myhelper: RD Sharma solution class 8 chapter 7 Factorization Exercise 7.4

Exercise 7.4

Page-7.12

Question 1:

Factorize each of the following expressions:
qr − pr + qs − ps

Answer 1:

$q r - p r + q s - p s = (q r - p r) + (q s - p s) [G r o u p i n g t h e e x p r e s s i o n s] = r (q - p) + s (q - p) = (r + s) (q - p) [T a k i n g (q - p) a s t h e c o m m o n f a c t o r]$

Question 2:

Factorize each of the following expressions:
p²q − pr² − pq + r²

Answer 2:

$p^{2} q - p r^{2} - p q + r^{2} = (p^{2} q - p q) + (r^{2} - p r^{2}) [G r o u p i n g t h e e x p r e s s i o n s] = p q (p - 1) + r^{2} (1 - p) = p q (p - 1) - r^{2} (p - 1) [∵ (1 - p) = - (p - 1)] = (p q - r^{2}) (p - 1) [T a k i n g (p - 1) a s t h e c o m m o n f a c t o r]$

Question 3:

Factorize each of the following expressions:
1 + x + xy + x²y

Answer 3:

$1 + x + x y + x^{2} y = (1 + x) + (x y + x^{2} y) [G r o u p i n g t h e e x p r e s s i o n s] = (1 + x) + x y (1 + x) = (1 + x y) (1 + x) [T a k i n g (1 + x) a s t h e c o m m o n f a c t o r]$

Question 4:

Factorize each of the following expressions:
ax + ay − bx − by

Answer 4:

$a x + a y - b x - b y = (a x + a y) - (b x + b y) [G r o u p i n g t h e e x p r e s s i o n s] = a (x + y) - b (x + y) = (a - b) (x + y) [T a k i n g (x + y) a s t h e c o m m o n f a c t o r]$

Question 5:

Factorize each of the following expressions:
xa² + xb² − ya² − yb²

Answer 5:

$x a^{2} + x b^{2} - y a^{2} - y b^{2} = (x a^{2} + x b^{2}) - (y a^{2} + y b^{2}) [G r o u p i n g t h e e x p r e s s i o n s] = x (a^{2} + b^{2}) - y (a^{2} + b^{2}) = (x - y) (a^{2} + b^{2}) [T a k i n g (a^{2} + b^{2}) a s t h e c o m m o n f a c t o r]$

Question 6:

Factorize each of the following expressions:
x² + xy + xz + yz

Answer 6:

$x^{2} + x y + x z + y z = (x^{2} + x y) + (x z + y z) [G r o u p i n g t h e e x p r e s s i o n s] = x (x + y) + z (x + y) = (x + z) (x + y) [T a k i n g (x + y) a s t h e c o m m o n f a c t o r] = (x + y) (x + z)$

Question 7:

Factorize each of the following expressions:
2ax + bx + 2ay + by

Answer 7:

$2 a x + b x + 2 a y + b y = (2 a x + b x) + (2 a y + b y) [G r o u p i n g t h e e x p r e s s i o n s] = x (2 a + b) + y (2 a + b) = (x + y) (2 a + b) [T a k i n g (2 a + b) a s t h e c o m m o n f a c t o r]$

Question 8:

Factorize each of the following expressions:
ab − by − ay + y²

Answer 8:

$ab - by - ay + y^{2} = (ab - ay) + (y^{2} - by) [Grouping the expressions] = a (b - y) + y (y - b) = a (b - y) - y (b - y) [∵ (y - b) = - (b - y)] = (a - y) (b - y) [Taking (b - y) as the common factor]$

Question 9:

Factorize each of the following expressions:
axy + bcxy − az − bcz

Answer 9:

$a x y + b c x y - a z - b c z = (a x y + b c x y) - (a z + b c z) [G r o u p i n g t h e e x p r e s s i o n s] = x y (a + b c) - z (a + b c) = (x y - z) (a + b c) [T a k i n g (a + b c) a s t h e c o m m o n f a c t o r]$

Question 10:

Factorize each of the following expressions:
lm² − mn² − lm + n²

Answer 10:

$l m^{2} - m n^{2} - l m + n^{2} = (l m^{2} - l m) + (n^{2} - m n^{2}) [R e g r o u p i n g t h e e x p r e s s i o n s] = l m (m - 1) + n^{2} (1 - m) = lm (m - 1) - n^{2} (m - 1) [∵ (1 - m) = - (m - 1)] = (lm - n^{2}) (m - 1) [Taking (m - 1) a s t h e common f a c t o r]$

Question 11:

Factorize each of the following expressions:
x³ − y² + x − x²y²

Answer 11:

$x^{3} - y^{2} + x - x^{2} y^{2} = (x^{3} + x) - (x^{2} y^{2} + y^{2}) [R e g r o u p i n g t h e e x p r e s s i o n s] = x (x^{2} + 1) - y^{2} (x^{2} + 1) = (x - y^{2}) (x^{2} + 1) [T a k i n g (x^{2} + 1) a s t h e c o m m o n f a c t o r]$

Question 12:

Factorize each of the following expressions:
6xy + 6 − 9y − 4x

Answer 12:

$6 x y + 6 - 9 y - 4 x = (6 x y - 4 x) + (6 - 9 y) [R e g r o u p i n g t h e e x p r e s s i o n s] = 2 x (3 y - 2) + 3 (2 - 3 y) = 2 x (3 y - 2) - 3 (3 y - 2) [∵ (2 - 3 y) = - (3 y - 2)] = (2 x - 3) (3 y - 2) [T a k i n g (3 y - 2) a s t h e c o m m o n f a c t o r]$

Question 13:

Factorize each of the following expressions:
x² − 2ax − 2ab + bx

Answer 13:

$x^{2} - 2 a x - 2 a b + b x = (x^{2} - 2 a x) + (b x - 2 a b) [R e g r o u p i n g t h e e x p r e s s i o n s] = x (x - 2 a) + b (x - 2 a) = (x + b) (x - 2 a) [T a k i n g (x - 2 a) a s t h e c o m m o n f a c t o r] = (x - 2 a) (x + b)$

Question 14:

Factorize each of the following expressions:
x³ − 2x²y + 3xy² − 6y³

Answer 14:

$x^{3} {- 2x}^{2} {y + 3xy}^{2} {- 6y}^{3} = (x^{3} - 2 x^{2} y) + (3 x y^{2} - 6 y^{3}) [G r o u p i n g t h e e x p r e s s i o n s] = x^{2} (x - 2 y) + 3 y^{2} (x - 2 y) = (x^{2} + 3 y^{2}) (x - 2 y) [T a k i n g (x - 2 y) a s t h e c o m m o n f a c t o r]$

Question 15:

Factorize each of the following expression:
abx² + (ay − b) x − y

Answer 15:

$a b x^{2} + (a y - b) x - y = a b x^{2} + a x y - b x - y = (a b x^{2} - b x) + (a x y - y) [R e g r o u p i n g t h e e x p r e s s i o n s] = b x (a x - 1) + y (a x - 1) = (b x + y) (a x - 1) [T a k i n g (a x - 1) a s t h e c o m m o n f a c t o r]$

Question 16:

Factorize each of the following expression:
(ax + by)² + (bx − ay)²

Answer 16:

$(a x + b y)^{2} + (b x - a y)^{2} = a^{2} x^{2} + 2 a b x y + b^{2} y^{2} + b^{2} x^{2} - 2 a b x y + a^{2} y^{2} = a^{2} x^{2} + b^{2} y^{2} + b^{2} x^{2} + a^{2} y^{2} = (a^{2} x^{2} + a^{2} y^{2}) + (b^{2} x^{2} + b^{2} y^{2}) [R e g r o u p i n g t h e e x p r e s s i o n s] = a^{2} (x^{2} + y^{2}) + b^{2} (x^{2} + y^{2}) = (a^{2} + b^{2}) (x^{2} + y^{2}) [T a k i n g (x^{2} + y^{2}) a s t h e c o m m o n f a c t o r]$

Question 17:

Factorize each of the following expression:
16(a − b)³ − 24 (a − b)²

Answer 17:

$16 (a - b)^{3} - 24 (a - b)^{2} = 8 (a - b)^{2} [2 (a - b) - 3] {T a k i n g [8 (a - b)^{2}] a s t h e c o m m o n f a c t o r} = 8 (a - b)^{2} (2 a - 2 b - 3)$

Question 18:

Factorize each of the following expression:
ab(x² + 1) + x(a² + b²)

Answer 18:

$a b (x^{2} + 1) + x (a^{2} + b^{2}) = a b x^{2} + a b + a^{2} x + b^{2} x = (a b x^{2} + a^{2} x) + (b^{2} x + a b) [R e g r o u p i n g t h e e x p r e s s i o n s] = a x (b x + a) + b (b x + a) = (a x + b) (b x + a) [T a k i n g (b x + a) a s t h e c o m m o n f a c t o r]$

Question 19:

Factorize each of the following expression:
a²x² + (ax² + 1)x + a

Answer 19:

$a^{2} x^{2} + (a x^{2} + 1) x + a = a^{2} x^{2} + a x^{3} + x + a = (a x^{3} + a^{2} x^{2}) + (x + a) [R e g r o u p i n g t h e e x p r e s s i o n s] = a x^{2} (x + a) + (x + a) = (a x^{2} + 1) (x + a) [T a k i n g (x + a) a s t h e c o m m o n f a c t o r]$

Question 20:

Factorize each of the following expression:
a(a − 2b − c) + 2bc

Answer 20:

$a (a - 2 b - c) + 2 b c = a^{2} - 2 a b - a c + 2 b c = (a^{2} - a c) + (2 b c - 2 a b) [R e g r o u p i n g t h e t e r m s] = a (a - c) + 2 b (c - a) = a (a - c) - 2 b (a - c) [∵ (c - a) = - (a - c)] = (a - 2 b) (a - c) [T a k i n g (a - c) a s t h e c o m m o n f a c t o r]$

Question 21:

Factorize each of the following expression:
a(a + b − c) − bc

Answer 21:

$a (a + b - c) - b c = a^{2} + a b - a c - b c = (a^{2} - a c) + (a b - b c) [R e g r o u p i n g t h e e x p r e s s i o n s] = a (a - c) + b (a - c) = (a + b) (a - c) [T a k i n g (a - c) a s t h e c o m m o n f a c t o r]$

Question 22:

Factorize each of the following expression:
x² − 11xy − x + 11y

Answer 22:

$x^{2} - 11 x y - x + 11 y = (x^{2} - x) + (11 y - 11 x y) [R e g r o u p i n g t h e e x p r e s s i o n s] = x (x - 1) + 11 y (1 - x) = x (x - 1) - 11 y (x - 1) [∵ (1 - x) = - (x - 1)] = (x - 11 y) (x - 1) [T a k i n g o u t t h e c o m m o n f a c t o r (x - 1)]$

Question 23:

Factorize each of the following expression:
ab − a − b + 1

Answer 23:

$a b - a - b + 1 = (a b - b) + (1 - a) [R e g r o u p i n g t h e e x p r e s s i o n s] = b (a - 1) + (1 - a) = b (a - 1) - (a - 1) [∵ (1 - a) = - (a - 1)] = (a - 1) (b - 1) [T a k i n g o u t t h e c o m m o n f a c t o r (a - 1)]$

Question 24:

Factorize each of the following expression:
x² + y − xy − x

Answer 24:

$x^{2} + y - x y - x = (x^{2} - x y) + (y - x) [R e g r o u p i n g t h e e x p r e s s i o n s] = x (x - y) + (y - x) = x (x - y) - (x - y) [∵ (y - x) = - (x - y)] = (x - 1) (x - y) [T a k i n g (x - y) a s t h e c o m m o n e x p r e s s i o n]$

RD Sharma solution class 8 chapter 7 Factorization Exercise 7.4