ID : in-10-polynomials [1] Class 10 Polynomials For more such worksheets visit www.edugain.com Answer t he quest ions (1) If α and β are the zeros of quadratic polynomial x 2 + px - 2q, f ind the value of α 2 + β 2. (2) If α and β are the zeros of quadratic polynomial x 2-3x - 4, f ind the value of α 3 + β 3. (3) If α and β are the zeros of polynomial x 2-5x + k, such that α - β = 3. Find the value of k. (4) If a and b are the zeros of quadratic polynomial x 2-2px + q, f ind the value of 1/a + 1/b. (5) If α and β are the zeros of polynomial x 2-5x + k, such that α 2 + β 2 = 13. Find the value of k. Choose correct answer(s) f rom given choice (6) Find the zeros of the polynomial f (x) = x 3 - x 2-4x + 4, if it is given that two of its zeros are equal in magnitude but opposite in sign. a. 1, -1 and -2 b. 2, -2 and 1 c. 2, -2 and -1 d. 1, -1 and 2 (7) Find a quadratic polynomial whose zeros are reciprocals of the zeros of the polynomial x 2-4x + 3. a. b. c. d. (8) If α and β are the zeros of polynomial x 2 - x - 2, f ind a polynomial whose zeros are 3α+2 and 3β+2. a. k [x 2-7x - 8] b. k [x 2-8x + 7] c. k [x 2 + 7x - 10] d. k [x 2 + 8x - 8] (9) Find the zeros of the polynomial f (x) = x 3-9x 2 + 26x - 24, if it is given that sum of its two zeros is 5. a. 3, 2 and -4 b. 4, 1 and 2 c. 4, 1 and 3 d. 3, 2 and 4 (10) If α and β are the zeros of quadratic polynomial x 2-2x + 1, f ind the value of 1/α 3 + 1/β 3. a. 0 b. 4 c. 3 d. 2
ID : in-10-polynomials [2] (11) Find the zeros of the polynomial f (x) = x 3-18x 2 + 92x - 120, if it is given that the zeros are in arithmetic progression. a. 1, 5 and 9 b. 3, 7 and 11 c. 0, 4 and 8 d. 2, 6 and 10 (12) If two zeros of polynomial x 3 + bx 2 + cx + d are 1+ 2 and 1-2, f ind its third zero. a. -b - 2 b. c - 2 c. - c - 2 d. b - 2 (13) Find the quadratic polynomial such that sum of its zeros is 19 and dif f erence between zeros is 5. a. k [x 2 + 19x + 84] b. k [x 2-20x + 84] c. k [x 2-19x - 84] d. k [x 2-19x + 84] (14) If α and β are the zeros of polynomial x 2 - x - 2, f ind a polynomial whose zeros are α 2 /β 2 and β 2 /α 2. a. b. c. d. 2016 Edugain (www.edugain.com). All Rights Reserved Many more such worksheets can be generated at www.edugain.com
Answers ID : in-10-polynomials [3] (1) p 2 + 4q Sum of zeros = α + β = -(p/1) = -p Product of zeros = αβ = -2p/1 = -2q α 2 + β 2 = (α + β) 2-2αβ = (-p) 2-2(-2q) α 2 + β 2 = p 2 + 4q (2) 63 Sum of zeros = α + β = -(-3/1) = 3 Product of zeros = αβ = -4/1 = -4 α 3 + β 3 = (α + β) 3-3αβ(α+β) = (3) 3-3(-4)(3) = 63
(3) 4 ID : in-10-polynomials [4] Sum of zeros α + β = -(-5/1) = 5 Product of zeros αβ = (k/1) = k (α - β) 2 = (3) 2 (α + β) 2-4αβ = 9 (5) 2-4k = 9 (5) 2-4k = 9 25-4k = 9 Step 8 4k = 16 Step 9 k = 16/4 = 4 (4) 2q Sum of zeros = a + b = -(-2p/1) = 2p Product of zeros = ab = p/1 = q 1/a + 1/b = (a + b)/ab = (2p)/(q) = 2q
(5) 6 ID : in-10-polynomials [5] Sum of zeros α + β = -(-5/1) = 5 Product of zeros αβ = (k/1) = k (α + β) 2 = (5) 2 (α 2 + β 2 ) + 2αβ = 25 13 + 2k = 25 2k = 12 k = 6
(6) b. 2, -2 and 1 ID : in-10-polynomials [6] Let α, β and γ be the zeros of polynomial It is given than β = -α α + β + γ = 1 α + (-α) + γ = 1 γ = 1 Product of zeros = αβγ = -4 (α)(-α)(1) = -4 Step 8 α 2 = -4/(-1) = 4 Step 9 α = 2 or -2 0 β = -2 or 2 1 Hence zeros are 2, -2 and 1
ID : in-10-polynomials [7] (7) c. Let α and β be the zeros of polynomial x 2-4x + 3 Sum of zeros α + β = -(-4/1) = 4 Product of zeros αβ = (3/1) = 3 Let S and P respectively be the sum and products of zeros of the required polynomial Required polynomial will be k [x 2 - Sx + P] = (k is a constant)
ID : in-10-polynomials [8] (8) a. k [x 2-7x - 8] Sum of zeros α + β = -(-1/1) = 1 Product of zeros αβ = (-2/1) = -2 Let S and P respectively be the sum and products of zeros of the required polynomial S = 3α + 2 + 3β + 2 = 3(α + β) + 2(2) S = 3(1) + 2(2) = 7 P = (3α + 2) (3β + 2) = 9αβ + 6α + 6β + 4 = 9αβ + 6(α + β) + 4 P = 9-2 + 6 1 + 4 = -8 Step 8 Required polynomial will be k [x 2 - Sx +P] = k [x 2-7x - 8] (k is a constant)
(9) d. 3, 2 and 4 ID : in-10-polynomials [9] Let α, β and γ be the zeros of polynomial α + β = 5 α + β + γ = 9 γ = 9-5 = 4 Product of zeros = αβγ = 24 αβ = 24/4 = 6 α (5 - α) = 6 Step 8 5α - α 2 = 6 Step 9 α 2-5α + 6 = 0 0 (α - 3) (α - 2) = 0 1 α = 3 or 2 2 β = (5-3) = 2 or (5-2) = 3 3 Hence zeros are 3, 2 and 4
(10) d. 2 ID : in-10-polynomials [10] Sum of zeros = α + β = -(-2/1) = 2 Product of zeros = αβ = 1/1 = 1 = 2
(11) d. 2, 6 and 10 ID : in-10-polynomials [11] Let α = (a - d), β = a and γ = (a + d) be the zeros of polynomial α + β + γ = 18 (a - d) + a + (a + d) = 18 3a = 18 a = 18/3 = 6 Product of zeros = αβγ = 120 (a - d)(a)(a + d) = 120 Step 8 a(a 2 - d 2 ) = 120 Step 9 6(6 2 - d 2 ) = 120 0 6 2 - d 2 = 120/6 = 20 1 d 2 = 16 2 d = 4 or - 4 3 Hence zeros are (6-4), 6 and (6 + 4) => 2, 6 and 10 OR (6 + 4), 6 and (6-4) => 10, 6 and 2
(12) a. -b - 2 ID : in-10-polynomials [12] Let third zero be x Sum of zeros = -b/1 = -b (1+ 2) + (1-2) + x = -b 2 + x = -b x = -b - 2 (13) d. k [x 2-19x + 84] Let α and β be the zeros of the required polynomial Sum of zeros α + β = 19 Dif f erence of zeros α - β = 5 (α + β) 2 - (α - β) 2 = 4αβ (19) 2 - (5) 2 = 4αβ 336 = 4αβ αβ = 336/4 = 84 Step 8 Required polynomial will be k [x 2 - (α + β)x + αβ] = k [x 2-19x + 84] (k is a constant)
ID : in-10-polynomials [13] (14) a. Sum of zeros α + β = -(-1/1) = 1 Product of zeros αβ = (-2/1) = -2 Let S and P respectively be the sum and products of zeros of the required polynomial P = (α 2 /β 2 ) (β 2 /α 2 ) = 1 Required polynomial will be k [x 2 - Sx + P] = (k is a constant)