{"id":1393,"date":"2015-11-12T18:35:29","date_gmt":"2015-11-12T18:35:29","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1393"},"modified":"2017-03-31T22:59:02","modified_gmt":"2017-03-31T22:59:02","slug":"use-descartes-rule-of-signs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/use-descartes-rule-of-signs\/","title":{"raw":"Use Descartes\u2019 Rule of Signs","rendered":"Use Descartes\u2019 Rule of Signs"},"content":{"raw":"<p id=\"fs-id1165135177655\">There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order,<strong> Descartes\u2019 Rule of Signs<\/strong> tells us of a relationship between the number of sign changes in [latex]f\\left(x\\right)[\/latex] and the number of positive real zeros. For example, the polynomial function below has one sign change.<\/p>\r\n<span id=\"fs-id1165134378690\" data-type=\"media\" data-alt=\"The function, f(x)=x^4+x^3+x^2+x-1, has one sign change between x and -1.&lt;img&gt;&lt;\/span&gt;&lt;\/p&gt;&lt;p id=\">This tells us that the function must have 1 positive real zero.<\/span>\r\n<p id=\"fs-id1165135206084\">There is a similar relationship between the number of sign changes in [latex]f\\left(-x\\right)[\/latex] and the number of negative real zeros.<\/p>\r\n<span id=\"fs-id1165135152070\" data-type=\"media\" data-alt=\"The function, f(-x)=(-x)^4+(-x)^3+(-x)^2+(-x)-1=+ x^4-x^3+x^2-x-1, has three sign changes between x^4 and x^3, x^3 and x^2, and x^2 and x.&lt;img&gt;&lt;\/span&gt;&lt;\/p&gt;&lt;p id=\">In this case, [latex]f\\left(\\mathrm{-x}\\right)[\/latex] has 3 sign changes. This tells us that [latex]f\\left(x\\right)[\/latex] could have 3 or 1 negative real zeros.<\/span>\r\n<div id=\"fs-id1165137844271\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\r\n<h3 class=\"title\" data-type=\"title\">A General Note: Descartes\u2019 Rule of Signs<\/h3>\r\n<p id=\"fs-id1165137844280\">According to <strong>Descartes\u2019 Rule of Signs<\/strong>, if we let [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]\u00a0be a polynomial function with real coefficients:<\/p>\r\n\r\n<ul id=\"fs-id1165134351104\"><li>The number of positive real zeros is either equal to the number of sign changes of [latex]f\\left(x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\r\n\t<li>The number of negative real zeros is either equal to the number of sign changes of [latex]f\\left(-x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\r\n<\/ul><\/div>\r\n<div id=\"Example_03_06_08\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-id1165134149118\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-id1165134149120\" class=\"problem textbox shaded\" data-type=\"problem\">\r\n<h3 data-type=\"title\">Example 7: Using Descartes\u2019 Rule of Signs<\/h3>\r\n<p>Use Descartes\u2019 Rule of Signs to determine the possible numbers of positive and negative real zeros for [latex]f\\left(x\\right)=-{x}^{4}-3{x}^{3}+6{x}^{2}-4x - 12[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137932378\" class=\"solution textbox shaded\" data-type=\"solution\">\r\n<h3>Solution<\/h3>\r\n<p id=\"fs-id1165137932381\">Begin by determining the number of sign changes.<\/p>\r\n<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-04-at-12.31.54-PM.png\"><img class=\"aligncenter size-full wp-image-11813\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201555\/Screen-Shot-2015-08-04-at-12.31.54-PM.png\" alt=\"Screen Shot 2015-08-04 at 12.31.54 PM\" width=\"534\" height=\"57\"\/><\/a>\r\n\r\nThere are two sign changes, so there are either 2 or 0 positive real roots. Next, we examine [latex]f\\left(-x\\right)[\/latex] to determine the number of negative real roots.\r\n<p id=\"fs-id1165137696389\" style=\"text-align: center;\">[latex]\\begin{cases}f\\left(-x\\right)=-{\\left(-x\\right)}^{4}-3{\\left(-x\\right)}^{3}+6{\\left(-x\\right)}^{2}-4\\left(-x\\right)-12\\hfill \\\\ f\\left(-x\\right)=-{x}^{4}+3{x}^{3}+6{x}^{2}+4x - 12\\hfill \\end{cases}[\/latex]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-04-at-12.32.40-PM.png\"><img class=\"aligncenter size-full wp-image-11814\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201557\/Screen-Shot-2015-08-04-at-12.32.40-PM.png\" alt=\"Screen Shot 2015-08-04 at 12.32.40 PM\" width=\"536\" height=\"52\"\/><\/a><\/p>\r\n<span data-type=\"media\" data-alt=\"The function, f(-x)=-x^4+3x^3+6x^2+4x-12, has two sign change between -x^4 and 3x^3, and 4x and -12.&lt;img&gt;&lt;\/span&gt;&lt;br&gt;&lt;\/figure&gt;&lt;p id=\"><span data-type=\"media\" data-alt=\"The function, f(-x)=-x^4+3x^3+6x^2+4x-12, has two sign change between -x^4 and 3x^3, and 4x and -12.&lt;img&gt;&lt;\/span&gt;&lt;br&gt;&lt;\/figure&gt;&lt;p id=\">Again, there are two sign changes, so there are either 2 or 0 negative real roots.<\/span><\/span>\r\n<p id=\"fs-id1165135383102\">There are four possibilities, as we can see below.<\/p>\r\n\r\n<table id=\"Table_03_06_01\" summary=\"..\"><colgroup><col data-align=\"center\"\/><col data-align=\"center\"\/><col data-align=\"center\"\/><col data-align=\"center\"\/><\/colgroup><thead><tr><th>Positive Real\r\nZeros<\/th>\r\n<th>Negative Real\r\nZeros<\/th>\r\n<th>Complex\r\nZeros<\/th>\r\n<th>Total\r\nZeros<\/th>\r\n<\/tr><\/thead><tbody><tr><td>2<\/td>\r\n<td>2<\/td>\r\n<td>0<\/td>\r\n<td>4<\/td>\r\n<\/tr><tr><td>2<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr><tr><td>0<\/td>\r\n<td>2<\/td>\r\n<td>2<\/td>\r\n<td>4<\/td>\r\n<\/tr><tr><td>0<\/td>\r\n<td>0<\/td>\r\n<td>4<\/td>\r\n<td>4<\/td>\r\n<\/tr><\/tbody><\/table><\/div>\r\n<div id=\"fs-id1165135628637\" class=\"commentary\" data-type=\"commentary\">\r\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\r\n<p id=\"fs-id1165135628642\">We can confirm the numbers of positive and negative real roots by examining a graph of the function.\u00a0We can see from the graph in Figure 3 that the function has 0 positive real roots and 2 negative real roots.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201558\/CNX_Precalc_Figure_03_06_0072.jpg\" alt=\"Graph of f(x)=-x^4-3x^3+6x^2-4x-12 with x-intercepts at -4.42 and -1.\" width=\"487\" height=\"403\" data-media-type=\"image\/jpg\"\/><b>Figure 3<\/b>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It 6<\/h3>\r\n<p id=\"fs-id1165135177622\">Use Descartes\u2019 Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for [latex]f\\left(x\\right)=2{x}^{4}-10{x}^{3}+11{x}^{2}-15x+12[\/latex].\u00a0Use a graph to verify the numbers of positive and negative real zeros for the function.<\/p>\r\n<a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-15\/\" target=\"_blank\">Solution<\/a>\r\n\r\n<\/div>","rendered":"<p id=\"fs-id1165135177655\">There is a straightforward way to determine the possible numbers of positive and negative real zeros for any polynomial function. If the polynomial is written in descending order,<strong> Descartes\u2019 Rule of Signs<\/strong> tells us of a relationship between the number of sign changes in [latex]f\\left(x\\right)[\/latex] and the number of positive real zeros. For example, the polynomial function below has one sign change.<\/p>\n<p><span id=\"fs-id1165134378690\" data-type=\"media\" data-alt=\"The function, f(x)=x^4+x^3+x^2+x-1, has one sign change between x and -1.&lt;img&gt;&lt;\/span&gt;&lt;\/p&gt;&lt;p id=\">This tells us that the function must have 1 positive real zero.<\/span><\/p>\n<p id=\"fs-id1165135206084\">There is a similar relationship between the number of sign changes in [latex]f\\left(-x\\right)[\/latex] and the number of negative real zeros.<\/p>\n<p><span id=\"fs-id1165135152070\" data-type=\"media\" data-alt=\"The function, f(-x)=(-x)^4+(-x)^3+(-x)^2+(-x)-1=+ x^4-x^3+x^2-x-1, has three sign changes between x^4 and x^3, x^3 and x^2, and x^2 and x.&lt;img&gt;&lt;\/span&gt;&lt;\/p&gt;&lt;p id=\">In this case, [latex]f\\left(\\mathrm{-x}\\right)[\/latex] has 3 sign changes. This tells us that [latex]f\\left(x\\right)[\/latex] could have 3 or 1 negative real zeros.<\/span><\/p>\n<div id=\"fs-id1165137844271\" class=\"note textbox\" data-type=\"note\" data-has-label=\"true\" data-label=\"A General Note\">\n<h3 class=\"title\" data-type=\"title\">A General Note: Descartes\u2019 Rule of Signs<\/h3>\n<p id=\"fs-id1165137844280\">According to <strong>Descartes\u2019 Rule of Signs<\/strong>, if we let [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]\u00a0be a polynomial function with real coefficients:<\/p>\n<ul id=\"fs-id1165134351104\">\n<li>The number of positive real zeros is either equal to the number of sign changes of [latex]f\\left(x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\n<li>The number of negative real zeros is either equal to the number of sign changes of [latex]f\\left(-x\\right)[\/latex] or is less than the number of sign changes by an even integer.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_03_06_08\" class=\"example\" data-type=\"example\">\n<div id=\"fs-id1165134149118\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-id1165134149120\" class=\"problem textbox shaded\" data-type=\"problem\">\n<h3 data-type=\"title\">Example 7: Using Descartes\u2019 Rule of Signs<\/h3>\n<p>Use Descartes\u2019 Rule of Signs to determine the possible numbers of positive and negative real zeros for [latex]f\\left(x\\right)=-{x}^{4}-3{x}^{3}+6{x}^{2}-4x - 12[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165137932378\" class=\"solution textbox shaded\" data-type=\"solution\">\n<h3>Solution<\/h3>\n<p id=\"fs-id1165137932381\">Begin by determining the number of sign changes.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-04-at-12.31.54-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-11813\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201555\/Screen-Shot-2015-08-04-at-12.31.54-PM.png\" alt=\"Screen Shot 2015-08-04 at 12.31.54 PM\" width=\"534\" height=\"57\" \/><\/a><\/p>\n<p>There are two sign changes, so there are either 2 or 0 positive real roots. Next, we examine [latex]f\\left(-x\\right)[\/latex] to determine the number of negative real roots.<\/p>\n<p id=\"fs-id1165137696389\" style=\"text-align: center;\">[latex]\\begin{cases}f\\left(-x\\right)=-{\\left(-x\\right)}^{4}-3{\\left(-x\\right)}^{3}+6{\\left(-x\\right)}^{2}-4\\left(-x\\right)-12\\hfill \\\\ f\\left(-x\\right)=-{x}^{4}+3{x}^{3}+6{x}^{2}+4x - 12\\hfill \\end{cases}[\/latex]<a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/Screen-Shot-2015-08-04-at-12.32.40-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-11814\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201557\/Screen-Shot-2015-08-04-at-12.32.40-PM.png\" alt=\"Screen Shot 2015-08-04 at 12.32.40 PM\" width=\"536\" height=\"52\" \/><\/a><\/p>\n<p><span data-type=\"media\" data-alt=\"The function, f(-x)=-x^4+3x^3+6x^2+4x-12, has two sign change between -x^4 and 3x^3, and 4x and -12.&lt;img&gt;&lt;\/span&gt;&lt;br&gt;&lt;\/figure&gt;&lt;p id=\"><span data-type=\"media\" data-alt=\"The function, f(-x)=-x^4+3x^3+6x^2+4x-12, has two sign change between -x^4 and 3x^3, and 4x and -12.&lt;img&gt;&lt;\/span&gt;&lt;br&gt;&lt;\/figure&gt;&lt;p id=\">Again, there are two sign changes, so there are either 2 or 0 negative real roots.<\/span><\/span><\/p>\n<p id=\"fs-id1165135383102\">There are four possibilities, as we can see below.<\/p>\n<table id=\"Table_03_06_01\" summary=\"..\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/>\n<col data-align=\"center\" \/><\/colgroup>\n<thead>\n<tr>\n<th>Positive Real<br \/>\nZeros<\/th>\n<th>Negative Real<br \/>\nZeros<\/th>\n<th>Complex<br \/>\nZeros<\/th>\n<th>Total<br \/>\nZeros<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2<\/td>\n<td>2<\/td>\n<td>0<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>2<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>2<\/td>\n<td>2<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<td>4<\/td>\n<td>4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165135628637\" class=\"commentary\" data-type=\"commentary\">\n<h3 data-type=\"title\">Analysis of the Solution<\/h3>\n<p id=\"fs-id1165135628642\">We can confirm the numbers of positive and negative real roots by examining a graph of the function.\u00a0We can see from the graph in Figure 3 that the function has 0 positive real roots and 2 negative real roots.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><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\/25201558\/CNX_Precalc_Figure_03_06_0072.jpg\" alt=\"Graph of f(x)=-x^4-3x^3+6x^2-4x-12 with x-intercepts at -4.42 and -1.\" width=\"487\" height=\"403\" data-media-type=\"image\/jpg\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It 6<\/h3>\n<p id=\"fs-id1165135177622\">Use Descartes\u2019 Rule of Signs to determine the maximum possible numbers of positive and negative real zeros for [latex]f\\left(x\\right)=2{x}^{4}-10{x}^{3}+11{x}^{2}-15x+12[\/latex].\u00a0Use a graph to verify the numbers of positive and negative real zeros for the function.<\/p>\n<p><a href=\"https:\/\/courses.candelalearning.com\/osprecalc\/chapter\/solutions-15\/\" target=\"_blank\">Solution<\/a><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1393\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/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 For Free at : http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.<\/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":276,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Precalculus\",\"author\":\"Jay Abramson, et 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http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175.\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1393","chapter","type-chapter","status-publish","hentry"],"part":1376,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1393","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1393\/revisions"}],"predecessor-version":[{"id":2937,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1393\/revisions\/2937"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1376"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1393\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/media?parent=1393"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1393"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1393"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/wp-json\/wp\/v2\/license?post=1393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}