{"id":23017,"date":"2023-01-30T07:54:10","date_gmt":"2023-01-30T07:54:10","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=23017"},"modified":"2023-02-20T08:47:33","modified_gmt":"2023-02-20T08:47:33","slug":"remainder-theorem","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem","title":{"rendered":"Remainder Theorem"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-b8fd060b-2a79-4178-b092-bffc9b44280c\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem#0-remainder-theorem-introduction>Remainder Theorem:&nbsp;Introduction<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem#1-what-is-the-remainder-theorem>What Is the Remainder Theorem?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem#7-facts-about-remainder-theorem>Facts about Remainder Theorem<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem#9-solved-examples>Solved Examples<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem#10-practice-problems>Practice Problems<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem#11-frequently-asked-questions>Frequently Asked Questions<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"0-remainder-theorem-introduction\">Remainder Theorem:&nbsp;Introduction<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Just like we divide a number by another number, we can also divide one polynomial by another.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">When we divide a polynomial by a linear polynomial, i.e., a polynomial of degree 1, we apply the<strong>&nbsp;<\/strong>remainder theorem to find the remainder.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Consider an example. The division $9 \\div 2$ can be written as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"617\" height=\"299\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-1.png\" alt=\"Remainder, dividend, divisor, quotient in the number division\" class=\"wp-image-23031\" title=\"Remainder, dividend, divisor, quotient in the number division\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-1.png 617w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-1-300x145.png 300w\" sizes=\"auto, (max-width: 617px) 100vw, 617px\" \/><\/figure>\n\n\n\n<p class=\" eplus-wrapper\">Similarly, if we divide a polynomial f(x) by a linear polynomial g(x) and get a remainder as r(x), we can express it as:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"617\" height=\"295\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-2.png\" alt=\"Identifying parts of a polynomial division\" class=\"wp-image-23032\" title=\"Identifying parts of a polynomial division\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-2.png 617w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-2-300x143.png 300w\" sizes=\"auto, (max-width: 617px) 100vw, 617px\" \/><\/figure>\n\n\n\n<p class=\" eplus-wrapper\">One way of finding the remainder is by long division of polynomials. This gives us the dividend and the quotient. Instead of this, we can use the theorem!<\/p>\n\n\n\n<p class=\" eplus-wrapper\">The&nbsp;remainder theorem&nbsp;of polynomials allows us to calculate the remainder of the division of a polynomial without carrying out the steps of long division.<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/choose-the-correct-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_remainder_c_pt.png\" alt=\"Choose the Correct Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Choose the Correct Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/complete-the-division-with-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_complete_div_2_pt.png\" alt=\"Complete the Division with Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Complete the Division with Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/determine-the-quotient-and-the-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_quotient_remainder_pt.png\" alt=\"Determine the Quotient and the Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Determine the Quotient and the Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/determine-the-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_remainder_pt.png\" alt=\"Determine the Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Determine the Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/divide-and-check-the-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_div_check_remainder_pt.png\" alt=\"Divide and Check the Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Divide and Check the Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/divide-and-fill-in-the-quotient-and-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_qr_3d_by_1d_pt.png\" alt=\"Divide and Fill in the Quotient and Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Divide and Fill in the Quotient and Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/fill-in-the-correct-quotient-and-remainder-after-dividing-2-digit-number-by-1-digit-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_qr_4d_by_1d_pt.png\" alt=\"Fill in the Correct Quotient and Remainder After Dividing 2-Digit Number by 1-Digit Number Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Fill in the Correct Quotient and Remainder After Dividing 2-Digit Number by 1-Digit Number Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/find-the-quotient-and-the-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_qr_2d_by_1d_pt.png\" alt=\"Find the Quotient and the Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Find the Quotient and the Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/find-the-remainder\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_remainder_a_pt.png\" alt=\"Find the Remainder Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Find the Remainder Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/find-the-remainder-when-dividing-a-3-digit-number-by-1-digit-number\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/mult_div_pv_find_remainder_b_pt.png\" alt=\"Find the Remainder When Dividing a 3-Digit Number by 1-Digit Number Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Find the Remainder When Dividing a 3-Digit Number by 1-Digit Number Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    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\"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"eplus-wrapper wp-block-heading\" id=\"1-what-is-the-remainder-theorem\">What Is the Remainder Theorem?<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">The&nbsp;remainder<strong> <\/strong>theorem states that when we divide a polynomial p$(x)$ having a degree greater than or equal to 1 by a linear polynomial $(x \u2212 a)$, the remainder is given by r$(x) =$ p$(a)$.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">In simple words, if p$(x) = (x \u2212 a) q(x) + r(x)$, then $r(x) = p(a)$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Using this theorem, the remainder of the division of any polynomial by a linear polynomial can be easily calculated.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Note:<\/strong>&nbsp;<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The degree of the remainder polynomial is always 1 less than the degree of the divisor polynomial. So, if any polynomial is divided by a linear polynomial (polynomial with degree $= 1$), the remainder is always constant (degree $= 0$).<\/li>\n\n\n\n<li class=\" eplus-wrapper\">We can say that $(x \u2212 a)$ is the divisor of the polynomial P(x) if and only if P(a) $= 0$. So, the theorem is applied to factorize polynomials.<\/li>\n<\/ol>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-with-remainder-horizontal-division\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-with-remainder-horizontal-division.jpeg\" alt=\"Divide 2-Digit Numbers by 1-Digit Numbers with Remainder: Horizontal Division Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-with-remainder-horizontal-timed-practice\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-with-remainder-horizontal-timed-practice.jpeg\" alt=\"Divide 2-Digit Numbers by 1-Digit Numbers with Remainder: Horizontal Timed Practice\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-with-remainder-missing-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-with-remainder-missing-numbers.jpeg\" alt=\"Divide 2-Digit Numbers by 1-Digit Numbers with Remainder: Missing Numbers Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-without-remainder-horizontal-division\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-without-remainder-horizontal-division.jpeg\" alt=\"Divide 2-Digit Numbers by 1-Digit Numbers without Remainder: Horizontal Division Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-without-remainder-horizontal-timed-practice\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-without-remainder-horizontal-timed-practice.jpeg\" alt=\"Divide 2-Digit Numbers by 1-Digit Numbers without Remainder: Horizontal Timed Practice\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-without-remainder-missing-numbers\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/divide-2-digit-numbers-by-1-digit-numbers-without-remainder-missing-numbers.jpeg\" alt=\"Divide 2-Digit Numbers by 1-Digit Numbers without Remainder: Missing Numbers Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/divide-2-digit-numbers-by-2-digit-numbers-with-remainder-horizontal-division\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/divide-2-digit-numbers-by-2-digit-numbers-with-remainder-horizontal-division.jpeg\" alt=\"Divide 2-Digit Numbers by 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     checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"eplus-wrapper wp-block-heading\" id=\"2-remainder-theorem-statement\">Remainder Theorem: Statement<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">The<strong> <\/strong>remainder theorem states that the remainder when a polynomial p(x) is divided by a linear polynomial $(x \u2212 a)$ is p(a).<\/p>\n\n\n\n<p class=\" eplus-wrapper\">To find the remainder in a polynomial division, we can follow the given steps:<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Equate the linear polynomial or the divisor to 0 to find its \u201czero.\u201d&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\" eplus-wrapper\">So, $(x \u2212 a) = 0 \\Rightarrow x = a$<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">Substitute $x = a$ in the given polynomial p(x) to find the remainder.<\/li>\n<\/ol>\n\n\n\n<p class=\" eplus-wrapper\">Remainder $= p(a)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Example: Find the remainder when $p(x) = x^{2} + 4x + 4$ is divided by $(x \u2212 1)$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Method 1: Long division<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"542\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-3.png\" alt=\"Finding remainder of polynomial division using long division method\" class=\"wp-image-23033\" title=\"Finding remainder of polynomial division using long division method\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-3.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/01\/Remainder-Theorem-3-300x262.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\" eplus-wrapper\">Remainder $= 9$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Method 2: Remainder theorem<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Here, $x \u2212 1 = 0 \\Rightarrow x = 1$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Thus, $a = 1$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Remainder $= p(1) = 1^{2} + 4(1) + 4 = 9$<\/p>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"3-remainder-theorem-proof\">Remainder Theorem: Proof<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">By the division algorithm, we can write&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Dividend $=$ (Divisor $\\times$ Quotient) $+$ Remainder<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Let us assume that q(x) is the quotient and \u201cr\u201d is the remainder when a polynomial p(x) is divided by a linear polynomial $(x &#8211; a)$.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Applying this to polynomial division, we get:<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$p(x) = (x \u2212 a) q(x) + r$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">If we substitute $x = a$, we get&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$p(a) = (a \u2212 a) q(a) + r$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$p(a) = (0) \u00b7 q(a) + r$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$p(a) = r$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Therefore, the remainder $= p(a)$.<\/p>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"4-how-to-divide-a-polynomial-by-a-non-zero-polynomial\">How to Divide a Polynomial by a Non-Zero Polynomial?<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">The steps are as follows:<\/p>\n\n\n\n<p class=\" eplus-wrapper\">1. Arrange the polynomials (dividend and divisor) in decreasing order of their degree.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">2. Divide the dividend polynomial&#8217;s first term by the divisor\u2019s first term to obtain the first term of the quotient.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">3. Multiply the divisor polynomial by the quotient\u2019s first term.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">3. Subtract the resulting product from the dividend to obtain the remainder.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">4. This remainder is the dividend now, and the divisor will stay the same.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">5. Repeat the process from the first step again until the degree of the new dividend is less than the degree of the divisor.&nbsp;<\/p>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"5-how-to-use-the-remainder-theorem-of-polynomials\">How to Use the Remainder Theorem of Polynomials?<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Note the following important points when finding remainder using the remainder theorem.<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The remainder when p(x) is divided by $(x \u2212 a)$ is p(a)&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\" eplus-wrapper\">since $x \u2212 a = 0 \\Rightarrow x = a$.<\/p>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The remainder when p(x) is divided by $(ax + b)$ is $p( \\frac{\u2212 b}{a})$&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\" eplus-wrapper\">since $ax + b = 0 \\Rightarrow x = \\frac{\u2212 b}{a}$.<\/p>\n\n\n\n<ol start=\"3\" class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The remainder when p(x) is divided by $(ax \u2212 b)$ is $p(\\frac{b}{a})$&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\" eplus-wrapper\">since $ax \u2212 b = 0 \\Rightarrow x = \\frac{b}{a}$.<\/p>\n\n\n\n<ol start=\"4\" class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The remainder when p(x) is divided by $bx \u2212 a$ is $p(\\frac{a}{b})$&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\" eplus-wrapper\">since $bx \u2212 a = 0 \\Rightarrow x = \\frac{a}{b}$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Remainder Theorem examples:<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example 1:<\/strong> Find the remainder when you divide $3x^{2} + 2x + 5 by (x + 1)$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x \u2212 a = x + 1$&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Here, $a = \u2212 1$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Remainder $= p(a)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">So $r = p( \u2212 1)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Substituting $x = \u2212 1$, we get<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;$3( \u2212 1)^{2} + 2( \u2212 1) + 5&nbsp; = 3(1) + 2( \u2212 1) + 5 = 3 \u2212 2 + 5&nbsp; = 6$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">So, the remainder $= 6$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Example 2: <\/strong>Find the remainder when you divide $5x^{2} + x + 1$ by $(2x \u2212 1)$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Put $2x \u2212 1 = 0$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Here, $a = \\frac{1}{2}$ and $p(x) = 5x^{2} + x + 1$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Remainder $= p(a) = p(\\frac{1}{2}) = 5(\\frac{1}{2})^{2} + (\\frac{1}{2}) + 1 = \\frac{5}{4} + \\frac{1}{2} + 1 = \\frac{11}{4}$<\/p>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"6-remainder-theorem-vs-factor-theorem\">Remainder Theorem vs. Factor Theorem<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Take a look at the following table to understand the difference between remainder theorem and factor theorem.<\/p>\n\n\n\n<figure class=\"wp-block-table eplus-wrapper\"><table class=\"wj-table-class\"><tbody><tr><td><strong>Remainder Theorem<\/strong><\/td><td><strong>Factor Theorem<\/strong><\/td><\/tr><tr><td>The remainder theorem states that the remainder when p(x) is divided by a linear polynomial of the form $(x \u2212 a)$ is given by p(a).<\/td><td>The factor theorem states that $(x \u2212 a)$ is a factor of p(x) if and only if f(a) $= 0$.<\/td><\/tr><tr><td>Remainder theorem is used to find the remainder of the polynomial division only when the divisor polynomial is linear.<\/td><td>Factor theorem helps to decide if a linear polynomial is a factor of the given polynomial or not.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"7-facts-about-remainder-theorem\">Facts about Remainder Theorem<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">Here are some facts about the<strong>&nbsp;<\/strong>remainder theorem:<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\">The remainder polynomial is always 1 less degree than the degree of the divisor polynomial.<\/li>\n\n\n\n<li class=\" eplus-wrapper\">Based on this, we can say that when a polynomial is divided by a linear polynomial (whose degree is 1), the remainder is a constant (whose degree is 0).<\/li>\n\n\n\n<li class=\" eplus-wrapper\">The basis of the theorem is that we can find the zero of the divisor linear polynomial by setting it to 0. We say, if $x \u2212 a = 0$, then $x = a$.&nbsp;This is why the divisor polynomial plays an important role in the&nbsp;remainder theorem.&nbsp;<\/li>\n<\/ul>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"8-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\" eplus-wrapper\">The remainder theorem&nbsp;is important because it helps us divide a polynomial and find the remainder easily. Instead of going through the steps of a long division, we can directly plug in the remainder theorem formula. All we have to do is substitute the right value and solve the polynomial!<\/p>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"9-solved-examples\">Solved Examples<\/h2>\n\n\n\n<p class=\" eplus-wrapper\"><strong>1. If <\/strong>$p(x) = x^{3}<sup> <\/sup>+ 2x + 1$<strong> is divided by <\/strong>$(x \u2212 1)$<strong>, what is the remainder?<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong>Applying the theorem, we know that the remainder is $r = p(a)$, where p(x) is divided by a linear polynomial of the form $(x \u2212 a)$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Here, $x &#8211; 1 =  \\Rightarrow a = 1$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">So, the remainder $= p(1) = 1^{3} + 2(1) + 1$&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 1 + 2 + 1$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 4$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>2. Find the remainder when <\/strong>$f(x) = 2x^{2} + 4x \u2212 4$<strong> is divided by <\/strong>$(2x \u2212 1)$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:<\/strong>&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Here, $f(x) = 2x^{2} + 4x \u2212 4$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$2x \u2212 1 = 0<strong> <\/strong>\\Rightarrow&nbsp; a = \\frac{1}{2}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Remainder $= f(\\frac{1}{2}) =&nbsp; 2(\\frac{1}{2})^{2} + 4(\\frac{1}{2}) \u2212 4 = \\frac{1}{2} + 2 \u2212 4 =&nbsp; \\frac{\u2212 3}{2}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>3. Check if <\/strong>$(x + 3)$<strong> is a factor of <\/strong>$x^{2}+ 6x + 9$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution:&nbsp;<\/strong>$(x \u2212 a)$ is a factor of p(x) if and only if f(a) $= 0$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x + 3 = 0 \\Rightarrow a = \u2212 3$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$f( \u2212 3) = ( \u2212 3)^{2} + 6( \u2212 3) + 9 = 9 \u2212 18 + 9 = 18 \u2212 18 = 0$<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>4. If the polynomial <\/strong>$p(x)= 2x^{2} + 3x + 5$<strong> is divided by <\/strong>$(x \u2212 2)$<strong>, find the remainder.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution.&nbsp;<\/strong>By the theorem, we know that $r = p(a)$ where $x \u2212 a = 0$.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x \u2212 2 = 0 \\Rightarrow a = 2$.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">So, $r = p(a) = p(2)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$p(2) = 2(4) + 3(2) + 5$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 8 + 6 + 5$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;$= 19$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Therefore, the remainder is 19.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>5. Find the remainder when you divide <\/strong>$4x^{2} + 3x + 6$ by $(x \u2212 3)$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution<\/strong>. We know that dividend $p(x) = 4x^{2} + 3x + 6$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Remainder $r = p(a)$, where divisor is $x \u2212 a$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Since the divisor is $x \u2212 3 = 0 \\Rightarrow a = 3$.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">So $r = p(3)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= 4(3)^{2} + 3(3) + 6$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= 36 + 9 + 6$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= 51$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Hence, the remainder $= 51$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>6. Find the remainder when you divide <\/strong>$8x^{2} + 8x + 9$<strong> by <\/strong>$(x \u2212 10)$<strong>.<\/strong><\/p>\n\n\n\n<p class=\" eplus-wrapper\"><strong>Solution: <\/strong>p(x) $= 8x^{2} + 8x + 9$&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Remainder = p(a), where divisor is $(x \u2212 a)$.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$x \u2212 10 = 0, a = 10$.&nbsp;<\/p>\n\n\n\n<p class=\" eplus-wrapper\">So $r = p(10)$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= 8(10)^{2} + 8(10) + 9$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= 800 + 80 + 9$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">$= 889$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Hence, the remainder $= 889$.<\/p>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"10-practice-problems\">Practice Problems<\/h2>\n\n\n\n<p class=\" eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Remainder Theorem<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">If p(x), a polynomial, is divided by $(x - c)$, what is the remainder?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">c<\/div><div class=\"spq_answer_block\" data-value=\"1\">0<\/div><div class=\"spq_answer_block\" data-value=\"2\">x<\/div><div class=\"spq_answer_block\" data-value=\"3\">p(c)<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: p(c)<br\/>Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and $(x - a)$ is the divisor.<br>\r\nHere, $(x - c) = 0 \\Rightarrow a = c$.<br>\r\nSo the remainder $= p(c)$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">If p(x), a polynomial, is divided by $(x - 5)$, what is the remainder?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">5x<\/div><div class=\"spq_answer_block\" data-value=\"1\">0<\/div><div class=\"spq_answer_block\" data-value=\"2\">p(5)<\/div><div class=\"spq_answer_block\" data-value=\"3\">$p(-5)$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: p(5)<br\/>Applying the remainder theorem, we know that the remainder $r = p(a)$, where p(x) is a polynomial and $(x - a)$ is the divisor.<br>\r\nHere, the divisor is $(x-5)\\Rightarrow a = 5$.<br>\r\n$r = p(a) = p(5)$.<br>\r\nTherefore, the remainder is p(5).<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Find the remainder if the $x^{2} + 6$ is divided by $x - 2$.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$10$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$20$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$-10$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$8$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $10$<br\/>Here, $a = 2$.<br>\r\nRemainder $= p(2) = 2^{2} + 6 = 10$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">The remainder when p(x) is divided by $(ax-b)$ is<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">p$(\\frac{-b}{a})$<\/div><div class=\"spq_answer_block\" data-value=\"1\">p$(\\frac{b}{a})$<\/div><div class=\"spq_answer_block\" data-value=\"2\">p$(\\frac{a}{b})$<\/div><div class=\"spq_answer_block\" data-value=\"3\">p$(a)$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: p$(\\frac{b}{a})$<br\/>The remainder when p(x) is divided by $(ax - b)$ is p$(\\frac{b}{a})$ since $ax - b = 0 \\Rightarrow x = \\frac{b}{a}$ .<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Remainder Theorem\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Remainder Theorem\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"If p(x), a polynomial, is divided by $$(x - c)$$, what is the remainder?\",\n                    \"text\": \"If p(x), a polynomial, is divided by $$(x - c)$$, what is the remainder?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, $$(x - c) = 0 \\\\Rightarrow a = c$$.<br>\r\nSo the remainder $$= p(c)$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"c\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, $$(x - c) = 0 \\\\Rightarrow a = c$$.<br>\r\nSo the remainder $$= p(c)$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"0\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, $$(x - c) = 0 \\\\Rightarrow a = c$$.<br>\r\nSo the remainder $$= p(c)$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"x\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, $$(x - c) = 0 \\\\Rightarrow a = c$$.<br>\r\nSo the remainder $$= p(c)$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"p(c)\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, $$(x - c) = 0 \\\\Rightarrow a = c$$.<br>\r\nSo the remainder $$= p(c)$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Applying the theorem, we know that the remainder r = p(a), where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, $$(x - c) = 0 \\\\Rightarrow a = c$$.<br>\r\nSo the remainder $$= p(c)$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"If p(x), a polynomial, is divided by $$(x - 5)$$, what is the remainder?\",\n                    \"text\": \"If p(x), a polynomial, is divided by $$(x - 5)$$, what is the remainder?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Applying the remainder theorem, we know that the remainder $$r = p(a)$$, where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, the divisor is $$(x-5)\\\\Rightarrow a = 5$$.<br>\r\n$$r = p(a) = p(5)$$.<br>\r\nTherefore, the remainder is p(5).\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"5x\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Applying the remainder theorem, we know that the remainder $$r = p(a)$$, where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, the divisor is $$(x-5)\\\\Rightarrow a = 5$$.<br>\r\n$$r = p(a) = p(5)$$.<br>\r\nTherefore, the remainder is p(5).\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"0\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Applying the remainder theorem, we know that the remainder $$r = p(a)$$, where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, the divisor is $$(x-5)\\\\Rightarrow a = 5$$.<br>\r\n$$r = p(a) = p(5)$$.<br>\r\nTherefore, the remainder is p(5).\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$p(-5)$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Applying the remainder theorem, we know that the remainder $$r = p(a)$$, where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, the divisor is $$(x-5)\\\\Rightarrow a = 5$$.<br>\r\n$$r = p(a) = p(5)$$.<br>\r\nTherefore, the remainder is p(5).\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"p(5)\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Applying the remainder theorem, we know that the remainder $$r = p(a)$$, where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, the divisor is $$(x-5)\\\\Rightarrow a = 5$$.<br>\r\n$$r = p(a) = p(5)$$.<br>\r\nTherefore, the remainder is p(5).\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Applying the remainder theorem, we know that the remainder $$r = p(a)$$, where p(x) is a polynomial and $$(x - a)$$ is the divisor.<br>\r\nHere, the divisor is $$(x-5)\\\\Rightarrow a = 5$$.<br>\r\n$$r = p(a) = p(5)$$.<br>\r\nTherefore, the remainder is p(5).\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Find the remainder if the $$x^{2} + 6$$ is divided by $$x - 2$$.\",\n                    \"text\": \"Find the remainder if the $$x^{2} + 6$$ is divided by $$x - 2$$.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Here, $$a = 2$$.<br>\r\nRemainder $$= p(2) = 2^{2} + 6 = 10$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$20$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Here, $$a = 2$$.<br>\r\nRemainder $$= p(2) = 2^{2} + 6 = 10$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$-10$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Here, $$a = 2$$.<br>\r\nRemainder $$= p(2) = 2^{2} + 6 = 10$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$8$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Here, $$a = 2$$.<br>\r\nRemainder $$= p(2) = 2^{2} + 6 = 10$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$10$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Here, $$a = 2$$.<br>\r\nRemainder $$= p(2) = 2^{2} + 6 = 10$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Here, $$a = 2$$.<br>\r\nRemainder $$= p(2) = 2^{2} + 6 = 10$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The remainder when p(x) is divided by $$(ax-b)$$ is\",\n                    \"text\": \"The remainder when p(x) is divided by $$(ax-b)$$ is\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The remainder when p(x) is divided by $$(ax - b)$$ is p$$(\\\\frac{b}{a})$$ since $$ax - b = 0 \\\\Rightarrow x = \\\\frac{b}{a}$$ .\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"p$$(\\\\frac{-b}{a})$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The remainder when p(x) is divided by $$(ax - b)$$ is p$$(\\\\frac{b}{a})$$ since $$ax - b = 0 \\\\Rightarrow x = \\\\frac{b}{a}$$ .\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"p$$(\\\\frac{a}{b})$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The remainder when p(x) is divided by $$(ax - b)$$ is p$$(\\\\frac{b}{a})$$ since $$ax - b = 0 \\\\Rightarrow x = \\\\frac{b}{a}$$ .\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"p$$(a)$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The remainder when p(x) is divided by $$(ax - b)$$ is p$$(\\\\frac{b}{a})$$ since $$ax - b = 0 \\\\Rightarrow x = \\\\frac{b}{a}$$ .\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"p$$(\\\\frac{b}{a})$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The remainder when p(x) is divided by $$(ax - b)$$ is p$$(\\\\frac{b}{a})$$ since $$ax - b = 0 \\\\Rightarrow x = \\\\frac{b}{a}$$ .\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The remainder when p(x) is divided by $$(ax - b)$$ is p$$(\\\\frac{b}{a})$$ since $$ax - b = 0 \\\\Rightarrow x = \\\\frac{b}{a}$$ .\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"11-frequently-asked-questions\">Frequently Asked Questions<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\"><strong>When do we use the factor theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\">\n\n<p class=\" eplus-wrapper\">Factor theorem helps us to check if the linear polynomial is a factor of a given polynomial.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\"><strong>What is the origin of the remainder theorem?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\">\n\n<p class=\" eplus-wrapper\">The remainder theorem finds its origin in the work of Chinese mathematician Sun Zi.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\"><strong>What if we apply the remainder theorem\u00a0and the remainder turns out to be 0?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\">\n\n<p class=\" eplus-wrapper\">If the remainder is 0, the given divisor $(x \u2212 a)$ is a factor of the dividend polynomial. In other words, it divides it exactly.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\"><strong>What is the simplest way to apply the remainder formula?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\">\n\n<p class=\" eplus-wrapper\">Where $p(x)$ is the polynomial and $(x \u2212 a)$ is the divisor, we simply substitute a with $x$ and solve $p(x)$.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\"><strong>Can the remainder formula be used for non-linear polynomials?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-b30bd9ec-1bd5-4688-a492-e9f9f041ea4a\">\n\n<p class=\" eplus-wrapper\">No, to apply the formula, the divisor has to be a linear polynomial and cannot be a non-linear polynomial.<\/p>\n\n<\/div><\/div>\n<\/div>\n\n\n<h2 class=\"eplus-wrapper wp-block-heading\" id=\"12-related-article-links\">Related Article Links<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\" eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/multiplication\/long-multiplication\">Long multiplication<\/a><\/li>\n\n\n\n<li class=\" eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/multiplication\/factor\">Factors<\/a><\/li>\n\n\n\n<li class=\" eplus-wrapper\"><a href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/algebra\/order-of-operations\">Order of Operations<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Remainder Theorem:&nbsp;Introduction Just like we divide a number by another number, we can also divide one polynomial by another.&nbsp; When we divide a polynomial by a linear polynomial, i.e., a polynomial of degree 1, we apply the&nbsp;remainder theorem to find the remainder.&nbsp; Consider an example. The division $9 \\div 2$ can be written as: Similarly, &#8230; <a title=\"Remainder Theorem\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/remainder-theorem\" aria-label=\"More on Remainder Theorem\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-23017","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/23017","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=23017"}],"version-history":[{"count":12,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/23017\/revisions"}],"predecessor-version":[{"id":25174,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/23017\/revisions\/25174"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=23017"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=23017"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=23017"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}