{"id":67407,"date":"2026-10-21T10:19:23","date_gmt":"2026-10-21T07:19:23","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=67407"},"modified":"2026-10-21T10:19:23","modified_gmt":"2026-10-21T07:19:23","slug":"synthetic-division-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/synthetic-division-calculator\/","title":{"rendered":"Synthetic Division 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\">Divide a Polynomial by x \u2212 c Step by Step<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"67407\" data-favorite-title=\"Synthetic Division\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/synthetic-division-calculator\/\" data-favorite-excerpt=\"Divide a polynomial by a linear divisor x \u2212 c or x + c using synthetic division. Enter the polynomial as text or as a list of coefficients and get the quotient, remainder, step-by-step table and root check.\" 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\/synthetic-division-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-67407\" data-post=\"67407\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Enter the polynomial as<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"enter_the_polynomial_as\"><option value=\"poly\" selected>Polynomial text (e.g. 2x^3 - 6x^2 + 2x - 1)<\/option><option value=\"coeffs\">Coefficients (e.g. 2, -6, 2, -1)<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field is-single has-term\" id=\"input-2\"><label for=\"field-2\">Polynomial P(x)<\/label><div class=\"term\">P(x)<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"polynomial_px\" id=\"field-2\" aria-describedby=\"calc-info-2\" placeholder=\"e.g. 2x^3 - 6x^2 + 2x - 1\" step=\"any\" value=\"2x^3 - 6x^2 + 2x - 1\"\/><small id=\"calc-info-2\">Powers with ^: 2x^4 - 5x^2 + 7. Missing powers are filled with 0 automatically.<\/small><\/div><div class=\"field is-single\" id=\"input-3\" hidden><label for=\"field-3\">Coefficients of P(x)<\/label><textarea id=\"field-3\" name=\"coefficients_of_px\">2, -6, 2, -1<\/textarea><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">Divisor form<\/label><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"divisor_form\"><option value=\"minus\" selected>x \u2212 c (e.g. x \u2212 3)<\/option><option value=\"plus\">x + c (e.g. x + 3)<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">Value of c<\/label><div class=\"term\">c<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"value_of_c\" id=\"field-5\" aria-describedby=\"calc-info-5\" placeholder=\"e.g. 3\" step=\"any\" value=\"3\"\/><small id=\"calc-info-5\">For x + c the calculator divides by x \u2212 (\u2212c). The divisor must be linear; for higher-degree divisors use the Polynomial Long Division Calculator (link in the results).<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"67407\">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> Synthetic Division Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Quotient and remainder<\/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 is-column\" id=\"output-2\"><span class=\"field-title\"><span>Synthetic division table<\/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 is-column\" id=\"output-3\"><span class=\"field-title\"><span>Result: P(x) = (x \u2212 c)\u00b7Q(x) + R<\/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 is-column\" id=\"output-4\"><span class=\"field-title\"><span>Is c a root of P(x)?<\/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 is-column\" id=\"output-5\"><span class=\"field-title\"><span>Steps and check<\/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=\"67407\" 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>Synthetic Division 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\\[ P(x) = (x - c) \\cdot Q(x) + R \\]\n\\[ b_0 = a_0, \\quad b_i = a_i + c \\cdot b_{i-1} \\;\\; (i = 1, \\dots, n), \\quad R = b_n \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a_0, a_1, \\dots, a_n\\) \u2014 coefficients of \\(P(x)\\) written from the highest power down to the constant term<\/li>\n<li>\\(c\\) \u2014 the root of the divisor, i.e. the number for which the divisor equals \\(x - c\\)<\/li>\n<li>\\(b_0, b_1, \\dots, b_{n-1}\\) \u2014 the coefficients of the quotient \\(Q(x)\\)<\/li>\n<li>\\(R\\) \u2014 the remainder; each step multiplies the previous result by \\(c\\) and adds the next coefficient<\/li>\n<\/ul>\n<p>The formula expresses the division algorithm: bring down the leading coefficient, then repeatedly multiply by \\(c\\) and add the next coefficient. All results except the last form the quotient, and the last one is the remainder. By the remainder theorem, \\(R = P(c)\\).<\/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>Synthetic Division \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> divide \\(P(x) = 2x^3 - 6x^2 + 2x - 1\\) by \\(x - 3\\), so \\(c = 3\\). The coefficients are \\(2, -6, 2, -1\\).<\/p>\n<p>\\[ b_0 = 2 \\]<\/p>\n<p>\\[ b_1 = -6 + 3 \\cdot 2 = 0, \\quad b_2 = 2 + 3 \\cdot 0 = 2 \\]<\/p>\n<p>\\[ R = -1 + 3 \\cdot 2 = 5 \\]<\/p>\n<p><strong>Result:<\/strong> \\(Q(x) = 2x^2 + 0x + 2 = 2x^2 + 2\\) and remainder \\(R = 5\\), so \\(2x^3 - 6x^2 + 2x - 1 = (x - 3)(2x^2 + 2) + 5\\). Check: \\(P(3) = 54 - 54 + 6 - 1 = 5 = R\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Choose how to enter the polynomial: <em>Polynomial text<\/em> (e.g. 2x^3 - 6x^2 + 2x - 1) or <em>Coefficients<\/em> (e.g. 2, -6, 2, -1).<\/li>\n<li>Fill in the <em>Polynomial P(x)<\/em> field or the <em>Coefficients of P(x)<\/em> field, depending on the mode you chose. Write powers with ^, list coefficients from the highest power down, and use 0 for any missing power.<\/li>\n<li>Select the <em>Divisor form<\/em>: x \u2212 c or x + c. For x + c the calculator internally divides by \\(x - (-c)\\).<\/li>\n<li>Enter the <em>Value of c<\/em> (any real number, e.g. 3).<\/li>\n<li>Read the results: the quotient \\(Q(x)\\) and remainder \\(R\\), the synthetic division table, the identity \\(P(x) = (x - c)\\cdot Q(x) + R\\), the root check, and the detailed steps with a remainder-theorem verification.<\/li>\n<\/ol>\n<h2>What is synthetic division?<\/h2>\n<p>Synthetic division is a shortcut for dividing a polynomial by a linear divisor of the form \\(x - c\\). It replaces the full long-division layout with a compact table that uses only the coefficients of the polynomial, so no variables or powers have to be written or aligned. For linear divisors it produces exactly the same quotient and remainder as long division \u2014 just much faster and with fewer opportunities for sign errors.<\/p>\n<p>The method relies on the division identity \\(P(x) = (x - c) \\cdot Q(x) + R\\), where the quotient \\(Q(x)\\) has degree one less than \\(P(x)\\) and the remainder \\(R\\) is a constant. Writing the coefficients of \\(P(x)\\) in a row and repeating \u201cmultiply by \\(c\\), add the next coefficient\u201d fills in the coefficients of \\(Q(x)\\) plus the remainder \\(R\\) at the end. A nice by-product is the remainder theorem: the last value in the bottom row equals \\(P(c)\\), the polynomial evaluated at \\(c\\).<\/p>\n<p>Interpreting the result is straightforward. The bottom row gives the quotient, and the shaded final cell gives the remainder. If the remainder is 0, then \\(c\\) is a root of \\(P(x)\\) and \\(x - c\\) is a factor (the factor theorem); the calculator reports this automatically. If the remainder is not 0, the divisor does not divide the polynomial evenly \u2014 the \u201cleft-over\u201d is exactly \\(R\\).<\/p>\n<p>Synthetic division is used constantly when factoring polynomials, finding rational roots, sketching polynomial graphs, and partial fraction decomposition. Once one root is found, you can apply the quotient again to hunt for further roots, lowering the degree each time.<\/p>\n<h2>Limitations and practical tips<\/h2>\n<ul>\n<li><strong>Linear divisors only.<\/strong> Synthetic division works only for divisors of the form \\(x - c\\). To divide by a quadratic or higher-degree polynomial, such as \\(x^2 - 5x + 6\\), use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/polynomial-long-division-calculator\/\">Polynomial Long Division Calculator<\/a>.<\/li>\n<li><strong>Convert the divisor first.<\/strong> If the divisor is \\(x + c\\), rewrite it as \\(x - (-c)\\) and use the negated value as the root: dividing by \\(x + 3\\) means \\(c = -3\\). The calculator does this automatically when you select the x + c form.<\/li>\n<li><strong>Include zeros for missing powers.<\/strong> When entering coefficients, every power from the highest down to the constant must appear, so a polynomial like \\(x^4 - 5x^2 + 7\\) has coefficients 1, 0, \u22125, 0, 7. Entering the polynomial as text fills the gaps automatically.<\/li>\n<li><strong>Divisors with a leading coefficient.<\/strong> If the divisor is \\(ax - b\\) with \\(a \\neq 1\\), divide the whole divisor by \\(a\\) to get \\(x - b\/a\\), run synthetic division with \\(c = b\/a\\), and remember that the true quotient and remainder are the results divided by \\(a\\).<\/li>\n<li><strong>Degrees up to 30.<\/strong> The calculator accepts polynomials of degree 30 or less with real coefficients; roots and factors of polynomials often go hand in hand with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/factor-calculator-all-factors-of-a-number\/\">Factor Calculator<\/a> when testing rational-root candidates, and the quotient from synthetic division can be checked further with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/multiplying-binomials\/\">Multiplying Binomials Calculator<\/a> for simple cases.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Divide a polynomial by a linear divisor x \u2212 c or x + c using synthetic division. Enter the polynomial as text or as a list of coefficients and get the quotient, remainder, step-by-step table and root check.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-67407","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\/67407","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=67407"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67407\/revisions"}],"predecessor-version":[{"id":67408,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67407\/revisions\/67408"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=67407"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=67407"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=67407"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}