{"id":65109,"date":"2025-08-18T09:59:46","date_gmt":"2025-08-18T06:59:46","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65109"},"modified":"2026-09-30T20:24:59","modified_gmt":"2026-09-30T17:24:59","slug":"descartes-rule-of-signs","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/descartes-rule-of-signs\/","title":{"rendered":"Descartes\u2019 Rule of Signs Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Analyze Positive and Negative Real Roots of Polynomials<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"65109\" data-favorite-title=\"Descartes\u2019 Rule of Signs\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/descartes-rule-of-signs\/\" data-favorite-excerpt=\"Estimate the number of positive, negative and non-real roots of a polynomial using Descartes&#039; Rule of Signs. Enter the coefficients in descending order of powers to see all possible root counts.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/descartes-rule-of-signs\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65109\" data-post=\"65109\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Polynomial Coefficients<\/label><input type=\"text\" name=\"polynomial_coefficients\" id=\"field-1\" aria-describedby=\"calc-info-1\" step=\"any\" value=\"1 -6 11 -6\"\/><small id=\"calc-info-1\">Enter coefficients of the polynomial in descending order of powers, separated by space. For example: &quot;1 -6 11 -6&quot; represents x\u00b3 - 6x\u00b2 + 11x - 6<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65109\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Your polynomial<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Multiplicity of zero as root<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Number of positive roots<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Number of negative roots<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Minimum number of non-real roots<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65109\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Descartes\u2019 Rule of Signs Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[\nP(x) = a_nx^n + a_{n-1}x^{n-1} + \\dots + a_1x + a_0 \\\\[1em]\nV(x) = \\text{number of sign changes in the coefficients of } P(x) \\\\[1em]\nN_{\\text{pos}} \\le V(x), \\quad N_{\\text{pos}} \\equiv V(x) \\pmod{2} \\\\[1em]\nN_{\\text{neg}} \\le V(-x), \\quad N_{\\text{neg}} \\equiv V(-x) \\pmod{2}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(P(x)\\) \u2014 the polynomial, with coefficients written in descending order of powers<\/li>\n<li>\\(V(x)\\) \u2014 the number of sign changes between consecutive nonzero coefficients of \\(P(x)\\)<\/li>\n<li>\\(V(-x)\\) \u2014 the number of sign changes in \\(P(-x)\\), i.e. the polynomial with the signs of all odd-power coefficients flipped<\/li>\n<li>\\(N_{\\text{pos}}\\), \\(N_{\\text{neg}}\\) \u2014 the number of positive and negative real roots, counted with multiplicity<\/li>\n<\/ul>\n<p>The rule states that the number of positive real roots equals the number of sign changes in \\(P(x)\\) or is less than it by an even number. The same applies to negative roots after substituting \\(-x\\) for \\(x\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Descartes\u2019 Rule of Signs \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> take the polynomial \\(f(x) = x^3 - 6x^2 + 11x - 6\\), entered as coefficients <code>1 -6 11 -6<\/code>.<\/p>\n<p><strong>Step 1 \u2014 positive roots.<\/strong> The signs of the coefficients of \\(f(x)\\) are:<\/p>\n<p>\\[ +,\\ -,\\ +,\\ - \\quad \\Rightarrow \\quad 3 \\text{ sign changes} \\]<\/p>\n<p>So there are exactly 3 or exactly 1 positive real roots.<\/p>\n<p><strong>Step 2 \u2014 negative roots.<\/strong> Substitute \\(-x\\):<\/p>\n<p>\\[ f(-x) = -x^3 - 6x^2 - 11x - 6 \\]<\/p>\n<p>The signs are \\(-, -, -, -\\), giving 0 sign changes, so there are no negative real roots.<\/p>\n<p><strong>Result:<\/strong> 3 or 1 positive roots, 0 negative roots, and at least \\(3 - 3 = 0\\) non-real roots. In fact \\(f(x) = (x-1)(x-2)(x-3)\\), so all three roots (1, 2, 3) are positive.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>In the <strong>Polynomial Coefficients<\/strong> field, type the coefficients of your polynomial in descending order of powers, separated by spaces. For example, <code>1 -6 11 -6<\/code> stands for \\(x^3 - 6x^2 + 11x - 6\\).<\/li>\n<li>Include zero coefficients where a power is missing: <code>2 0 -5 1<\/code> means \\(2x^3 - 5x + 1\\). Leading zeros and trailing zeros are handled automatically.<\/li>\n<li>Press <strong>Calculate<\/strong>.<\/li>\n<li>Read the results: the calculator shows your polynomial in standard form, the multiplicity of zero as a root (if the constant term is 0), the possible numbers of positive real roots, the possible numbers of negative real roots, and the minimum number of non-real (complex) roots.<\/li>\n<\/ol>\n<h2>What Descartes\u2019 Rule of Signs tells you<\/h2>\n<p>Descartes\u2019 Rule of Signs, published by Ren\u00e9 Descartes in 1637, is a quick test that bounds how many positive and negative real roots a polynomial can have \u2014 without actually solving the equation. You write the polynomial in standard form and count how many times the consecutive nonzero coefficients switch sign. That count, reduced by an even number, is the exact number of positive roots. Repeating the count on \\(f(-x)\\) gives the possibilities for negative roots.<\/p>\n<p>The \u201cdiffer by an even number\u201d part matters. If there are 3 sign changes, the polynomial has 3 or 1 positive roots \u2014 never 2 or 0 is not guaranteed; only 3 or 1 are possible. If there is exactly 1 sign change, the polynomial has exactly 1 positive root, which is one of the most useful special cases of the rule.<\/p>\n<p>Combining the two counts with the degree of the polynomial also tells you the minimum number of non-real roots. Since a polynomial of degree \\(n\\) has exactly \\(n\\) roots in total (counting multiplicity, by the Fundamental Theorem of Algebra), and non-real roots always come in conjugate pairs, subtracting the maximum possible counts of real roots from the degree gives the minimum number of complex roots. To explore those, see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-root\/\">Complex Root Calculator<\/a> and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-conjugate\/\">Complex Conjugate Calculator<\/a>.<\/p>\n<h2>Interpreting the results<\/h2>\n<p>Each result line lists every possibility consistent with the rule. \u201c3 or 1\u201d for positive roots means both outcomes are compatible with the sign pattern; additional information (a graph, rational root tests, or numerical methods) is needed to pin down the actual count. A result of exactly 0 is definitive: if \\(f(-x)\\) has no sign changes, the polynomial has no negative real roots, period.<\/p>\n<p>If the constant term is zero, \\(x = 0\\) is a root, and the calculator reports its multiplicity by counting trailing zero coefficients. In that case, apply the rule to the remaining part of the polynomial after factoring out \\(x^k\\).<\/p>\n<h2>Assumptions, limitations and tips<\/h2>\n<ul>\n<li><strong>Coefficients must be real.<\/strong> The rule does not apply to polynomials with complex coefficients.<\/li>\n<li><strong>Zero coefficients are skipped<\/strong> when counting sign changes. For \\(x^3 + 1\\), the signs are \\(+, +\\) \u2014 one sign change, hence exactly 1 negative root after checking \\(f(-x)\\) as well.<\/li>\n<li><strong>The rule counts multiplicity.<\/strong> A double root at \\(x = 2\\) counts as two positive roots.<\/li>\n<li><strong>It gives possibilities, not exact values.<\/strong> The rule never tells you what the roots are, only how many of each sign can exist. Use it to narrow the search before applying other tools.<\/li>\n<li><strong>Zero roots give a quick check:<\/strong> if a polynomial of odd degree has no sign changes in \\(f(x)\\), it must have at least one negative root, because odd-degree polynomials always have at least one real root.<\/li>\n<li>The rule works especially well alongside equation solvers such as the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/cubic-equation\/\">Cubic Equation Calculator<\/a> and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quartic-equation\/\">Quartic Equation Calculator<\/a>, and it can confirm the root structure found with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-equation-2\/\">Quadratic Equation Calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Estimate the number of positive, negative and non-real roots of a polynomial using Descartes' Rule of Signs. Enter the coefficients in descending order of powers to see all possible root counts.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-65109","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65109","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65109"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65109\/revisions"}],"predecessor-version":[{"id":66221,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65109\/revisions\/66221"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65109"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65109"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65109"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}