Class 8 Maths Solution | Lesson 10 Algebraic Expression | Curriculum Development Centre (CDC)

Factorization

Exercise 10.1.1

1. Factorize the following algebraic expressions:

(a)

6x + 3

= 3 (2x + 1)

(b)

x² + 4x

= x (x + 4)

(c)

12a + 3b

= 3 (4a + b)

(d)

12p² + 6q²

= 6 (2p² + q²)

(e)

14xy + 7y

= 7y (2x + 1)

(f)

x + x³

= x (1 + x2)

(g)

12x² + xy + xz

= x (12x + y + z)

(h)

x³ + x² + x

= x (x² + x + 1)

(i)

2x² - 2x³ + 8x⁴

= 2x² (1 - x + 4x²)

2. Factorize the following by grouping the terms:

(a) ax + bx + ay + by

(ax + ay) + (bx + by)

= a (x + y) + b(x + y)

= (a + b)(x + y)

(b) 2ab + a²b - 2b - ab

= (2ab - 2b) + (a²b -

= 2b (a - 1) + ab(a -1)

= (2b + ab)(a -1)

= b (2 + a)(a -1)

(c) x²y - xy + 2x²y - 2xy

= 3x²y - 3xy

= 3xy (x - 1)

(d) x² + 3x + xy + 3y

= x(x + 3) + y(x + 3)

= (x + 3)(x + y)

(e) 2ab + 3a + 2b² + 3b

= (2ab + 3a) + (2b² + 3b)

= a (2b + 3) + b (2b + 3)

= (2b + 3)(a + b)

(f) 2ab + 3a + 2b² + 3b

= (a – b) + (a² – ab)

= 1 (a – b) + a(a – b)

= (a + 1)(a – b)

(g) 2ab + 3a + 2b² + 3b

= (2a² + 5a) + (6 - a - 15)

= a (2a + 5) - (a + 9)

= (a - 3)(2a + 5)

(h) 2xa – x²a + 2a – ax

= (2xa – ax) + (2a – x²a)

= ax (2 – x) + 2a (1 – x)

= a (2 – x)(x + 1)

(i) x²y + 4xy – xy² – 4y²

= (x²y + 4xy) + (–xy² – 4y²)

= xy (x + 4) – y² (x + 4)

= (x + 4)(xy – y²)

= y (x + 4)(x - y)

(j) 3x (x + y) + 3y (x + y)

= 3x (x + y) + 3y (x + y)

= 3 (x + y)(x + y)

= 3 (x + y)²

(k) 2x² + 3ax + 2ax + 3a²

= (2x² + 2ax) + (3ax + 3a²)

= 2x (x + a) + 3a (x + a)

= (x + a)(2x + 3a)

Exercise 10.1.2

(a) x² - 4

= (x)² - (2)²

= (x - 2)(x + 2)

(b) a² - 4b²

= (a)² - (2b)²

= (a - 2b)(a + 2b)

(c) 9x² - y²

= (3x)² - (y)²

= (3x - y)(3x + y)

(d) 5x² - 20y²

= 5(x² - 4y²)

= 5(x)² - (2y)²

= 5(x - 2y)(x + 2y)

(e) 13a² - 117b²

= 13(a² - 9b²)

= 13(a)² - (3b)²

= 13(a - 3b)(a + 3b)

(f) 25 - (1/9)y²

= (5/3)² - (1/3)y)²

= (5/3 - 1/3y)(5/3 + 1/3y)

(g) 121x² - (1/y²)

= (11x)² - (1/y)²

= (11x - 1/y)(11x + 1/y)

(h) 2p² - 50q²

= 2(p² - 25q²)

= 2(p)² - (5q)²

= 2(p - 5q)(p + 5q)

(i) 72 - 2b²

= 2(36 - b²)

= 2(6)² - (b)²

= 2(6 - b)(6 + b)

(j) 121 - 25y²

= (11)² - (5y)²

= (11 - 5y)(11 + 5y)

(k) 15/a² - 60a²

= 15(1/a² - 4a²)

= 15(1/a - 2a)(1/a + 2a)

= 15(1/a - 2a)(1/a + 2a)

(l) 81 - 64y²

= (9)² - (8y)²

= (9 - 8y)(9 + 8y)

(m) 4x³y - 81xy³

= xy(4x² - 81y²)

= xy(2x)² - (9y)²

= xy(2x - 9y)(2x + 9y)

(n) 4x³y - 81xy³

= xy(4x² - 81y²)

= xy(2x)² - (9y)²

= xy(2x - 9y)(2x + 9y)

(o) 169 - 196z²

= (13)² - (14z)²

= (13 - 14z)(13 + 14z)

(p) ab³ - 9a³b

= ab(a² - 9b²)

= ab(a - 3b)(a + 3b)

(q) 49/121 x² - 64/9 y²

= (7/11 x)² - (8/3 y)²

= (7/11 x - 8/3 y)(7/11 x + 8/3 y)

(r) zx² - zy²

= z(x² - y²)

= z(x - y)(x + y)

(s) (x + 2)² - 4

= (x + 2)² - (2)²

= (x + 2 - 2)(x + 2 + 2)

= (x)(x + 4)

(t) 256 - x²/4

= (16)² - (x/2)²

= (16 - x/2)(16 + x/2)

(u) 1 - 81p²/121q²

= (1)² - (9p/11q)²

= (1 - 9p/11q)(1 + 9p/11q)

(v) 3(x - y)² - 12

= 3(x - y)² - (4)

= 3[(x - y)² - 4]

= 3[(x - y - 2)(x - y + 2)]

(w) 9(x - 1)² - 16(x + 2)²

= 9(x - 1)² - 16(x + 2)²

= 9(x - 1)(x - 1) - 16(x + 2)(x + 2)

= [3(x - 1)]² - [4(x + 2)]²

= (3(x - 1) - 4(x + 2))(3(x - 1) + 4(x + 2))

= (3x - 3 - 4x - 8)(3x - 3 + 4x + 8)

= - (x + 11)(7x + 5)