{"id":2639,"date":"2016-07-15T19:19:30","date_gmt":"2016-07-15T19:19:30","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=2639"},"modified":"2018-05-17T00:57:36","modified_gmt":"2018-05-17T00:57:36","slug":"read-define-and-identify-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/read-define-and-identify-polynomial-functions\/","title":{"raw":"Polynomial Functions","rendered":"Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Introduction to polynomial functions\r\n<ul>\r\n \t<li>Identify polynomial functions<\/li>\r\n \t<li>Identify the degree and leading coefficient of a polynomial function<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Add and subtract polynomial functions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3>Identify polynomial functions<\/h3>\r\nWe have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions.\u00a0Polynomials are algebraic expressions that are created by summing\u00a0monomial terms, such as [latex]-3x^2[\/latex],\u00a0where the exponents are only integers. Functions are\u00a0a specific type of relation in which each input value has one and only one output value.\u00a0Polynomial functions have all of these characteristics as well as a domain and range, and corresponding graphs. In this section we will identify and evaluate polynomial functions.\u00a0Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable.\r\n\r\nWhen we introduced polynomials, we presented the following: [latex]4x^3-9x^2+6x[\/latex]. \u00a0We can turn this into a polynomial function by using function notation:\r\n<p style=\"text-align: center\">[latex]f(x)=4x^3-9x^2+6x[\/latex]<\/p>\r\n<p style=\"text-align: left\">Polynomial functions are written with the leading term first, and all other terms in descending order as a matter of convention. In the first example, we will identify some basic characteristics of polynomial functions.<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\r\n\r\n<div id=\"eip-id1165134474011\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{ccc}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/div>\r\n[reveal-answer q=\"83362\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"83362\"]\r\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they contain\u00a0powers that are non-negative integers and the coefficients are real numbers.<\/p>\r\n\r\n<ul id=\"fs-id1165137864157\">\r\n \t<li>[latex]f\\left(x\\right)[\/latex] can be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\r\n \t<li>[latex]g\\left(x\\right)[\/latex] can be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\r\n \t<li>[latex]h\\left(x\\right)=5\\sqrt{x}+2[\/latex] is not a polynomial because the variable is under a square root - therefore the exponent is not a positive integer.<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video you will see additional examples of how to identify a polynomial function using the definition.\r\n\r\nhttps:\/\/youtu.be\/w02qTLrJYiQ\r\n\r\n&nbsp;\r\n<h2>Define\u00a0the degree and leading coefficient\u00a0of a polynomial function<\/h2>\r\nJust as we identified the degree of a polynomial, we can identify the degree of a polynomial function.\u00a0\u00a0To review: the <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; the <strong>leading term<\/strong> is the term containing the highest power of the variable, or the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify the degree, leading term, and leading coefficient of the following polynomial functions.\r\n<div id=\"eip-id1165134242117\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{ccc} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\ g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\ h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left\">[reveal-answer q=\"200839\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"200839\"]<\/div>\r\n<div class=\"equation unnumbered\" style=\"text-align: left\">\r\n<p id=\"fs-id1165137722510\">For the function [latex]f\\left(x\\right)[\/latex], the highest power of <em>x<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, \u20134.<\/p>\r\n<p id=\"fs-id1165135457771\">For the function [latex]g\\left(t\\right)[\/latex], the highest power of <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.<\/p>\r\n<p id=\"fs-id1165135503949\">For the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, \u20131.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nIn the next video we will show more examples of how to identify the degree, leading term and leading coefficient of a polynomial function.\r\n\r\nhttps:\/\/youtu.be\/F_G_w82s0QA\r\n<h2>Graphs of Polynomial Functions<\/h2>\r\nPlotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. Additionally, the algebra of finding points like x-intercepts for higher degree polynomials can get very messy and oftentimes impossible to find\u00a0by hand. We have therefore developed some techniques for describing the general behavior of polynomial graphs.\r\n\r\nPolynomial functions of degree 2 or more have graphs that do not have sharp corners these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below\u00a0shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201418\/CNX_Precalc_Figure_03_04_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/>\r\n\r\nNow you try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhich of the graphs below\u00a0represents a polynomial function?\r\n<span>\r\n<\/span>\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201420\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/>\r\n\r\n[reveal-answer q=\"207827\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"207827\"]\r\n<p id=\"fs-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\r\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_03_04_01\" class=\"example\"><\/div>\r\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\r\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\r\n\r\n<\/div>\r\n<h2>Identifying\u00a0the shape of the graph of a polynomial function<\/h2>\r\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict what it's graph will look like.\u00a0Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the\u00a0graph\u00a0of the polynomial will match the end behavior of the term of highest degree.<\/p>\r\nAs an example we compare the outputs of a degree 2 polynomial and a degree 5 polynomial in the following table.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>x<\/td>\r\n<td>[latex]f(x)=2x^2-2x+4[\/latex]<\/td>\r\n<td>[latex]g(x)=x^5+2x^3-12x+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>4<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>10<\/td>\r\n<td>184<\/td>\r\n<td>98117<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>100<\/td>\r\n<td>19804<\/td>\r\n<td>9998001197<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1000<\/td>\r\n<td>1998004<\/td>\r\n<td>9999980000000000<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs the inputs for both functions get larger, the degree 5 polynomial outputs get much larger than the degree 2 polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of polynomial graphs.\r\n\r\nThere are two other important features of polynomials that influence the shape of it's graph. The first \u00a0is whether the degree is even or odd, and the second is whether the leading term is negative.\r\n<h3>Even degree polynomials<\/h3>\r\n<p id=\"fs-id1165135436540\">In the figure below, we show\u00a0the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex] and [latex]\\text{and}h\\left(x\\right)={x}^{6}[\/latex], which are all have even degrees. Notice that these graphs have similar shapes, very much like that of a\u00a0quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201324\/CNX_Precalc_Figure_03_03_0022.jpg\" alt=\"Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.\" width=\"487\" height=\"253\" \/>\r\n<h3>Odd degree polynomials<\/h3>\r\n<p id=\"fs-id1165137533222\">The next figure\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},\\text{and}h\\left(x\\right)={x}^{7}[\/latex], which are all odd degree functions. <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201325\/CNX_Precalc_Figure_03_03_0032.jpg\" alt=\"Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.\" width=\"312\" height=\"366\" \/><\/p>\r\n<p id=\"fs-id1165137533222\">Notice that one arm of the graph points down and the other points up. \u00a0This is because\u00a0when your input is negative, you will get a negative output if the degree is odd.\u00a0The following table of values shows this.<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>x<\/td>\r\n<td>[latex]f(x)=x^4[\/latex]<\/td>\r\n<td>[latex]h(x)=x^5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-1<\/td>\r\n<td>1<\/td>\r\n<td>-1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-2<\/td>\r\n<td>16<\/td>\r\n<td>-32<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-3<\/td>\r\n<td>81<\/td>\r\n<td>-243<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow you try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify whether\u00a0graph represents a polynomial function that has a degree that is even or odd.\r\n\r\na)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201335\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"231\" height=\"245\" \/>\r\n\r\nb)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201338\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"227\" height=\"241\" \/>\r\n[reveal-answer q=\"657906\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"657906\"]\r\n\r\na) Both arms of this polynomial point upward, similar to a quadratic polynomial, therefore the degree must be even. \u00a0If you apply negative inputs to an even degree polynomial you will get positive outputs back.\r\n\r\nb) As the inputs of this polynomial become more negative the outputs also become negative, the only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have odd degree.\r\n\r\nPut Answer Here[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>\u00a0The sign of the leading term<\/h3>\r\nWhat would happen if we change the sign of the leading term of an even degree polynomial? \u00a0For example, let's say that the leading term of a polynomial is [latex]-3x^4[\/latex]. \u00a0We will use a table of values to compare the outputs for a polynomial with leading term\u00a0[latex]-3x^4[\/latex], and\u00a0[latex]3x^4[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>x<\/td>\r\n<td>[latex]-3x^4[\/latex]<\/td>\r\n<td>[latex]3x^4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>-2<\/td>\r\n<td>-48<\/td>\r\n<td>48<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center\">-1<\/td>\r\n<td>-3<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>1<\/td>\r\n<td>-3<\/td>\r\n<td>3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2<\/td>\r\n<td>-48<\/td>\r\n<td>48<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlotting these points on a grid leads to this plot, the red points indicate a negative leading coefficient, and the blue points indicate a positive leading coefficient:\r\n\r\n<img class=\"wp-image-2649 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15212141\/Screen-Shot-2016-07-15-at-2.21.36-PM-140x300.png\" alt=\"Screen Shot 2016-07-15 at 2.21.36 PM\" width=\"266\" height=\"570\" \/>\r\n\r\nThe negative sign creates a reflection of [latex]3x^4[\/latex] across the x-axis. \u00a0The arms of a polynomial with a leading term of\u00a0[latex]-3x^4[\/latex] will point down, whereas the arms of a polynomial with leading term\u00a0[latex]3x^4[\/latex] will point up.\r\n\r\nNow you try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.\r\n\r\na)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201336\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" width=\"214\" height=\"227\" \/>\r\n\r\nb)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201339\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" width=\"217\" height=\"230\" \/>\r\n[reveal-answer q=\"317874\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"317874\"]\r\n\r\na) Both arms of this polynomial point in the same direction so it must have an even degree. \u00a0The leading term of the polynomial must be negative since the arms are pointing downward.\r\n\r\nb) The arms of this polynomial point in different directions, so the degree must be odd. As the inputs get really big and positive, the outputs get really big and negative, so the leading coefficient must be negative.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Add and subtract polynomial functions<\/h2>\r\nAdding and subtracting polynomial functions is the same as adding and subtracting polynomials. When you evaluate a sum or difference of functions, you can either evaluate first, or perform the operation on the functions first, as we will see. Our next examples describe the notation used to add and subtract polynomial functions.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFor [latex]f(x)=2x^3-5x+3[\/latex] and [latex]h(x)=x-5[\/latex],\r\n\r\nFind the following:\r\n\r\n[latex](f+h)(x)[\/latex] and [latex](h-f)(x)[\/latex]\r\n[reveal-answer q=\"295585\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"295585\"]\r\n\r\n[latex]\\begin{array}{ccc}(f+h)(x)=f(x)+ h(x)(2x^3-5x+3)+(x-5)\\\\=2x^3-5x+3+x-5\\,\\,\\,\\,\\,\\text{combine like terms}\\\\=2x^3-4x-2\\,\\,\\,\\,\\,\\text{simplify}\\end{array}[\/latex]\r\n\r\n[latex]\\begin{array}{ccc}(h-f)(x)=h(x)-f(x)=(x-5)-(2x^3-5x+3)\\\\=x-5-2x^2+5x-3\\,\\,\\,\\,\\,\\,\\text{combine like terms}\\\\=-2x^2+6x-8\\,\\,\\,\\,\\,\\text{simplify}\\end{array}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn our next example we will evaluate a sum and difference of functions and show that you can get to the same result in one of two ways.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFor [latex]f(x)=2x^3-5x+3[\/latex] and [latex]h(x)=x-5[\/latex]\r\n\r\nEvaluate:\u00a0[latex](f+h)(2)[\/latex]\r\n\r\nShow that you get the same result by\r\n\r\n1)evaluating the functions first, then performing the indicated operation on the result and\r\n\r\n2) performing the operation on the functions first, then evaluating the result\r\n[reveal-answer q=\"754772\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"754772\"]\r\n\r\n&nbsp;\r\n\r\n1)[latex](f+h)(2)[\/latex] First, we will evaluate the functions separately:\r\n\r\n[latex]f(2)=2(2)^3-5(2)+3=16-10+3=9[\/latex]\r\n\r\n[latex]h(2)=(2)-5=-3[\/latex]\r\n\r\nNow we will perform the indicated operation using the results:\r\n\r\n[latex](f+h)(2)=f(2)+h(2)=9+(-3)=6[\/latex]\r\n\r\n2) We can get the same result by adding the functions first, then evaluating the result at x=2\r\n\r\n[latex](f+h)(x)=f(x)+h(x)=2x^3-4x-2[\/latex] from above.\r\n\r\nNow we can evaluate this result at x=2\r\n\r\n[latex](f+h)(2)=2(2)^3-4(2)-2=16-8-2=6[\/latex]\r\n\r\nBoth methods give the same result, and both require about the same amount of work.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2><\/h2>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Introduction to polynomial functions\n<ul>\n<li>Identify polynomial functions<\/li>\n<li>Identify the degree and leading coefficient of a polynomial function<\/li>\n<\/ul>\n<\/li>\n<li>Add and subtract polynomial functions<\/li>\n<\/ul>\n<\/div>\n<h3>Identify polynomial functions<\/h3>\n<p>We have introduced polynomials and functions, so now we will combine these ideas to describe polynomial functions.\u00a0Polynomials are algebraic expressions that are created by summing\u00a0monomial terms, such as [latex]-3x^2[\/latex],\u00a0where the exponents are only integers. Functions are\u00a0a specific type of relation in which each input value has one and only one output value.\u00a0Polynomial functions have all of these characteristics as well as a domain and range, and corresponding graphs. In this section we will identify and evaluate polynomial functions.\u00a0Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable.<\/p>\n<p>When we introduced polynomials, we presented the following: [latex]4x^3-9x^2+6x[\/latex]. \u00a0We can turn this into a polynomial function by using function notation:<\/p>\n<p style=\"text-align: center\">[latex]f(x)=4x^3-9x^2+6x[\/latex]<\/p>\n<p style=\"text-align: left\">Polynomial functions are written with the leading term first, and all other terms in descending order as a matter of convention. In the first example, we will identify some basic characteristics of polynomial functions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n<div id=\"eip-id1165134474011\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{ccc}f\\left(x\\right)=2{x}^{3}\\cdot 3x+4\\hfill \\\\ g\\left(x\\right)=-x\\left({x}^{2}-4\\right)\\hfill \\\\ h\\left(x\\right)=5\\sqrt{x}+2\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q83362\">Show Answer<\/span><\/p>\n<div id=\"q83362\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they contain\u00a0powers that are non-negative integers and the coefficients are real numbers.<\/p>\n<ul id=\"fs-id1165137864157\">\n<li>[latex]f\\left(x\\right)[\/latex] can be written as [latex]f\\left(x\\right)=6{x}^{4}+4[\/latex].<\/li>\n<li>[latex]g\\left(x\\right)[\/latex] can be written as [latex]g\\left(x\\right)=-{x}^{3}+4x[\/latex].<\/li>\n<li>[latex]h\\left(x\\right)=5\\sqrt{x}+2[\/latex] is not a polynomial because the variable is under a square root &#8211; therefore the exponent is not a positive integer.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video you will see additional examples of how to identify a polynomial function using the definition.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Determine if a Function is a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/w02qTLrJYiQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<h2>Define\u00a0the degree and leading coefficient\u00a0of a polynomial function<\/h2>\n<p>Just as we identified the degree of a polynomial, we can identify the degree of a polynomial function.\u00a0\u00a0To review: the <strong>degree<\/strong> of the polynomial is the highest power of the variable that occurs in the polynomial; the <strong>leading term<\/strong> is the term containing the highest power of the variable, or the term with the highest degree. The <strong>leading coefficient<\/strong> is the coefficient of the leading term.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<div id=\"eip-id1165134242117\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{ccc} f\\left(x\\right)=3+2{x}^{2}-4{x}^{3} \\\\ g\\left(t\\right)=5{t}^{5}-2{t}^{3}+7t\\\\ h\\left(p\\right)=6p-{p}^{3}-2\\end{array}[\/latex]<\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q200839\">Show Answer<\/span><\/p>\n<div id=\"q200839\" class=\"hidden-answer\" style=\"display: none\"><\/div>\n<div class=\"equation unnumbered\" style=\"text-align: left\">\n<p id=\"fs-id1165137722510\">For the function [latex]f\\left(x\\right)[\/latex], the highest power of <em>x<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-4{x}^{3}[\/latex]. The leading coefficient is the coefficient of that term, \u20134.<\/p>\n<p id=\"fs-id1165135457771\">For the function [latex]g\\left(t\\right)[\/latex], the highest power of <em>t<\/em>\u00a0is 5, so the degree is 5. The leading term is the term containing that degree, [latex]5{t}^{5}[\/latex]. The leading coefficient is the coefficient of that term, 5.<\/p>\n<p id=\"fs-id1165135503949\">For the function [latex]h\\left(p\\right)[\/latex], the highest power of <em>p<\/em>\u00a0is 3, so the degree is 3. The leading term is the term containing that degree, [latex]-{p}^{3}[\/latex]; the leading coefficient is the coefficient of that term, \u20131.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next video we will show more examples of how to identify the degree, leading term and leading coefficient of a polynomial function.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Degree, Leading Term, and Leading Coefficient of a Polynomial Function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/F_G_w82s0QA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Graphs of Polynomial Functions<\/h2>\n<p>Plotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. Additionally, the algebra of finding points like x-intercepts for higher degree polynomials can get very messy and oftentimes impossible to find\u00a0by hand. We have therefore developed some techniques for describing the general behavior of polynomial graphs.<\/p>\n<p>Polynomial functions of degree 2 or more have graphs that do not have sharp corners these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below\u00a0shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201418\/CNX_Precalc_Figure_03_04_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/><\/p>\n<p>Now you try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Which of the graphs below\u00a0represents a polynomial function?<br \/>\n<span><br \/>\n<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201420\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q207827\">Show Answer<\/span><\/p>\n<div id=\"q207827\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_04_01\" class=\"example\"><\/div>\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have as their domain all real numbers?<\/strong><\/p>\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\n<\/div>\n<h2>Identifying\u00a0the shape of the graph of a polynomial function<\/h2>\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict what it&#8217;s graph will look like.\u00a0Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the\u00a0graph\u00a0of the polynomial will match the end behavior of the term of highest degree.<\/p>\n<p>As an example we compare the outputs of a degree 2 polynomial and a degree 5 polynomial in the following table.<\/p>\n<table>\n<tbody>\n<tr>\n<td>x<\/td>\n<td>[latex]f(x)=2x^2-2x+4[\/latex]<\/td>\n<td>[latex]g(x)=x^5+2x^3-12x+3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>4<\/td>\n<td>8<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>184<\/td>\n<td>98117<\/td>\n<\/tr>\n<tr>\n<td>100<\/td>\n<td>19804<\/td>\n<td>9998001197<\/td>\n<\/tr>\n<tr>\n<td>1000<\/td>\n<td>1998004<\/td>\n<td>9999980000000000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As the inputs for both functions get larger, the degree 5 polynomial outputs get much larger than the degree 2 polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of polynomial graphs.<\/p>\n<p>There are two other important features of polynomials that influence the shape of it&#8217;s graph. The first \u00a0is whether the degree is even or odd, and the second is whether the leading term is negative.<\/p>\n<h3>Even degree polynomials<\/h3>\n<p id=\"fs-id1165135436540\">In the figure below, we show\u00a0the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex] and [latex]\\text{and}h\\left(x\\right)={x}^{6}[\/latex], which are all have even degrees. Notice that these graphs have similar shapes, very much like that of a\u00a0quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201324\/CNX_Precalc_Figure_03_03_0022.jpg\" alt=\"Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.\" width=\"487\" height=\"253\" \/><\/p>\n<h3>Odd degree polynomials<\/h3>\n<p id=\"fs-id1165137533222\">The next figure\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},\\text{and}h\\left(x\\right)={x}^{7}[\/latex], which are all odd degree functions. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201325\/CNX_Precalc_Figure_03_03_0032.jpg\" alt=\"Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.\" width=\"312\" height=\"366\" \/><\/p>\n<p id=\"fs-id1165137533222\">Notice that one arm of the graph points down and the other points up. \u00a0This is because\u00a0when your input is negative, you will get a negative output if the degree is odd.\u00a0The following table of values shows this.<\/p>\n<table>\n<tbody>\n<tr>\n<td>x<\/td>\n<td>[latex]f(x)=x^4[\/latex]<\/td>\n<td>[latex]h(x)=x^5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>-1<\/td>\n<td>1<\/td>\n<td>-1<\/td>\n<\/tr>\n<tr>\n<td>-2<\/td>\n<td>16<\/td>\n<td>-32<\/td>\n<\/tr>\n<tr>\n<td>-3<\/td>\n<td>81<\/td>\n<td>-243<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now you try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify whether\u00a0graph represents a polynomial function that has a degree that is even or odd.<\/p>\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201335\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"231\" height=\"245\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201338\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"227\" height=\"241\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q657906\">Show Answer<\/span><\/p>\n<div id=\"q657906\" class=\"hidden-answer\" style=\"display: none\">\n<p>a) Both arms of this polynomial point upward, similar to a quadratic polynomial, therefore the degree must be even. \u00a0If you apply negative inputs to an even degree polynomial you will get positive outputs back.<\/p>\n<p>b) As the inputs of this polynomial become more negative the outputs also become negative, the only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have odd degree.<\/p>\n<p>Put Answer Here<\/p><\/div>\n<\/div>\n<\/div>\n<h3>\u00a0The sign of the leading term<\/h3>\n<p>What would happen if we change the sign of the leading term of an even degree polynomial? \u00a0For example, let&#8217;s say that the leading term of a polynomial is [latex]-3x^4[\/latex]. \u00a0We will use a table of values to compare the outputs for a polynomial with leading term\u00a0[latex]-3x^4[\/latex], and\u00a0[latex]3x^4[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td>x<\/td>\n<td>[latex]-3x^4[\/latex]<\/td>\n<td>[latex]3x^4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>-2<\/td>\n<td>-48<\/td>\n<td>48<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center\">-1<\/td>\n<td>-3<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>1<\/td>\n<td>-3<\/td>\n<td>3<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>-48<\/td>\n<td>48<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plotting these points on a grid leads to this plot, the red points indicate a negative leading coefficient, and the blue points indicate a positive leading coefficient:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2649 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15212141\/Screen-Shot-2016-07-15-at-2.21.36-PM-140x300.png\" alt=\"Screen Shot 2016-07-15 at 2.21.36 PM\" width=\"266\" height=\"570\" \/><\/p>\n<p>The negative sign creates a reflection of [latex]3x^4[\/latex] across the x-axis. \u00a0The arms of a polynomial with a leading term of\u00a0[latex]-3x^4[\/latex] will point down, whereas the arms of a polynomial with leading term\u00a0[latex]3x^4[\/latex] will point up.<\/p>\n<p>Now you try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.<\/p>\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201336\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" width=\"214\" height=\"227\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201339\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" width=\"217\" height=\"230\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q317874\">Show Answer<\/span><\/p>\n<div id=\"q317874\" class=\"hidden-answer\" style=\"display: none\">\n<p>a) Both arms of this polynomial point in the same direction so it must have an even degree. \u00a0The leading term of the polynomial must be negative since the arms are pointing downward.<\/p>\n<p>b) The arms of this polynomial point in different directions, so the degree must be odd. As the inputs get really big and positive, the outputs get really big and negative, so the leading coefficient must be negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Add and subtract polynomial functions<\/h2>\n<p>Adding and subtracting polynomial functions is the same as adding and subtracting polynomials. When you evaluate a sum or difference of functions, you can either evaluate first, or perform the operation on the functions first, as we will see. Our next examples describe the notation used to add and subtract polynomial functions.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>For [latex]f(x)=2x^3-5x+3[\/latex] and [latex]h(x)=x-5[\/latex],<\/p>\n<p>Find the following:<\/p>\n<p>[latex](f+h)(x)[\/latex] and [latex](h-f)(x)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q295585\">Show Answer<\/span><\/p>\n<div id=\"q295585\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\begin{array}{ccc}(f+h)(x)=f(x)+ h(x)(2x^3-5x+3)+(x-5)\\\\=2x^3-5x+3+x-5\\,\\,\\,\\,\\,\\text{combine like terms}\\\\=2x^3-4x-2\\,\\,\\,\\,\\,\\text{simplify}\\end{array}[\/latex]<\/p>\n<p>[latex]\\begin{array}{ccc}(h-f)(x)=h(x)-f(x)=(x-5)-(2x^3-5x+3)\\\\=x-5-2x^2+5x-3\\,\\,\\,\\,\\,\\,\\text{combine like terms}\\\\=-2x^2+6x-8\\,\\,\\,\\,\\,\\text{simplify}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In our next example we will evaluate a sum and difference of functions and show that you can get to the same result in one of two ways.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>For [latex]f(x)=2x^3-5x+3[\/latex] and [latex]h(x)=x-5[\/latex]<\/p>\n<p>Evaluate:\u00a0[latex](f+h)(2)[\/latex]<\/p>\n<p>Show that you get the same result by<\/p>\n<p>1)evaluating the functions first, then performing the indicated operation on the result and<\/p>\n<p>2) performing the operation on the functions first, then evaluating the result<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q754772\">Show Answer<\/span><\/p>\n<div id=\"q754772\" class=\"hidden-answer\" style=\"display: none\">\n<p>&nbsp;<\/p>\n<p>1)[latex](f+h)(2)[\/latex] First, we will evaluate the functions separately:<\/p>\n<p>[latex]f(2)=2(2)^3-5(2)+3=16-10+3=9[\/latex]<\/p>\n<p>[latex]h(2)=(2)-5=-3[\/latex]<\/p>\n<p>Now we will perform the indicated operation using the results:<\/p>\n<p>[latex](f+h)(2)=f(2)+h(2)=9+(-3)=6[\/latex]<\/p>\n<p>2) We can get the same result by adding the functions first, then evaluating the result at x=2<\/p>\n<p>[latex](f+h)(x)=f(x)+h(x)=2x^3-4x-2[\/latex] from above.<\/p>\n<p>Now we can evaluate this result at x=2<\/p>\n<p>[latex](f+h)(2)=2(2)^3-4(2)-2=16-8-2=6[\/latex]<\/p>\n<p>Both methods give the same result, and both require about the same amount of work.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2><\/h2>\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-2639\">\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 a Function is a Polynomial Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/w02qTLrJYiQ\">https:\/\/youtu.be\/w02qTLrJYiQ<\/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>Degree, Leading Term, and Leading Coefficient of a Polynomial Function. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/F_G_w82s0QA\">https:\/\/youtu.be\/F_G_w82s0QA<\/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 Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay, et al. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download fro free at : http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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":21,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Determine if a 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