{"id":308,"date":"2020-12-07T07:22:40","date_gmt":"2020-12-07T01:52:40","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=308"},"modified":"2020-12-07T12:21:19","modified_gmt":"2020-12-07T06:51:19","slug":"division-algorithm-for-polynomials","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/division-algorithm-for-polynomials\/","title":{"rendered":"Division Algorithm For Polynomials"},"content":{"rendered":"<h2><strong>Division Algorithm For Polynomials<\/strong><\/h2>\n<p>If p(x) and g(x) are any two <a href=\"https:\/\/cbselibrary.com\/polynomials\/\">polynomials<\/a> with<br \/>\ng(x) \u2260\u00a00, then we can find polynomials q(x) and r(x) such that<br \/>\np(x) = q(x) \u00d7 g(x) + r(x)<br \/>\nwhere r(x) = 0 or degree of r(x) &lt; degree of g(x).<br \/>\nThe result is called Division Algorithm for polynomials.<br \/>\n<strong>Dividend = Quotient \u00d7 Divisor + Remainder<\/strong><\/p>\n<h2><a href=\"https:\/\/cbselibrary.com\/polynomials\/\">Polynomials<\/a> &#8211; Long Division<\/h2>\n<p><strong>Working rule to Divide a Polynomial <\/strong><strong>by Another Polynomial:<\/strong><br \/>\n<strong>Step 1:<\/strong> First arrange the term of dividend and the divisor in the decreasing order of their degrees.<br \/>\n<strong>Step 2:<\/strong> To obtain the first term of quotient divide the highest degree term of the dividend by the highest degree term of the divisor.<br \/>\n<strong>Step 3:<\/strong> To obtain the second term of the quotient, divide the highest degree term of the new dividend obtained as remainder by the highest degree term of the divisor.<br \/>\n<strong>Step 4:<\/strong> Continue this process till the degree of remainder is less than the degree of divisor.<\/p>\n<p><img loading=\"lazy\" class=\"wp-image-11932 aligncenter\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/division-algorithm-for-polynomials.jpg\" alt=\"\" width=\"647\" height=\"336\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/division-algorithm-for-polynomials.jpg 910w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/division-algorithm-for-polynomials-300x156.jpg 300w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2016\/08\/division-algorithm-for-polynomials-768x399.jpg 768w\" sizes=\"(max-width: 647px) 100vw, 647px\" \/><\/p>\n<p><strong>People also ask<\/strong><\/p>\n<ul>\n<li><a href=\"https:\/\/cbselibrary.com\/factorization-of-polynomials-using-factor-theorem\/\">Factorization of polynomials using factor theorem<\/a><\/li>\n<li><a href=\"https:\/\/cbselibrary.com\/algebraic-identities-of-polynomials\/\">Algebraic Identities Of Polynomials<\/a><\/li>\n<\/ul>\n<p><iframe loading=\"lazy\" title=\"Long Division of Polynomials\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/znsUfxzpiGE?start=317&#038;feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<h2><strong>Division Algorithm For Polynomials With Examples<\/strong><\/h2>\n<p><strong>Example 1:<\/strong>\u00a0 \u00a0 Divide 3x<sup>3<\/sup> + 16x<sup>2<\/sup>\u00a0+ 21x + 20\u00a0 by\u00a0 x + 4.<br \/>\n<strong>Sol.\u00a0<\/strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-126085\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-1.png\" alt=\"Division Algorithm For Polynomials 1\" width=\"464\" height=\"266\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-1.png 464w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-1-300x172.png 300w\" sizes=\"(max-width: 464px) 100vw, 464px\" \/><br \/>\nQuotient = 3x<sup>2<\/sup>\u00a0+ 4x + 5<br \/>\nRemainder = 0<\/p>\n<p><strong>Example 2:<\/strong>\u00a0 \u00a0 Apply the division algorithm to find the quotient and remainder on dividing p(x) by g(x) as given below :<br \/>\np(x) = x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + 5x \u2013 3 and g(x) = x<sup>2<\/sup>\u00a0\u2013 2<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong>We have,<br \/>\np(x) = x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + 5x \u2013 3 and g(x) = x<sup>2<\/sup>\u00a0\u2013 2<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-126087\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-2.png\" alt=\"Division Algorithm For Polynomials 2\" width=\"472\" height=\"180\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-2.png 472w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-2-300x114.png 300w\" sizes=\"(max-width: 472px) 100vw, 472px\" \/><br \/>\nWe stop here since<br \/>\ndegree of (7x \u2013 9) &lt; degree of (x<sup>2<\/sup> \u2013 2)<br \/>\nSo, quotient = x \u2013 3, remainder = 7x \u2013 9<br \/>\nTherefore,<br \/>\nQuotient \u00d7 Divisor + Remainder<br \/>\n= \u00a0 \u00a0 (x \u2013 3) (x<sup>2<\/sup> \u2013 2) + 7x \u2013 9<br \/>\n= \u00a0 \u00a0 x3\u00a0\u2013 2x \u2013 3x<sup>2<\/sup> + 6 + 7x \u2013 9<br \/>\n= \u00a0 \u00a0\u00a0x<sup>3<\/sup> \u2013 3x<sup>2<\/sup> + 5x \u2013 3 = Dividend<br \/>\nTherefore, the division algorithm is verified.<\/p>\n<p><strong>Example 3:<\/strong>\u00a0 \u00a0 Apply the division algorithm to find the quotient and remainder on dividing p(x) by g(x) as given below<br \/>\np(x) = x<sup>4<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + 4x + 5, g (x) = x<sup>2<\/sup>\u00a0+ 1 \u2013 x<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong>We have,<br \/>\np(x) = x<sup>4<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + 4x + 5, g (x) = x<sup>2<\/sup>\u00a0+ 1 \u2013 x<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-126088\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-3.png\" alt=\"Division Algorithm For Polynomials 3\" width=\"265\" height=\"271\" \/><br \/>\nWe stop here since<br \/>\ndegree of (8) &lt; degree of (x<sup>2<\/sup> \u2013 x + 1).<br \/>\nSo, quotient = x<sup>2<\/sup>\u00a0+ x \u2013 3, remainder = 8<br \/>\nTherefore,<br \/>\nQuotient \u00d7 Divisor + Remainder<br \/>\n= \u00a0 (x<sup>2<\/sup> + x \u2013 3) (x<sup>2<\/sup> \u2013 x + 1) + 8<br \/>\n= \u00a0\u00a0x<sup>4<\/sup>\u00a0\u2013 x<sup>3<\/sup>\u00a0+ x<sup>2<\/sup>\u00a0+ x<sup>3<\/sup>\u00a0\u2013 x<sup>2<\/sup>\u00a0+ x \u2013 3x<sup>2<\/sup> + 3x \u2013 3 + 8<br \/>\n= \u00a0\u00a0x<sup>4<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + 4x + 5 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 = Dividend<br \/>\nTherefore the Division Algorithm is verified.<\/p>\n<p><strong>Example 4:<\/strong>\u00a0 \u00a0 Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm. t<sup>2<\/sup>\u00a0\u2013 3; 2t<sup>4<\/sup> + 3t<sup>3<\/sup> \u2013 2t<sup>2<\/sup> \u2013 9t \u2013 12.<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong>We divide \u00a02t<sup>4<\/sup> + 3t<sup>3<\/sup> \u2013 2t<sup>2<\/sup> \u2013 9t \u2013 12 \u00a0by \u00a0t<sup>2<\/sup>\u00a0\u2013 3<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-126089\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/03\/Division-Algorithm-For-Polynomials-4.png\" alt=\"Division Algorithm For Polynomials 4\" width=\"272\" height=\"278\" \/><br \/>\nHere, remainder is 0, so t<sup>2<\/sup>\u00a0\u2013 3\u00a0is a factor of 2t<sup>4<\/sup> + 3t<sup>3<\/sup> \u2013 2t<sup>2<\/sup> \u2013 9t \u2013 12.<br \/>\n2t<sup>4<\/sup> + 3t<sup>3<\/sup> \u2013 2t<sup>2<\/sup> \u2013 9t \u2013 12 = (2t<sup>2<\/sup> + 3t + 4) (t<sup>2<\/sup>\u00a0\u2013 3)<\/p>\n<p><strong>Example 5:<\/strong>\u00a0 \u00a0 Obtain all the zeroes of\u00a03x<sup>4<\/sup> + 6x<sup>3<\/sup> \u2013 2x<sup>2<\/sup> \u2013 10x \u2013 5, if two of its zeroes are \\(\\sqrt{\\frac{5}{3}}\\) \u00a0and \u00a0\u00a0\\(-\\sqrt{\\frac{5}{3}}\\).<br \/>\n<strong>Sol. \u00a0 \u00a0<\/strong>Since two zeroes are\u00a0\\(\\sqrt{\\frac{5}{3}}\\) \u00a0and \u00a0\u00a0\\(-\\sqrt{\\frac{5}{3}}\\)<br \/>\nx = \\(\\sqrt{\\frac{5}{3}}\\), x =\u00a0\\(-\\sqrt{\\frac{5}{3}}\\)<br \/>\n\\(\\Rightarrow \\left( \\text{x}-\\sqrt{\\frac{5}{3}} \\right)\\left( \\text{x +}\\sqrt{\\frac{5}{3}} \\right)={{\\text{x}}^{2}}-\\frac{5}{3}\\) \u00a0 Or \u00a03x<sup>2<\/sup>\u00a0\u2013 5 is a factor of the given polynomial.<br \/>\nNow, we apply the division algorithm to the given polynomial and 3x<sup>2<\/sup>\u00a0\u2013 5.<br \/>\n<img loading=\"lazy\" class=\"\" src=\"https:\/\/c1.staticflickr.com\/9\/8838\/28876489186_62587caeb4_o.png\" alt=\"Division-Algorithm-For-Polynomials-5\" width=\"222\" height=\"227\" \/><br \/>\nSo, 3x<sup>4<\/sup> + 6x<sup>3<\/sup> \u2013 2x<sup>2<\/sup> \u2013 10x \u2013 5<br \/>\n= (3x<sup>2<\/sup> \u2013 5) (x<sup>2<\/sup> + 2x + 1) + 0<br \/>\nQuotient = x<sup>2<\/sup>\u00a0+ 2x + 1 = (x + 1)<sup>2<\/sup><br \/>\nZeroes of (x + 1)<sup>2<\/sup> are \u20131, \u20131.<br \/>\nHence, all its zeroes are \\(\\sqrt{\\frac{5}{3}}\\), \u00a0\\(-\\sqrt{\\frac{5}{3}}\\), \u20131, \u20131.<\/p>\n<p><strong>Example 6:<\/strong>\u00a0 \u00a0 On dividing x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + x + 2 by a polynomial g(x), the quotient and remainder were \u00a0 \u00a0 \u00a0 \u00a0 \u00a0x \u2013 2 and \u20132x + 4, respectively. Find g(x).<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong>p(x) = x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + x + 2 \u00a0 \u00a0q(x) = x \u2013 2\u00a0 \u00a0\u00a0and \u00a0 \u00a0 r (x) = \u20132x + 4<br \/>\nBy Division Algorithm, we know that<br \/>\np(x) = q(x) \u00d7 g(x) + r(x)<br \/>\nTherefore,<br \/>\nx<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + x + 2 = (x \u2013 2) \u00d7 g(x) + (\u20132x + 4)<br \/>\n\u21d2 x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + x + 2 + 2x \u2013 4 = (x \u2013 2) \u00d7 g(x)<br \/>\n\\(\\Rightarrow g(\\text{x})=\\frac{{{\\text{x}}^{3}}-3{{\\text{x}}^{2}}+3\\text{x}-2}{\\text{x}-2}\\)<br \/>\nOn dividing \u00a0x<sup>3<\/sup>\u00a0\u2013 3x<sup>2<\/sup> + x + 2 \u00a0by x \u2013 2,<br \/>\nwe get g(x)<br \/>\n<img loading=\"lazy\" class=\"\" src=\"https:\/\/c1.staticflickr.com\/9\/8678\/28802916492_a26ff97d61_o.png\" alt=\"Division-Algorithm-For-Polynomials-6\" width=\"403\" height=\"217\" \/><br \/>\nHence, g(x) = x<sup>2<\/sup>\u00a0\u2013 x + 1.<\/p>\n<p><strong>Example 7:<\/strong>\u00a0 \u00a0 Give examples of polynomials p(x), q(x) and r(x), which satisfy the division algorithm and<br \/>\n(i) deg p(x) = deg q(x)<br \/>\n(ii) deg q(x) = deg r(x)<br \/>\n(iii) deg q(x) = 0<br \/>\n<strong>Sol. \u00a0\u00a0<\/strong><br \/>\n<strong>(i)<\/strong>\u00a0 \u00a0Let q(x) = 3x<sup>2<\/sup> + 2x + 6, degree of q(x) = 2<br \/>\np(x) = 12x<sup>2<\/sup> + 8x + 24, degree of p(x) = 2<br \/>\nHere, deg p(x) = deg q(x)<br \/>\n<strong>(ii)<\/strong>\u00a0 \u00a0p(x) = x<sup>5<\/sup>\u00a0+ 2x<sup>4<\/sup> + 3x<sup>3<\/sup>+ 5x<sup>2<\/sup> + 2<br \/>\nq(x) = x<sup>2<\/sup>\u00a0+ x + 1, degree of q(x) = 2<br \/>\ng(x) = x<sup>3<\/sup>\u00a0+ x<sup>2<\/sup>\u00a0+ x + 1<br \/>\nr(x) = 2x<sup>2<\/sup> \u2013 2x + 1, degree of r(x) = 2<br \/>\nHere, deg q(x) = deg r(x)<br \/>\n<strong>(iii) \u00a0<\/strong> Let p(x) = 2x<sup>4<\/sup> + x<sup>3<\/sup>\u00a0+ 6x<sup>2<\/sup> + 4x + 12<br \/>\nq(x) = 2, degree of q(x) = 0<br \/>\ng(x) = x<sup>4<\/sup>\u00a0+ 4x<sup>3<\/sup> + 3x<sup>2<\/sup> + 2x + 6<br \/>\nr(x) = 0<br \/>\nHere, deg q(x) = 0<\/p>\n<p><strong>Example 8:<\/strong>\u00a0 \u00a0 If the zeroes of polynomial x<sup>3<\/sup> \u2013 3x<sup>2<\/sup> + x + 1 are a \u2013 b, a , a + b. Find a and b.<br \/>\n<strong>Sol.\u00a0\u00a0\u00a0\u00a0\u2235\u00a0<\/strong>\u00a0a \u2013 b, a, a + b are zeros<br \/>\n\u2234 \u00a0product (a \u2013 b) a(a + b) = \u20131<br \/>\n\u21d2\u00a0(a<sup>2<\/sup> \u2013 b<sup>2<\/sup>) a = \u20131 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026(1)<br \/>\nand sum of zeroes is (a \u2013 b) + a + (a + b) = 3<br \/>\n\u21d2\u00a03a = 3 \u21d2\u00a0a = 1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u2026(2)<br \/>\nby (1) and (2)<br \/>\n(1 \u2013 b<sup>2<\/sup>)1 = \u20131<br \/>\n\u21d2\u00a02 = b<sup>2<\/sup>\u00a0\u21d2 b = \u00b1\u00a0\u221a2<br \/>\n\u2234 \u00a0a = \u20131 &amp; b = \u00b1 \u221a2<\/p>\n<p><strong>Example 9:<\/strong>\u00a0 \u00a0 If two zeroes of the polynomial\u00a0x<sup>4<\/sup> \u2013 6x<sup>3<\/sup> \u201326x<sup>2<\/sup> + 138x \u2013 35 are 2 \u00b1 \u221a3, find other zeroes.<br \/>\n<strong>Sol.\u00a0\u00a0\u00a0\u00a0<\/strong><strong>\u2235 \u00a0<\/strong>2 \u00b1 \u221a3 are zeroes.<br \/>\n\u2234 \u00a0x = 2\u00a0\u00b1 \u221a3<br \/>\n\u21d2 \u00a0x \u2013 2 = \u00b1(squaring both sides)<br \/>\n\u21d2 \u00a0(x \u2013 2)<sup>2<\/sup> = 3 \u00a0 \u00a0 \u00a0\u21d2\u00a0 \u00a0x<sup>2<\/sup> + 4 \u2013 4x \u2013 3 = 0<br \/>\n\u21d2 \u00a0x<sup>2<\/sup> \u2013 4x + 1 = 0 , is a factor of given polynomial<br \/>\n\u2234 \u00a0other factors\u00a0\\(=\\frac{{{\\text{x}}^{4}}-6{{\\text{x}}^{3}}-26{{\\text{x}}^{2}}+138\\text{x}-35}{{{\\text{x}}^{2}}-4\\text{x}+1}\\)<br \/>\n<img loading=\"lazy\" class=\"\" src=\"https:\/\/c1.staticflickr.com\/9\/8683\/28292457643_28fb1cab21_o.png\" alt=\"Division-Algorithm-For-Polynomials-7\" width=\"285\" height=\"199\" \/><br \/>\n\u2234 \u00a0other factors = x<sup>2<\/sup> \u2013 2x \u2013 35<br \/>\n= x<sup>2<\/sup> \u2013 7x + 5x \u2013 35 = x(x \u2013 7) + 5(x \u2013 7)<br \/>\n= (x \u2013 7) (x + 5)<br \/>\n\u2234 \u00a0other zeroes are (x \u2013 7) = 0 \u21d2\u00a0x = 7<br \/>\nx + 5 = 0 \u21d2 \u00a0x = \u2013 5<\/p>\n<p><strong>Example 10:<\/strong>\u00a0 \u00a0 \u00a0If the polynomial x<sup>4<\/sup> \u2013 6x<sup>3<\/sup> + 16x<sup>2<\/sup> \u201325x + 10 is divided by another\u00a0 polynomial x<sup>2<\/sup> \u20132x + k, the remainder comes out to be x + a, find k &amp; a.<br \/>\n<strong>Sol.\u00a0\u00a0\u00a0\u00a0<\/strong><br \/>\n<img loading=\"lazy\" class=\"\" src=\"https:\/\/c1.staticflickr.com\/9\/8808\/28876489056_9265392803_o.png\" alt=\"Division-Algorithm-For-Polynomials-8\" width=\"361\" height=\"209\" \/><br \/>\nAccording to questions, remainder is x + a<br \/>\n\u2234 \u00a0coefficient of x = 1<br \/>\n\u21d2\u00a0\u00a02k\u00a0 \u2013 9 = 1<br \/>\n\u21d2\u00a0\u00a0k = (10\/2) = 5<br \/>\nAlso constant term = a<br \/>\n\u21d2\u00a0\u00a0k<sup>2<\/sup> \u2013 8k + 10 = a \u00a0\u21d2\u00a0\u00a0(5)<sup>2<\/sup> \u2013 8(5) + 10 = a<br \/>\n\u21d2\u00a0\u00a0a = 25 \u2013 40 + 10<br \/>\n\u21d2\u00a0\u00a0a = \u2013 5<br \/>\n\u2234 \u00a0k = 5, a = \u20135<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Division Algorithm For Polynomials If p(x) and g(x) are any two polynomials with g(x) \u2260\u00a00, then we can find polynomials q(x) and r(x) such that p(x) = q(x) \u00d7 g(x) + r(x) where r(x) = 0 or degree of r(x) &lt; degree of g(x). The result is called Division Algorithm for polynomials. Dividend = Quotient &#8230; <a title=\"Division Algorithm For Polynomials\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/division-algorithm-for-polynomials\/\" aria-label=\"More on Division Algorithm For Polynomials\">Read more<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[39,41,24],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Division Algorithm For Polynomials - CBSE Library<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cbselibrary.com\/division-algorithm-for-polynomials\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Division Algorithm For Polynomials\" \/>\n<meta property=\"og:description\" content=\"Division Algorithm For Polynomials If p(x) and g(x) are any two polynomials with g(x) \u2260\u00a00, then we can find polynomials q(x) and r(x) such that p(x) = q(x) \u00d7 g(x) + r(x) where r(x) = 0 or degree of r(x) &lt; degree of g(x). The result is called Division Algorithm for polynomials. Dividend = Quotient ... 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