{"id":34559,"date":"2018-12-26T06:55:46","date_gmt":"2018-12-26T06:55:46","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=34559"},"modified":"2020-11-27T13:03:24","modified_gmt":"2020-11-27T07:33:24","slug":"ncert-solutions-for-class-10-maths-chapter-2-ex-2-4","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/ncert-solutions-for-class-10-maths-chapter-2-ex-2-4\/","title":{"rendered":"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4"},"content":{"rendered":"<p>NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4 are part of <a class=\"button-red\" href=\"https:\/\/cbselibrary.com\/ncert-solutions-for-class-10-maths\/\">NCERT Solutions for Class 10 Maths<\/a>. Here are we have given Chapter 2 <strong>Polynomials\u00a0Class 10 NCERT Solutions Ex 2.4.\u00a0<\/strong><\/p>\n<ul>\n<li><a title=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.1\" href=\"https:\/\/cbselibrary.com\/ncert-solutions-for-class-10-maths-chapter-2\/\" target=\"_blank\" rel=\"noopener\">Polynomials Class 10 Ex 2.1<\/a><\/li>\n<li><a title=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.2\" href=\"https:\/\/cbselibrary.com\/ncert-solutions-for-class-10-maths-chapter-2-ex-2-2\/\" target=\"_blank\" rel=\"noopener\">Polynomials Class 10 Ex 2.2<\/a><\/li>\n<li><a title=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.3\" href=\"https:\/\/cbselibrary.com\/ncert-solutions-for-class-10-maths-chapter-2-ex-2-3\/\" target=\"_blank\" rel=\"noopener\">Polynomials Class 10 Ex 2.3<\/a><\/li>\n<\/ul>\n<table style=\"table-layout: fixed; width: 650px;\" border=\"1\">\n<tbody>\n<tr>\n<td><strong>Board<\/strong><\/td>\n<td>CBSE<\/td>\n<\/tr>\n<tr>\n<td><strong>Textbook<\/strong><\/td>\n<td>NCERT<\/td>\n<\/tr>\n<tr>\n<td><strong>Class<\/strong><\/td>\n<td>Class 10<\/td>\n<\/tr>\n<tr>\n<td><strong>Subject<\/strong><\/td>\n<td>Maths<\/td>\n<\/tr>\n<tr>\n<td><strong>Chapter<\/strong><\/td>\n<td>Chapter 2<\/td>\n<\/tr>\n<tr>\n<td><strong>Chapter Name<\/strong><\/td>\n<td>Polynomials<\/td>\n<\/tr>\n<tr>\n<td><strong>Exercise<\/strong><\/td>\n<td>Ex 2.4<\/td>\n<\/tr>\n<tr>\n<td><strong>Number of Questions Solved<\/strong><\/td>\n<td>5<\/td>\n<\/tr>\n<tr>\n<td><strong>Category<\/strong><\/td>\n<td>NCERT Solutions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4<\/h2>\n<div class=\"solnBox\">\n<p><strong>Question 1.<\/strong>Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and the coefficients in each case:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100103\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-27.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 27\" width=\"446\" height=\"38\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-27.jpg 446w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-27-300x26.jpg 300w\" sizes=\"(max-width: 446px) 100vw, 446px\" \/><\/p>\n<div class=\"ttl\"><strong>Solution:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100106\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-28.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 28\" width=\"410\" height=\"181\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-28.jpg 410w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-28-300x132.jpg 300w\" sizes=\"(max-width: 410px) 100vw, 410px\" \/><br \/>\n<\/strong>On comparing the given polynomial with the polynomial ax<sup>3<\/sup>\u00a0+ bx<sup>2<\/sup>\u00a0+ cx + d, we obtain a = 2, b = 1, c = -5, d = 2<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100108\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-29.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 29\" width=\"291\" height=\"147\" \/><br \/>\nThus, the relationship between the zeroes and the coefficients is verified.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100117\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-30.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 30\" width=\"367\" height=\"103\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-30.jpg 367w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-30-300x84.jpg 300w\" sizes=\"(max-width: 367px) 100vw, 367px\" \/><br \/>\nOn comparing the given polynomial with the polynomial ax<sup><sub>3<\/sub><\/sup>\u00a0+ bx<sub><sup>2<\/sup><\/sub>\u00a0+ cx + d, we obtain a = 1, b = -4, c = 5, d = -2.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100119\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-31.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 31\" width=\"425\" height=\"128\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-31.jpg 425w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-31-300x90.jpg 300w\" sizes=\"(max-width: 425px) 100vw, 425px\" \/><br \/>\nThus, the relationship between the zeroes and the coefficients is verified.<\/div>\n<div class=\"table-responsive\">\n<p>Concept Insight: The zero of a polynomial is that value of the variable which makes the polynomial 0. Remember that there are three relationships between the zeroes of a cubic polynomial and its coefficients which involve the sum of zeroes, product of all zeroes and the product of zeroes taken two at a time.<\/p>\n<\/div>\n<\/div>\n<div class=\"solnBox\">\n<p><strong>Question 2.\u00a0<\/strong>Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.<\/p>\n<div class=\"ttl\"><strong>Solution:<br \/>\n<\/strong>Let the polynomial be ax<sup><sub>3<\/sub><\/sup>\u00a0 + bx<sup><sub>2<\/sub><\/sup>\u00a0+ cx + d and its zeroes be \u03b1, \u03b2, \u03b3.<br \/>\nAccording to the given information:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100122\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-32.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 32\" width=\"176\" height=\"112\" \/><br \/>\nIf a = k, then b = -2k, c = -7k, d = 14k<br \/>\nThus, the required cubic polynomial is k(x<sup><sub>3<\/sub><\/sup>\u00a0&#8211; 2x<sup><sub>2<\/sub><\/sup>\u00a0&#8211; 7x + 14), where k is any real number.<br \/>\nThe simplest polynomial will be obtained by taking k = 1.<\/div>\n<div class=\"table-responsive\">\n<p>Concept Insight:\u00a0A cubic polynomial involves four unknowns and we have three relations involving these unknowns, so the coefficients of the cubic polynomial can be found by assuming the value of one unknown and then finding the other three unknowns from the three relations.<\/p>\n<\/div>\n<\/div>\n<div class=\"solnBox\">\n<p><strong>Question 3.\u00a0<\/strong>If the zeroes of the polynomial x<sup>3<\/sup>\u00a0&#8211; 3x<sup>2<\/sup>\u00a0+ x + 1 are\u00a0 a &#8211; b, a, a + b, find a and b.<\/p>\n<div class=\"ttl\"><strong>Solution:<br \/>\n<\/strong>Let p(x) = x<sup>3<\/sup>\u00a0&#8211; 3x<sup>2<\/sup>\u00a0+ x + 1<br \/>\nThe zeroes of the polynomial p(x) are given as a &#8211; b, a, a + b.<br \/>\nComparing the given polynomial with dx<sup>3<\/sup>\u00a0+ ex<sup>2<\/sup>\u00a0+ fx + g, we can observe that<br \/>\nd = 1, e = -3, f = 1, g = 1<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100124\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-33.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 33\" width=\"209\" height=\"130\" \/><br \/>\nTherefore, the zeroes of p(x) are 1-b, 1, 1+b .<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100126\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-34.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 34\" width=\"211\" height=\"191\" \/><\/p>\n<div class=\"table-responsive\">\n<p>Concept Insight:\u00a0When the polynomial and its zeroes are given then, remember to apply relationships between the zeroes and coefficients. These relations involve the sum of the zeroes, product of zeroes and the product of zeroes taken two at a time.<\/p>\n<\/div>\n<\/div>\n<div class=\"solnBox\">\n<p><strong>Question 4.\u00a0<\/strong>If two zeroes of the polynomial x<sup>4<\/sup>\u00a0&#8211; 6x<sup>3<\/sup>\u00a0&#8211; 26x<sup>2<\/sup>\u00a0+ 138x &#8211; 35 are\u00a02 \u00b1\u221a3, find other zeroes.<\/p>\n<div class=\"ttl\"><strong>Solution:<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100127\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-35.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 35\" width=\"370\" height=\"49\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-35.jpg 370w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-35-300x40.jpg 300w\" sizes=\"(max-width: 370px) 100vw, 370px\" \/><br \/>\n<\/strong>Therefore, x<sup><sub>2<\/sub><\/sup>\u00a0&#8211; 4x + 1 is a factor of the given polynomial. For finding the remaining zeroes of the given polynomial, we will carry out the division of<strong><br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100128\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-36.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 36\" width=\"420\" height=\"385\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-36.jpg 420w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-36-300x275.jpg 300w\" sizes=\"(max-width: 420px) 100vw, 420px\" \/><br \/>\nHence, 7 and -5 are also zeroes of the given polynomial.<\/strong><\/div>\n<div class=\"table-responsive\">\n<p>Concept Insight:\u00a0If a is a zero of a polynomial p(x), where degree of p(x) is greater than 1, then (x &#8211; a) will be a factor of p(x) that is when p(x) is divided by (x &#8211; a), then the remainder obtained will be 0 and the quotient will be a factor of p(x) using division algorithm.\u00a0 Remember that if (x &#8211; a) and (x &#8211; b) are factors of a polynomial, then (x &#8211; a)(x &#8211; b) will also be a factor of that polynomial.<\/p>\n<\/div>\n<\/div>\n<div class=\"solnBox\">\n<p><strong>Question 5.\u00a0<\/strong>If the polynomial x<sup><sub>4<\/sub><\/sup>\u00a0&#8211; 6x<sup><sub>3<\/sub><\/sup>\u00a0+ 16x<sup><sub>2<\/sub><\/sup>\u00a0&#8211; 25x + 10 is divided by another polynomial x<sup><sub>2<\/sub><\/sup>\u00a0&#8211; 2x + k, the remainder comes out to be x + a, find k and a.<\/p>\n<div class=\"ttl\"><strong>Solution:<br \/>\n<\/strong>By division algorithm,<br \/>\nDividend = Divisor x Quotient + Remainder<br \/>\nDivisor x Quotient = Dividend &#8211; Remainder<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100129\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-37.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 37\" width=\"347\" height=\"48\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-37.jpg 347w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-37-300x41.jpg 300w\" sizes=\"(max-width: 347px) 100vw, 347px\" \/><br \/>\nIt will be perfectly divisible by x<sup><sub>2<\/sub><\/sup>\u00a0&#8211; 2x + k.<br \/>\n<img loading=\"lazy\" class=\"alignnone size-full wp-image-100130\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-38.jpg\" alt=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials 38\" width=\"565\" height=\"437\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-38.jpg 565w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2018\/12\/NCERT-Solutions-for-Class-10-Maths-Chapter-2-Polynomials-38-300x232.jpg 300w\" sizes=\"(max-width: 565px) 100vw, 565px\" \/><br \/>\n10 &#8211; a &#8211; 8 x 5 + 25 = 0<br \/>\n10 &#8211; a &#8211; 40 + 25 = 0<br \/>\n&#8211; 5 &#8211; a = 0<br \/>\na = -5<br \/>\nThus, k = 5 and a = -5.<\/div>\n<div class=\"table-responsive\">\n<p>Concept Insight:\u00a0 When a polynomial is divided by any non-zero polynomial, then it satisfies the division algorithm which is as below:<br \/>\nDividend = Divisor x Quotient + Remainder<br \/>\nDivisor x Quotient = Dividend &#8211; Remainder<br \/>\nSo, from this relation, it can be said that the result (Dividend &#8211; Remainder) will be completely divisible by the divisor.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We hope the NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4 help you. If you have any query regarding NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4, drop a comment below and we will get back to you at the earliest.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4 are part of NCERT Solutions for Class 10 Maths. Here are we have given Chapter 2 Polynomials\u00a0Class 10 NCERT Solutions Ex 2.4.\u00a0 Polynomials Class 10 Ex 2.1 Polynomials Class 10 Ex 2.2 Polynomials Class 10 Ex 2.3 Board CBSE Textbook NCERT Class Class 10 &#8230; <a title=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/ncert-solutions-for-class-10-maths-chapter-2-ex-2-4\/\" aria-label=\"More on NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4\">Read more<\/a><\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[6805],"tags":[36685],"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>NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4 - ApluseTopper.com<\/title>\n<meta name=\"description\" content=\"NCERT Solutions for Class 10 Maths Chapter 2 Polynomials Ex 2.4 helpful to finish your Home Work. 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