not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions

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1 POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the x-coordintes of the points where the grph of y = p(x) intersects the x-xis. Reltion between the zeroes nd coefficients of polynomil: If α nd β re the zeroes of qudrtic polynomil x + bx + c, then α + β b, αβ If α, β nd γ re the zeroes of cubic polynomil x + bx + cx + d, then α+β+γ b, α β + β γ + γ α c nd α β γ d. The division lgorithm sttes tht given ny polynomil p(x) nd ny non-zero polynomil g(x), there re polynomils q(x) nd r(x) such tht p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x). (B) Multiple Choice Questions CHAPTER Choose the correct nswer from the given four options: Smple Question 1: If one zero of the qudrtic polynomil x + x + k is, then the vlue of k is (A) 10 (B) 10 (C) 5 (D) 5 Solution : Answer (B) c.

2 POLYNOMIALS 9 Smple Question : Given tht two of the zeroes of the cubic polynomil x + bx + cx + d re 0, the third zero is b b c (A) (B) (C) (D) d Solution : Answer (A). [Hint: Becuse if third zero is α, sum of the zeroes b = α = ] EXERCISE.1 Choose the correct nswer from the given four options in the following questions: 1. If one of the zeroes of the qudrtic polynomil (k 1) x + k x + 1 is, then the vlue of k is 4 4 (A) (B) (C). A qudrtic polynomil, whose zeroes re nd 4, is (A) x x + 1 (B) x + x + 1 (C) x x 6 (D) x + x 4. If the zeroes of the qudrtic polynomil x + ( + 1) x + b re nd, then (A) = 7, b = 1 (B) = 5, b = 1 (C) =, b = 6 (D) = 0, b = 6 4. The number of polynomils hving zeroes s nd 5 is (A) 1 (B) (C) (D) more thn 5. Given tht one of the zeroes of the cubic polynomil x + bx + cx + d is zero, the product of the other two zeroes is (A) c c (B) (C) 0 (D) b 6. If one of the zeroes of the cubic polynomil x + x + bx + c is 1, then the product of the other two zeroes is (A) b + 1 (B) b 1 (C) b + 1 (D) b 1 (D)

3 10 EXEMPLAR PROBLEMS 7. The zeroes of the qudrtic polynomil x + 99x + 17 re (A) both positive (B) both negtive (C) one positive nd one negtive (D) both equl 8. The zeroes of the qudrtic polynomil x + kx + k, k 0, (A) cnnot both be positive (C) re lwys unequl (B) cnnot both be negtive (D) re lwys equl 9. If the zeroes of the qudrtic polynomil x + bx + c, c 0 re equl, then (A) c nd hve opposite signs (C) c nd hve the sme sign (B) c nd b hve opposite signs (D) c nd b hve the sme sign 10. If one of the zeroes of qudrtic polynomil of the form x +x + b is the negtive of the other, then it (A) hs no liner term nd the constnt term is negtive. (B) hs no liner term nd the constnt term is positive. (C) cn hve liner term but the constnt term is negtive. (D) cn hve liner term but the constnt term is positive. 11. Which of the following is not the grph of qudrtic polynomil? (A) (B) (C) (D)

4 POLYNOMIALS 11 (C) Short Answer Questions with Resoning Smple Question 1: Cn x 1 be the reminder on division of polynomil p (x) by x +? Justify your nswer. Solution : No, since degree (x 1) = 1 = degree (x + ). Smple Question : Is the following sttement True or Flse? Justify your nswer. If the zeroes of qudrtic polynomil x + bx + c re both negtive, then, b nd c ll hve the sme sign. Solution : True, becuse b = sum of the zeroes < 0, so tht b of the zeroes = c > 0. EXERCISE. > 0. Also the product 1. Answer the following nd justify: (i) Cn x 1 be the quotient on division of x 6 + x + x 1 by polynomil in x of degree 5? (ii) Wht will the quotient nd reminder be on division of x + bx + c by px + qx + rx + s, p 0? (iii) If on division of polynomil p (x) by polynomil g (x), the quotient is zero, wht is the reltion between the degrees of p (x) nd g (x)? (iv) If on division of non-zero polynomil p (x) by polynomil g (x), the reminder is zero, wht is the reltion between the degrees of p (x) nd g (x)? (v) Cn the qudrtic polynomil x + kx + k hve equl zeroes for some odd integer k > 1?. Are the following sttements True or Flse? Justify your nswers. (i) If the zeroes of qudrtic polynomil x + bx + c re both positive, then, b nd c ll hve the sme sign. (ii) If the grph of polynomil intersects the x-xis t only one point, it cnnot be qudrtic polynomil. (iii) If the grph of polynomil intersects the x-xis t exctly two points, it need not be qudrtic polynomil. (iv) If two of the zeroes of cubic polynomil re zero, then it does not hve liner nd constnt terms.

5 1 EXEMPLAR PROBLEMS (v) (vi) (vii) If ll the zeroes of cubic polynomil re negtive, then ll the coefficients nd the constnt term of the polynomil hve the sme sign. If ll three zeroes of cubic polynomil x + x bx + c re positive, then t lest one of, b nd c is non-negtive. The only vlue of k for which the qudrtic polynomil kx + x + k hs equl zeros is 1 (D) Short Answer Questions Smple Question 1:Find the zeroes of the polynomil x x, nd verify the reltion between the coefficients nd the zeroes of the polynomil. Solution : x x = 1 6 (6x + x 1) = 1 6 [6x + 9x 8x 1] = 1 6 [x (x + ) 4 (x + )] = 1 6 (x 4) (x + ) Hence, 4 nd re the zeroes of the given polynomil. The given polynomil is x x. The sum of zeroes = Coefficient of x = 6 Coefficient of x the product of zeroes = 4 EXERCISE. Constnt term Coefficient of x = Find the zeroes of the following polynomils by fctoristion method nd verify the reltions between the zeroes nd the coefficients of the polynomils: 1. 4x x 1. x + 4x 4 nd

6 POLYNOMIALS 1. 5t + 1t t t 15t 5. x + 7 x x + 5 x 7. s (1 + )s + 8. v + 4 v y + 5 y y 11 y (E) Long Answer Questions Smple Question 1: Find qudrtic polynomil, the sum nd product of whose zeroes re nd, respectively. Also find its zeroes. Solution : A qudrtic polynomil, the sum nd product of whose zeroes re nd is x x x x = 1 [x x ] Hence, the zeroes re = 1 [x + x x ] = 1 [ x ( x + 1) ( x + 1)] = 1 [ x + 1] [ x ] 1 nd. Smple Question : If the reminder on division of x + x + kx + by x is 1, find the quotient nd the vlue of k. Hence, find the zeroes of the cubic polynomil x + x + kx 18.

7 14 EXEMPLAR PROBLEMS Solution : Let p(x) = x + x + kx + Then, p() = + + k + = 1 i.e., k = 7 i.e., k = 9 Hence, the given polynomil will become x + x 9x +. Now, x ) x + x 9x +(x + 5x +6 x x 5x 9x + 5x 15x 6x + 6x 18 So, x + x 9x + = (x + 5x + 6) (x ) + 1 i.e., x + x 9x 18 = (x ) (x + 5x + 6) So, the zeroes of x x kx 1 = (x ) (x + ) (x + ) re,,. EXERCISE.4 1. For ech of the following, find qudrtic polynomil whose sum nd product respectively of the zeroes re s given. Also find the zeroes of these polynomils by fctoristion. (i) 8, 4 (iii), 9 (ii) (iv) 1 8, , 1. Given tht the zeroes of the cubic polynomil x 6x + x + 10 re of the form, + b, + b for some rel numbers nd b, find the vlues of nd b s well s the zeroes of the given polynomil.

8 POLYNOMIALS 15. Given tht is zero of the cubic polynomil 6x + x 10x 4, find its other two zeroes. 4. Find k so tht x + x + k is fctor of x 4 + x 14 x + 5x + 6. Also find ll the zeroes of the two polynomils. 5. Given tht x 5 is fctor of the cubic polynomil x 5x + 1x 5, find ll the zeroes of the polynomil. 6. For which vlues of nd b, re the zeroes of q(x) = x + x + lso the zeroes of the polynomil p(x) = x 5 x 4 4x + x + x + b? Which zeroes of p(x) re not the zeroes of q(x)?

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