ID : in-8-factorisation [1] Class 8 Factorisation For more such worksheets visit www.edugain.com Answer the questions (1) Find factors of following polynomial A) xy - 7y + 9x - 63 B) xy - 5y + 6x - 30 (2) Write the following polynomials in factored form. A) 90p 3 q 3 r 2 + 18p 3 q 3 r B) 48pqr + 96qr 3 C) 21x 2 z 2 + 30y 3 z D) 36x 2 y 3 z 2 + 72x 2 y 3 z 3 (3) Find factors of following polynomial A) y 2-2y - 24 B) y 2 - y - 12 (4) Factorize (64p 2-25). (5) Factorize (5a + 6) 2-120a. (6) Find factors of following polynomial A) y 2-6xy - 2y + 12x B) 3q 2 + 5pq + 18q + 30p (7) Write the following polynomials in factored form: A) 30a 3 b + 90a 2 b B) 108p 2 q 3 + 72pq C) 28x 3 + 35x 2 y 2 D) 15p + 27p 2 q 2 (8) Factorize (49b 2-126b + 81). (9) Find factors of polynomial (p 2 + 6pq + 9q 2-9r 2 ). (10) Factorize (625p 4-200p 2 + 16). 2017 Edugain (www.edugain.com). All Rights Reserved Many more such worksheets can be generated at www.edugain.com
Answers ID : in-8-factorisation [2] (1) A) (x - 7)(y + 9) The factors of the polynomial, xy - 7y + 9x - 63 can be found as, xy - 7y + 9x - 63 = y(x - 7) + 9(x - 7) = (x - 7)(y + 9) Thus, the factors of the polynomial, xy - 7y + 9x - 63 are ( x - 7)(y + 9). B) (x - 5)(y + 6) The factors of the polynomial, xy - 5y + 6x - 30 can be found as, xy - 5y + 6x - 30 = y(x - 5) + 6(x - 5) = (x - 5)(y + 6) Thus, the factors of the polynomial, xy - 5y + 6x - 30 are ( x - 5)(y + 6). (2) A) 18p 3 q 3 r(5r + 1) If we look at the given polynomial carefully, we observe that 18p 3 q 3 r is common to both the terms. By taking out the common factor 18p 3 q 3 r from the polynomial 90p 3 q 3 r 2 + 18p 3 q 3 r, we get: 18p 3 q 3 r(5r + 1) Thus, the polynomial 90p 3 q 3 r 2 + 18p 3 q 3 r can be written in the factored form as 18p 3 q 3 r(5r + 1).
ID : in-8-factorisation [3] B) 48qr(p + 2r 2 ) If we look at the given polynomial carefully, we observe that 48qr is common to both the terms. By taking out the common factor 48qr from the polynomial 48pqr + 96qr 3, we get: 48qr(p + 2r 2 ) Thus, the polynomial 48pqr + 96qr 3 can be written in the factored form as 48qr(p + 2r 2 ). C) 3z(7x 2 z + 10y 3 ) If we look at the given polynomial carefully, we observe that 3z is common to both the terms. By taking out the common factor 3z from the polynomial 21x 2 z 2 + 30y 3 z, we get: 3z(7x 2 z + 10y 3 ) Thus, the polynomial 21x 2 z 2 + 30y 3 z can be written in the factored form as 3z(7x 2 z + 10y 3 ). D) 36x 2 y 3 z 2 (1 + 2z) If we look at the given polynomial carefully, we observe that 36x 2 y 3 z 2 is common to both the terms. By taking out the common factor 36x 2 y 3 z 2 from the polynomial 36x 2 y 3 z 2 + 72x 2 y 3 z 3, we get: 36x 2 y 3 z 2 (1 + 2z) Thus, the polynomial 36x 2 y 3 z 2 + 72x 2 y 3 z 3 can be written in the factored form as 36x 2 y 3 z 2 (1 + 2z).
(3) A) (y - 6)(y + 4) ID : in-8-factorisation [4] In order to find factors of polynomial y 2-2y - 24, we need to find two numbers whose sum is -2 and product is (-24 1 = -24) We notice that -6 and 4 are such numbers, since (-6) (4) = -24, and (-6) + (4) = -2 Now we can re-write polynomial as following, y 2-2y - 24 = y 2-6y + 4y - 24 = y(y - 6) + 4(y - 6) = (y - 6)(y + 4) Step 4 Thus, the factors of the polynomial y 2-2y - 24 are (y - 6) and (y + 4). B) (y + 3)(y - 4) In order to find factors of polynomial y 2 - y - 12, we need to find two numbers whose sum is -1 and product is (-12 1 = -12) We notice that 3 and -4 are such numbers, since (3) (-4) = -12, and (3) + (-4) = -1 Now we can re-write polynomial as following, y 2 - y - 12 = y 2 + 3y - 4y - 12 = y(y + 3) -4(y + 3) = (y + 3)(y - 4) Step 4 Thus, the factors of the polynomial y 2 - y - 12 are (y + 3) and (y - 4).
(4) (8p + 5) (8p - 5) ID : in-8-factorisation [5] We know that, a 2 - b 2 = (a + b)(a - b). Let's write (64p 2-25) as: (8p) 2 - (5) 2 = (8p + 5)(8p - 5) (5) (5a - 6) (5a - 6) Using algebraic identity (a + b) 2 = a 2 + 2ab + b 2 (5a + 6) 2-120a = (5a) 2 + 60a + (6) 2-120a (5a + 6) 2-120a = (5a) 2-60a + (6) 2 (5a + 6) 2-120a = (5a) 2-2(6)(5a) + (6) 2 Now using algebraic identity (a - b) 2 = a 2-2ab + b 2 (5a + 6) 2-120a = (5a - 6) 2 (5a + 6) 2-120a = (5a - 6) (5a - 6) (6) A) (y - 2)(y - 6x) Re-order some of the terms as following, y 2-6xy - 2y + 12x = y 2-2y - 6xy + 12x Now we can see that y is common in first two terms, and -6x is common in last two terms. Lets rewrite the expression as following, = y(y - 2) -6x(y - 2) = (y - 2)(y - 6x) Thus, the factors of the polynomial, y 2-6xy - 2y + 12x are (y - 6x)( y - 2).
B) (q + 6)(3q + 5p) ID : in-8-factorisation [6] Re-order some of the terms as following, 3q 2 + 5pq + 18q + 30p = 3q 2 + 18q + 5pq + 30p Now we can see that 3q is common in first two terms, and 5p is common in last two terms. Lets rewrite the expression as following, = 3q(q + 6) + 5p(q + 6) = (q + 6)(3q + 5p) Thus, the factors of the polynomial, 3q 2 + 5pq + 18q + 30p are (3q + 5p)(q + 6). (7) A) 30a 2 b(a + 3) If we look at the given polynomial carefully, we observe that the factor 30a 2 b is common to both terms of the polynomial 30a 3 b + 90a 2 b. By taking out the common factor 30a 2 b from the polynomial 30a 3 b + 90a 2 b, we get the following factorization: 30a 3 b + 90a 2 b = 30a 2 b(a + 3) Thus, the polynomial 30a 3 b + 90a 2 b can be written in factored form as 30a 2 b(a + 3). B) 36pq(3pq 2 + 2) If we look at the given polynomial carefully, we observe that the factor 36pq is common to both terms of the polynomial 108p 2 q 3 + 72pq. By taking out the common factor 36pq from the polynomial 108p 2 q 3 + 72pq, we get the following factorization: 108p 2 q 3 + 72pq = 36pq(3pq 2 + 2) Thus, the polynomial 108p 2 q 3 + 72pq can be written in factored form as 36pq(3pq 2 + 2).
ID : in-8-factorisation [7] C) 7x 2 (4x + 5y 2 ) If we look at the given polynomial carefully, we observe that the factor 7x 2 is common to both terms of the polynomial 28x 3 + 35x 2 y 2. By taking out the common factor 7x 2 from the polynomial 28x 3 + 35x 2 y 2, we get the following factorization: 28x 3 + 35x 2 y 2 = 7x 2 (4x + 5y 2 ) Thus, the polynomial 28x 3 + 35x 2 y 2 can be written in factored form as 7x 2 (4x + 5y 2 ). D) 3p(5 + 9pq 2 ) If we look at the given polynomial carefully, we observe that the factor 3p is common to both terms of the polynomial 15p + 27p 2 q 2. By taking out the common factor 3p from the polynomial 15p + 27p 2 q 2, we get the following factorization: 15p + 27p 2 q 2 = 3p(5 + 9pq 2 ) Thus, the polynomial 15p + 27p 2 q 2 can be written in factored form as 3p(5 + 9pq 2 ). (8) (7b - 9) 2 We know that, (a - b) 2 = a 2-2ab + b 2. (49b 2-126b + 81) can be factorized as, 49b 2-126b + 81 = (7b) 2-2(9)(7b) + (9) 2 = (7b - 9) 2
(9) (p + 3q + 3r) (p + 3q - 3r) ID : in-8-factorisation [8] We know that, (a + b) 2 = a 2 + 2ab + b 2, a 2 - b 2 = (a + b)(a - b) The factors of the polynomial (p 2 + 6pq + 9q 2-9r 2 ) can be found using above identities as following, p 2 + 6pq + 9q 2-9r 2 = {(p) 2 + 6pq + (3q) 2 } - (3r) 2 = (p + 3q) 2 - (3r) 2 = (p + 3q + 3r) (p + 3q - 3r) (10) (5p + 2) (5p + 2) (5p - 2) (5p - 2) Let's factorize 625p 4-200p 2 + 16 = 625p 4-200p 2 + 16 = (25p 2 ) 2 - (2 4 25) + 4 2 Since we know that (a - b) 2 = a 2-2ab + b 2 ] = (25p 2-4) 2 = {(5p) 2-2 2 } 2 = {(5p + 2)(5p - 2)} 2...[Since we know that a 2 - b 2 = (a + b)(a - b)] = (5p + 2)(5p + 2)(5p - 2)(5p - 2) Thus, the factors of (625p 4-200p 2 + 16) are (5p + 2), (5p + 2), (5p - 2) and (5p - 2).