latex]t^2+2t-3[\/latex] is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \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 terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \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.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146073[\/ohm_question]\r\n\r\n<\/div>\r\nIn the following video, you will be shown more examples of how to identify and categorize polynomials.\r\n\r\nhttps:\/\/youtu.be\/nPAqfuoSbPI\r\n<h2>Determining the Degree of Polynomials<\/h2>\r\nWe can find the <strong>degree<\/strong> of a 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 term. So the degree of [latex]2x^{3}+3x^{2}+8x+5[\/latex] is 3.\r\n\r\nA polynomial is said to be written in standard form when the terms are arranged from the highest degree to the lowest degree. 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 <strong>leading term<\/strong> because it is written first in standard form. The coefficient of the leading term is called the <strong>leading coefficient<\/strong>.\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\">\r\n<h3>How to: Given a polynomial expression, identify the degree and leading coefficient<\/h3>\r\n<ol>\r\n \t<li>Find the highest power of the variable (usually x) to determine the degree.<\/li>\r\n \t<li>Identify the term containing the highest power of the variable to find the leading term.<\/li>\r\n \t<li>Identify the coefficient of the leading term.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Degree of a 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], you usually do not write the term at all (because\u00a0[latex]0[\/latex] times anything is\u00a0[latex]0[\/latex], and adding\u00a0[latex]0[\/latex] does not change the value).\r\n\r\nA term without a variable is called a <b>constant <\/b>term, and the degree of that term is\u00a0[latex]0[\/latex].\u00a0 In the polynomial [latex]3x+13[\/latex], we could have written the polynomial 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 The degree of this binomial is 1.\r\n\r\nIf a polynomial does not have a constant term, like in the polynomial [latex]14x^{3}+3x[\/latex] we would 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 base written without an exponent has an implied exponent of [latex]1[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224331\/CNX_BMath_Figure_10_01_002.png\" alt=\"A table is shown. The top row is titled \" width=\"696\" height=\"523\" \/>\r\n\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[reveal-answer q=\"162218\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"162218\"]\r\n\r\nSolution\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[\/hidden-answer]\r\n\r\n<\/div>\r\nWorking with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form. Look back at the polynomials in the previous example. Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.\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, 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}-2{t}^{3}+7t[\/latex]<\/li>\r\n \t<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"753071\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"753071\"]\r\n<ol>\r\n \t<li>The highest power of <em>x<\/em> is\u00a0[latex]3[\/latex], so the degree is\u00a0[latex]3[\/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\r\n \t<li>The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\r\n \t<li>The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, we will identify the terms, leading coefficient, and degree of a polynomial.\r\n\r\nhttps:\/\/youtu.be\/3u16B2PN9zk\r\n\r\n&nbsp;","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify polynomials, monomials, binomials, and trinomials<\/li>\n<li>Determine the degree of polynomials<\/li>\n<\/ul>\n<\/div>\n<p>Polynomials come in many forms. They can vary by how many terms, or monomials, make up the polynomial and they also can vary by the degrees of the monomials in the polynomial. 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 monomial with the largest exponent.<\/p>\n<h2>Identify Polynomials, Monomials, Binomials, and Trinomials<\/h2>\n<p>A monomial, or a sum and\/or difference of monomials, is called a polynomial.\u00a0\u00a0A polynomial containing two terms, such as [latex]2x - 9[\/latex], is called a <strong>binomial<\/strong>.\u00a0 A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[\/latex], is called a <strong>trinomial<\/strong>.<\/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 (&#8220;poly&#8221; means many)<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:<\/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. They are special members of the family of polynomials and so they have special names. We use the words \u2018monomial\u2019, \u2018binomial\u2019, and \u2018trinomial\u2019 when referring to these special polynomials and just call all the rest \u2018polynomials\u2019.<\/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<p>Solution<\/p>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q239104\">Show Solution<\/span><\/p>\n<div id=\"q239104\" class=\"hidden-answer\" style=\"display: none\">\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.<\/li>\n<li>[latex]t^2+2t-3[\/latex] is a polynomial because it is an expression whose terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \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 terms are connected by addition and subtraction, and there are no variables under a root or in the denominator of a fraction. \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.<\/li>\n<\/ol>\n<\/div>\n<\/div>\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>In the following video, you will be shown 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<h2>Determining the Degree of Polynomials<\/h2>\n<p>We can find the <strong>degree<\/strong> of a 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 term. So the degree of [latex]2x^{3}+3x^{2}+8x+5[\/latex] is 3.<\/p>\n<p>A polynomial is said to be written in standard form when the terms are arranged from the highest degree to the lowest degree. 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 <strong>leading term<\/strong> because it is written first in standard form. The coefficient of the leading term is called the <strong>leading coefficient<\/strong>.<\/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\">\n<h3>How to: Given a polynomial expression, identify the degree and leading coefficient<\/h3>\n<ol>\n<li>Find the highest power of the variable (usually x) to determine the degree.<\/li>\n<li>Identify the term containing the highest power of the variable to find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Degree of a 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], you usually do not write the term at all (because\u00a0[latex]0[\/latex] times anything is\u00a0[latex]0[\/latex], and adding\u00a0[latex]0[\/latex] does not change the value).<\/p>\n<p>A term without a variable is called a <b>constant <\/b>term, and the degree of that term is\u00a0[latex]0[\/latex].\u00a0 In the polynomial [latex]3x+13[\/latex], we could have written the polynomial 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 The degree of this binomial is 1.<\/p>\n<p>If a polynomial does not have a constant term, like in the polynomial [latex]14x^{3}+3x[\/latex] we would 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 base written without an exponent has an implied exponent of [latex]1[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224331\/CNX_BMath_Figure_10_01_002.png\" alt=\"A table is shown. The top row is titled\" width=\"696\" height=\"523\" \/><\/p>\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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q162218\">Show Solution<\/span><\/p>\n<div id=\"q162218\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\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<\/div>\n<\/div>\n<p>Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form. Look back at the polynomials in the previous example. Notice that they are all written in standard form. Get in the habit of writing the term with the highest degree first.<\/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, 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}-2{t}^{3}+7t[\/latex]<\/li>\n<li>[latex]6p-{p}^{3}-2[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q753071\">Show Solution<\/span><\/p>\n<div id=\"q753071\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>The highest power of <em>x<\/em> is\u00a0[latex]3[\/latex], so the degree is\u00a0[latex]3[\/latex]. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, [latex]-4[\/latex].<\/li>\n<li>The highest power of <em>t<\/em> is [latex]5[\/latex], so the degree is [latex]5[\/latex]. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, [latex]5[\/latex].<\/li>\n<li>The highest power of <em>p<\/em> is [latex]3[\/latex], so the degree is [latex]3[\/latex]. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex], The leading coefficient is the coefficient of that term, [latex]-1[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, we will 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<p>&nbsp;<\/p>\n","protected":false},"author":253111,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"4f4d10a080b94c69bc274009a30cab84, b5da00e0cab64b1daab019fc06285197, 425e3448043c42ef9fbda9eeecbc888f","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-17796","chapter","type-chapter","status-publish","hentry"],"part":8336,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17796","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/users\/253111"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17796\/revisions"}],"predecessor-version":[{"id":20390,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17796\/revisions\/20390"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/parts\/8336"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapters\/17796\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/media?parent=17796"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/pressbooks\/v2\/chapter-type?post=17796"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/contributor?post=17796"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/wp-json\/wp\/v2\/license?post=17796"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}