ID : in-8-multiplication-of-polynomials [1] Class 8 Multiplication of Polynomials For more such worksheets visit www.edugain.com Answer t he quest ions (1) If (2pq + 2p) ( -2pq + 2p + 2) = ( -4p 2 q 2 + apq 2 + 4pq + 4p 2 + 4p), f ind value of a. (2) Simplif y the f ollowing polynomial expressions: A) [(2b 2 - b + 4) (3b 2-2b)] + (2b 2 + 3b + 5) B) [(q 2 + q - 5) (4q 2-4q - 3)] + ( -2q 2 + 2q + 3) (3) Find product of f ollowing polynomials A) ( -2a 2-4a + 2) and ( -5a 2-5a - 4) B) ( -3x 2-4x + 5) and (3x 2-4x + 4) (4) If F1 = -8y - 5, F2 = -y - 7 and F3 = 9y - 6, simplif y F1 F2 + F3 in terms of y. (5) Find product of (q 2 + 2q - 1) and ( - 2q - 1) f or q = -3. (6) Find product of ( -2xy - 3x - y - 3) and (2xy - 2x + 3). Choose correct answer(s) f rom given choice (7) Simplif y the f ollowing expressions: A) ( -a - 2) (2a + 3) ( -a - 2) a. 2a 3 + 11a 2 + 20a + 12 b. 2a 3 + 11a 2 + 20a - 12 c. -2a 3 + 11a 2 + 20a + 12 d. 2a 3-11a 2 + 20a + 12 B) (x - 1) (x + 1) ( -2x - 1) a. -2x 3 + x 2 + 2x + 1 b. -2x 3 - x 2 + 2x - 1 c. -2x 3 - x 2-2x + 1 d. -2x 3 - x 2 + 2x + 1 (8) Find the product of the f ollowing polynomials A) (0.5q 2 + q + 7) and (0.5q 2-5q - 0.2) a. 0.25q 4-2q 3-1.6q 2-35.2q + 1.4 b. 0.25q 4 + 2q 3-1.6q 2-35.2q - 1.4 c. 0.25q 4-2q 3-1.6q 2-35.2q - 1.4 d. 0.25q 4-2q 3 + 1.6q 2-35.2q - 1.4
B) ( -a 2-2a + 8) and (a 2 + 0.3a) a. a 4-2.3a 3 + 7.4a 2 + 2.4a b. -a 4-2.3a 3 + 7.4a 2 + 2.4a c. - a 4 + 2.3a 3 + 7.4a 2 + 2.4a d. - a 4-2.3a 3 + 7.4a 2-2.4a ID : in-8-multiplication-of-polynomials [2] (9) Simplif y the f ollowing polynomial expressions A) [( -9y - 3) (7y - 5)] - (6y + 3) a. - 18y - 12 b. 18y - 12 c. -63y 2 + 18y + 12 d. - 18y + 12 B) [( -8x + 3) ( -6x - 7)] + (2x + 1) a. - 40x - 20 b. 48x 2 + 40x - 20 c. - 40x + 20 d. 40x + 20 (10) If (2q 2 + 3q - 8) ( -6q 2-7q - 1) = -12q 4-32q 3 + 25q 2 + aq + 8, f ind the value of a. a. 48 b. 43 c. 56 d. 53 (11) If the base of a triangle is ( -8p 2 + 2p + 8) and its height is ( -4p 2-4p - 8), then what is its area? a. 16p 4 + 12p 3 + 12p 2-24p + 32 b. - 16p 4 + 12p 3 + 12p 2-24p - 32 c. 16p 4 + 12p 3 + 12p 2-24p - 32 d. 16p 4 + 12p 3-12p 2-24p - 32 (12) If the length and width of a rectangle are (y 2-9y - 7) and ( -2y 2 + 2y - 9), f ind the area of the rectangle. a. 2y 4 + 20y 3-13y 2 + 67y + 63 b. - 2y 4 + 20y 3-13y 2 + 67y + 63 c. - 2y 4 + 20y 3-13y 2-67y + 63 d. - 2y 4 + 20y 3 + 13y 2 + 67y + 63 2016 Edugain (www.edugain.com). All Rights Reserved Many more such worksheets can be generated at www.edugain.com
Answers ID : in-8-multiplication-of-polynomials [3] (1) 0 (2) A) 6b 4-7b 3 + 16b 2-5b + 5 Let's f irst multiply f irst two polynomials, (2b 2 - b + 4) (3b 2-2b) = (2b 2 )(3b 2-2b) + ( - b)(3b 2-2b) + (4)(3b 2-2b) = (6b 4-4b 3 ) + ( -3b 3 + 2b 2 ) + (12b 2-8b) = 6b 4-7b 3 + 14b 2-8b Now, on adding 6b 4-7b 3 + 14b 2-8b and 2b 2 + 3b + 5, we get: (6b 4-7b 3 + 14b 2-8b) + (2b 2 + 3b + 5) = 6b 4-7b 3 + 16b 2-5b + 5 Theref ore, the simplif ied f orm of [(2b 2 - b + 4) (3b 2-2b)] + (2b 2 + 3b + 5) is 6b 4-7b 3 + 16b 2-5b + 5. B) 4q 4-29q 2 + 19q + 18 Let's f irst multiply f irst two polynomials, (q 2 + q - 5) (4q 2-4q - 3) = (q 2 )(4q 2-4q - 3) + (q)(4q 2-4q - 3) + ( - 5)(4q 2-4q - 3) = (4q 4-4q 3-3q 2 ) + (4q 3-4q 2-3q) + ( -20q 2 + 20q + 15) = 4q 4-27q 2 + 17q + 15 Now, on adding 4q 4-27q 2 + 17q + 15 and -2q 2 + 2q + 3, we get: (4q 4-27q 2 + 17q + 15) + ( -2q 2 + 2q + 3) = 4q 4-29q 2 + 19q + 18 Theref ore, the simplif ied f orm of [(q 2 + q - 5) (4q 2-4q - 3)] + ( -2q 2 + 2q + 3) is 4q 4-29q 2 + 19q + 18.
ID : in-8-multiplication-of-polynomials [4] (3) A) 10a 4 + 30a 3 + 18a 2 + 6a - 8 In order to multiply two polynomials, lets multiply each term of f irst polynomial to second polynomial as f ollowing, ( -2a 2-4a + 2) ( -5a 2-5a - 4) = ( -2a 2 )( -5a 2-5a - 4) + ( - 4a)( -5a 2-5a - 4) + (2)( -5a 2-5a - 4) = (10a 4 + 10a 3 + 8a 2 ) + (20a 3 + 20a 2 + 16a) + ( -10a 2-10a - 8) = 10a 4 + 30a 3 + 18a 2 + 6a - 8 Theref ore, the product of ( -2a 2-4a + 2) and ( -5a 2-5a - 4) is 10a 4 + 30a 3 + 18a 2 + 6a - 8. B) -9x 4 + 19x 2-36x + 20 In order to multiply two polynomials, lets multiply each term of f irst polynomial to second polynomial as f ollowing, ( -3x 2-4x + 5) (3x 2-4x + 4) = ( -3x 2 )(3x 2-4x + 4) + ( - 4x)(3x 2-4x + 4) + (5)(3x 2-4x + 4) = ( -9x 4 + 12x 3-12x 2 ) + ( -12x 3 + 16x 2-16x) + (15x 2-20x + 20) = -9x 4 + 19x 2-36x + 20 Theref ore, the product of ( -3x 2-4x + 5) and (3x 2-4x + 4) is -9x 4 + 19x 2-36x + 20. (4) 8y 2 + 70y + 29 T his is a straightf orward case of polynomial simplif ication We know that F1 = -8y - 5, F2 = -y - 7, and F3 = 9y - 6 So (F1 x F2 + F3) is (F1 x F2 + F3) = (( -8y - 5) x ( -y - 7)) + (9y - 6) This simplif ies to 8y 2 + 70y + 29
(5) 10 ID : in-8-multiplication-of-polynomials [5] It is given that q = -3 The product of (q 2 + 2q - 1) and ( - 2q - 1) = (q 2 + 2q - 1) ( - 2q - 1) By putting q = -3, we get: (q 2 + 2q - 1) ( - 2q - 1) = { (-3) 2 + 2(-3) - 1 } { - 2(-3) - 1 } = (2) (5) = 10 Thus, the product of (q 2 + 2q - 1) and ( - 2q - 1) f or q = -3 is 10. (6) -4x 2 y 2-2x 2 y - 2xy 2-10xy + 6x 2-3x - 3y - 9 (7) A) a. 2a 3 + 11a 2 + 20a + 12 Let's f irst multiply f irst two polynomials, ( -a - 2) (2a + 3) = ( -a)(2a + 3) + ( -2)(2a + 3) = ( -2a 2-3a) + ( -4a - 6) = -2a 2-7a - 6 Now, multiply polynomials ( -a - 2) and ( -2a 2-7a - 6), we get: ( -a - 2) ( -2a 2-7a - 6) = ( -a)( -2a 2-7a - 6) + ( -2)( -2a 2-7a - 6) = (2a 3 + 7a 2 + 6a) + (4a 2 + 14a + 12) = 2a 3 + 11a 2 + 20a + 12 Theref ore, the simplif ied f orm of [( -a - 2) (2a + 3)] ( -a - 2) is 2a 3 + 11a 2 + 20a + 12.
ID : in-8-multiplication-of-polynomials [6] B) d. -2x 3 - x 2 + 2x + 1 Let's f irst multiply f irst two polynomials, (x - 1) (x + 1) = (x)(x + 1) + ( -1)(x + 1) = (x 2 + x) + ( -x - 1) = x 2-1 Now, multiply polynomials ( -2x - 1) and (x 2-1), we get: ( -2x - 1) (x 2-1) = ( -2x)(x 2-1) + ( -1)(x 2-1) = ( -2x 3 + 2x) + ( -x 2 + 1) = -2x 3 - x 2 + 2x + 1 Theref ore, the simplif ied f orm of [(x - 1) (x + 1)] ( -2x - 1) is -2x 3 - x 2 + 2x + 1. (8) A) c. 0.25q 4-2q 3-1.6q 2-35.2q - 1.4 In order to multiply two polynomials, lets multiply each term of f irst polynomial to second polynomial as f ollowing, (0.5q 2 + q + 7) (0.5q 2-5q - 0.2) = (0.5q 2 )(0.5q 2-5q - 0.2) + (q)(0.5q 2-5q - 0.2) + (7)(0.5q 2-5q - 0.2) = (0.25q 4-2.5q 3-0.1q 2 ) + (0.5q 3-5q 2-0.2q) + (3.5q 2-35q - 1.4) = 0.25q 4-2q 3-1.6q 2-35.2q - 1.4 Theref ore, the product of (0.5q 2 + q + 7) and (0.5q 2-5q - 0.2) is 0.25q 4-2q 3-1.6q 2-35.2q - 1.4.
ID : in-8-multiplication-of-polynomials [7] B) b. -a 4-2.3a 3 + 7.4a 2 + 2.4a In order to multiply two polynomials, lets multiply each term of f irst polynomial to second polynomial as f ollowing, ( -a 2-2a + 8) (a 2 + 0.3a) = ( -a 2 )(a 2 + 0.3a) + ( - 2a)(a 2 + 0.3a) + (8)(a 2 + 0.3a) = ( -a 4-0.3a 3 ) + ( -2a 3-0.6a 2 ) + (8a 2 + 2.4a) = -a 4-2.3a 3 + 7.4a 2 + 2.4a Theref ore, the product of ( -a 2-2a + 8) and (a 2 + 0.3a) is -a 4-2.3a 3 + 7.4a 2 + 2.4a. (9) A) c. -63y 2 + 18y + 12 Let's f irst multiply two given polynomials, ( -9y - 3) (7y - 5) = ( -9y)(7y - 5) + ( -3)(7y - 5) = ( -63y 2 + 45y) + ( -21y + 15) = -63y 2 + 24y + 15 Now, subtract 6y + 3 f rom -63y 2 + 24y + 15, ( -63y 2 + 24y + 15) - (6y + 3) = -63y 2 + 18y + 12 Theref ore, the simplif ied f orm of [( -9y - 3) (7y - 5)] - (6y + 3) is -63y 2 + 18y + 12.
ID : in-8-multiplication-of-polynomials [8] B) b. 48x 2 + 40x - 20 Let's f irst multiply two given polynomials, ( -8x + 3) ( -6x - 7) = ( -8x)( -6x - 7) + (3)( -6x - 7) = (48x 2 + 56x) + ( -18x - 21) = 48x 2 + 38x - 21 Now, add 48x 2 + 38x - 21 to 2x + 1, (48x 2 + 38x - 21) + (2x + 1) = 48x 2 + 40x - 20 Theref ore, the simplif ied f orm of [( -8x + 3) ( -6x - 7)] + (2x + 1) is 48x 2 + 40x - 20. (10) d. 53 It is given that, (2q 2 + 3q - 8) ( -6q 2-7q - 1) = -12q 4-32q 3 + 25q 2 + aq + 8 On right hand side of above equation, a is coef f icient of q. If we can compute coef f icient of q on lef t hand side multiplication, we can f ind value of a by comparing coef f icients. Multiplication of f ollowing terms on lef t hand side can contribute to coef f icient of q (-8-7) (3-1) Step 4 Coef f icients of q f rom lef t hand side multiplication, = (-8-7) + (3-1) = 53 Step 5 On comparing the coef f icients of q, we can f ind that value of a is 53. (11) c. 16p 4 + 12p 3 + 12p 2-24p - 32 If a is the base of a triangle and b is the height of the triangle, we know that the area of the rectangle is obtained as 1/2 x (a x b) The same thing applies even if the base and height of the triangle are given by equations So the area of this triangle is Area = 1/2 x ( -8p 2 + 2p + 8) x ( -4p 2-4p - 8) Area = 16p 4 + 12p 3 + 12p 2-24p - 32
ID : in-8-multiplication-of-polynomials [9] (12) b. - 2y 4 + 20y 3-13y 2 + 67y + 63 If a and b are the length and width of a rectangle, we know that the area of the rectangle is obtained by multiplying them, i.e. area = a x b The same thing applies even if the length and width of a rectangle are given by equations So the area of this rectangle is Area = (y 2-9y - 7) x ( -2y 2 + 2y - 9) Area = - 2y 4 + 20y 3-13y 2 + 67y + 63