Rd Sharma 2019 2020 for Class 8 Maths Chapter 7 - Factorization

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Rd Sharma 2019 2020 Solutions for Class 8 Maths Chapter 7 Factorization are provided here with simple step-by-step explanations. These solutions for Factorization are extremely popular among Class 8 students for Maths Factorization Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2019 2020 Book of Class 8 Maths Chapter 7 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2019 2020 Solutions. All Rd Sharma 2019 2020 Solutions for class Class 8 Maths are prepared by experts and are 100% accurate.

Page No 7.12:

Question 1:

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

Answer:

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

Page No 7.12:

Question 2:

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

Answer:

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

Page No 7.12:

Question 3:

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

Answer:

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

Page No 7.12:

Question 4:

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

Answer:

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

Page No 7.12:

Question 5:

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

Answer:

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

Page No 7.12:

Question 6:

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

Answer:

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

Page No 7.12:

Question 7:

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

Answer:

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

Page No 7.12:

Question 8:

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

Answer:

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

Page No 7.12:

Question 9:

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

Answer:

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

Page No 7.12:

Question 10:

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

Answer:

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

Page No 7.12:

Question 11:

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

Answer:

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

Page No 7.12:

Question 12:

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

Answer:

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

Page No 7.12:

Question 13:

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

Answer:

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

Page No 7.12:

Question 14:

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

Answer:

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

Page No 7.12:

Question 15:

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

Answer:

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

Page No 7.12:

Question 16:

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

Answer:

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

Page No 7.12:

Question 17:

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

Answer:

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

Page No 7.12:

Question 18:

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

Answer:

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

Page No 7.12:

Question 19:

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

Answer:

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

Page No 7.12:

Question 20:

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

Answer:

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

Page No 7.12:

Question 21:

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

Answer:

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

Page No 7.12:

Question 22:

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

Answer:

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

Page No 7.12:

Question 23:

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

Answer:

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

Page No 7.12:

Question 24:

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

Answer:

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

Page No 7.17:

Question 1:

Factorize each of the following expression:
16x² − 25y²

Answer:

$16 x^{2} - 25 y^{2} = (4 x)^{2} - (5 y)^{2} = (4 x - 5 y) (4 x + 5 y)$

Page No 7.17:

Question 2:

Factorize each of the following expression:
27x² − 12y²

Answer:

$27 x^{2} - 12 y^{2} = 3 (9 x^{2} - 4 y^{2}) = 3 [(3 x)^{2} - (2 y)^{2}] = 3 (3 x - 2 y) (3 x + 2 y)$

Page No 7.17:

Question 3:

Factorize each of the following expression:
144a² − 289b²

Answer:

$144 a^{2} - 289 b^{2} = (12 a)^{2} - (17 b)^{2} = (12 a - 17 b) (12 a + 17 b)$

Page No 7.17:

Question 4:

Factorize each of the following expression:
12m² − 27

Answer:

$12 m^{2} - 27 = 3 (4 m^{2} - 9) = 3 [(2 m)^{2} - 3^{2}] = 3 (2 m - 3) (2 m + 3)$

Page No 7.17:

Question 5:

Factorize each of the following expression:
125x² − 45y²

Answer:

$125 x^{2} - 45 y^{2} = 5 (25 x^{2} - 9 y^{2}) = 5 [(5 x)^{2} - (3 y)^{2}] = 5 (5 x - 3 y) (5 x + 3 y)$

Page No 7.17:

Question 6:

Factorize each of the following expression:
144a² − 169b²

Answer:

$144 a^{2} - 169 b^{2} = (12 a)^{2} - (13 b)^{2} = (12 a - 13 b) (12 a + 13 b)$

Page No 7.17:

Question 7:

Factorize each of the following expression:
(2a − b)² − 16c²

Answer:

$(2 a - b)^{2} - 16 c^{2} = (2 a - b)^{2} - (4 c)^{2} = [(2 a - b) - 4 c] [(2 a - b) + 4 c] = (2 a - b - 4 c) (2 a - b + 4 c)$

Page No 7.17:

Question 8:

Factorize each of the following expression:
(x + 2y)² − 4(2x − y)²

Answer:

$(x + 2 y)^{2} - 4 (2 x - y)^{2} = (x + 2 y)^{2} - [2 (2 x - y)]^{2} = [(x + 2 y) - 2 (2 x - y)] [(x + 2 y) + 2 (2 x - y)] = (x + 2 y - 4 x + 2 y) (x + 2 y + 4 x - 2 y) = 5 x (4 y - 3 x)$

Page No 7.17:

Question 9:

Factorize each of the following expression:
3a⁵ − 48a³

Answer:

$3 a^{5} - 48 a^{3} = 3 a^{3} (a^{2} - 16) = 3 a^{3} (a^{2} - 4^{2}) = 3 a^{3} (a - 4) (a + 4)$

Page No 7.17:

Question 10:

Factorize each of the following expression:
a⁴ − 16b⁴

Answer:

$a^{4} - 16 b^{4} = a^{4} - 2^{4} b^{4} = (a^{2})^{2} - (2^{2} b^{2})^{2} = (a^{2} - 2^{2} b^{2}) (a^{2} + 2^{2} b^{2}) = [a^{2} - (2 b)^{2}] (a^{2} + 4 b^{2}) = (a - 2 b) (a + 2 b) (a^{2} + 4 b^{2})$

Page No 7.17:

Question 11:

Factorize each of the following expression:
x⁸ − 1

Answer:

$x^{8} - 1 = (x^{4})^{2} - 1^{2} = (x^{4} - 1) (x^{4} + 1) = [(x^{2})^{2} - 1^{2}] (x^{4} + 1) = (x^{2} - 1) (x^{2} + 1) (x^{4} + 1) = (x^{2} - 1^{2}) (x^{2} + 1) (x^{4} + 1) = (x - 1) (x + 1) (x^{2} + 1) (x^{4} + 1)$

Page No 7.17:

Question 12:

Factorize each of the following expression:
64 − (a + 1)²

Answer:

$64 - (a + 1)^{2} = (8)^{2} - (a + 1)^{2} = [8 - (a + 1)] [8 + (a + 1)] = (8 - a - 1) (8 + a + 1) = (7 - a) (9 + a)$

Page No 7.17:

Question 13:

Factorize each of the following expression:
36l² − (m + n)²

Answer:

$36 l^{2} - (m + n)^{2} = (6 l)^{2} - (m + n)^{2} = [6 l - (m + n)] [6 l + (m + n)] = (6 l - m - n) (6 l + m + n)$

Page No 7.17:

Question 14:

Factorize each of the following expression:
25x⁴y⁴ − 1

Answer:

$25 x^{4} y^{4} - 1 = (5 x^{2} y^{2})^{2} - 1 = (5 x^{2} y^{2} - 1) (5 x^{2} y^{2} + 1)$

Page No 7.17:

Question 15:

Factorize each of the following expression:
$a^{4} - \frac{1}{b^{4}}$

Answer:

$a$ ⁴- 1/b⁴
= (a²)²- 1/(b²)²
= a²- 1/b²a²+ 1/b²
= a - 1/ba + 1/ba²+ 1/b²

Page No 7.17:

Question 16:

Factorize each of the following expression:
x³ − 144x

Answer:

$x^{3} - 144 x = x (x^{2} - 144) = x (x^{2} - 12^{2}) = x (x - 12) (x + 12)$

Page No 7.17:

Question 17:

Factorize each of the following expression:
(x - 4y)² − 625

Answer:

$(x - 4 y)^{2} - 625 = (x - 4 y)^{2} - 25^{2} = [(x - 4 y) - 25] [(x - 4 y) + 25] = (x - 4 y - 25) (x - 4 y + 25)$

Page No 7.17:

Question 18:

Factorize each of the following expression:
9(a − b)² − 100(x − y)²

Answer:

$9 (a - b)^{2} - 100 (x - y)^{2} = [3 (a - b)]^{2} - [10 (x - y)]^{2} = [3 (a - b) - 10 (x - y)] [3 (a - b) + 10 (x - y)] = (3 a - 3 b - 10 x + 10 y) (3 a - 3 b + 10 x - 10 y)$

Page No 7.17:

Question 19:

Factorize each of the following expression:
(3 + 2a)² − 25a²

Answer:

$(3 + 2 a)^{2} - 25 a^{2} = (3 + 2 a)^{2} - (5 a)^{2} = [(3 + 2 a) - 5 a] [(3 + 2 a) + 5 a] = (3 + 2 a - 5 a) (3 + 2 a + 5 a) = (3 - 3 a) (3 + 7 a) = 3 (1 - a) (3 + 7 a)$

Page No 7.17:

Question 20:

Factorize each of the following expression:
(x + y)² − (a − b)²

Answer:

$(x + y)^{2} - (a - b)^{2} = [(x + y) - (a - b)] [(x + y) + (a - b)] = (x + y - a + b) (x + y + a - b)$

Page No 7.17:

Question 21:

Factorize each of the following expression:
$\frac{1}{16} x^{2} y^{2} - \frac{4}{49} y^{2} z^{2}$

Answer:

$\frac{1}{16} x^{2} y^{2} - \frac{4}{49} y^{2} z^{2} = y^{2} (\frac{1}{16} x^{2} - \frac{4}{49} z^{2}) = y^{2} [{(\frac{1}{4} x)}^{2} - {(\frac{2}{7} z)}^{2}] = y^{2} (\frac{1}{4} x - \frac{2}{7} z) (\frac{1}{4} x + \frac{2}{7} z) = y^{2} (\frac{x}{4} - \frac{2}{7} z) (\frac{x}{4} + \frac{2}{7} z)$

Page No 7.17:

Question 22:

Factorize each of the following expression:
75a³b² - 108ab⁴

Answer:

$75 a^{3} b^{2} - 108 a b^{4} = 3 a b^{2} (25 a^{2} - 36 b^{2}) = 3 a b^{2} [(5 a)^{2} - (6 b)^{2}] = 3 a b^{2} (5 a - 6 b) (5 a + 6 b)$

Page No 7.17:

Question 23:

Factorize each of the following expression:
x⁵ − 16x³

Answer:

$x^{5} - 16 x^{3} = x^{3} (x^{2} - 16) = x^{3} (x^{2} - 4^{2}) = x^{3} (x - 4) (x + 4)$

Page No 7.17:

Question 24:

Factorize each of the following expression:
$\frac{50}{x^{2}} - \frac{2 x^{2}}{81}$

Answer:

$\frac{50}{x^{2}} - \frac{2 x^{2}}{81} = 2 (\frac{25}{x^{2}} - \frac{x^{2}}{81}) = 2 \{{(\frac{5}{x})}^{2} - {(\frac{x}{9})}^{2}\} = 2 (\frac{5}{x} - \frac{x}{9}) (\frac{5}{x} + \frac{x}{9})$

Page No 7.17:

Question 25:

Factorize each of the following expression:
256x⁵ − 81x

Answer:

$256 x^{5} - 81 x = x (256 x^{4} - 81) = x [(16 x^{2})^{2} - 9^{2}] = x (16 x^{2} + 9) (16 x^{2} - 9) = x (16 x^{2} + 9) [(4 x)^{2} - 3^{2}] = x (16 x^{2} + 9) (4 x + 3) (4 x - 3)$

Page No 7.17:

Question 26:

Factorize each of the following expression:
a⁴ − (2b + c)⁴

Answer:

$a^{4} - (2 b + c)^{4} = (a^{2})^{2} - [(2 b + c)^{2}]^{2} = [a^{2} + (2 b + c)^{2}] [a^{2} - (2 b + c)^{2}] = [a^{2} + (2 b + c)^{2}] {[a + (2 b + c)] [a - (2 b + c)]} = [a^{2} + (2 b + c)^{2}] (a + 2 b + c) (a - 2 b - c)$

Page No 7.17:

Question 27:

Factorize each of the following expression:
(3x + 4y)⁴ − x⁴

Answer:

$(3 x + 4 y)^{4} - x^{4} = [(3 x + 4 y)^{2}]^{2} - (x^{2})^{2} = [(3 x + 4 y)^{2} + x^{2}] [(3 x + 4 y)^{2} - x^{2}] = [(3 x + 4 y)^{2} + x^{2}] [(3 x + 4 y) + x] [(3 x + 4 y) - x] = {(3 x + 4 y)^{2} + x^{2}} (3 x + 4 y + x) (3 x + 4 y - x) = \{{(3 x + 4 y)}^{2} + x^{2}\} (4 x + 4 y) (2 x + 4 y) = \{{(3 x + 4 y)}^{2} + x^{2}\} 4 (x + y) 2 (x + 2 y) = 8 \{{(3 x + 4 y)}^{2} + x^{2}\} (x + y) (x + 2 y)$

Page No 7.17:

Question 28:

Factorize each of the following expression:
p²q² − p⁴q⁴

Answer:

$p^{2} q^{2} - p^{4} q^{4} = p^{2} q^{2} (1 - p^{2} q^{2}) = p^{2} q^{2} [1 - (p q)^{2}] = p^{2} q^{2} (1 - p q) (1 + p q)$

Page No 7.17:

Question 29:

Factorize each of the following expression:
3x³y − 243xy³

Answer:

$3 x^{3} y - 243 x y^{3} = 3 x y (x^{2} - 81 y^{2}) = 3 x y [x^{2} - (9 y)^{2}] = 3 x y (x - 9 y) (x + 9 y)$

Page No 7.17:

Question 30:

Factorize each of the following expression:
a⁴b⁴− 16c⁴

Answer:

$a^{4} b^{4} - 16 c^{4} = [(a^{2} b^{2})^{2} - (4 c^{2})^{2}] = (a^{2} b^{2} + 4 c^{2}) (a^{2} b^{2} - 4 c^{2}) = (a^{2} b^{2} + 4 c^{2}) [(a b)^{2} - (2 c)^{2}] = (a^{2} b^{2} + 4 c^{2}) (a b + 2 c) (a b - 2 c)$

Page No 7.17:

Question 31:

Factorize each of the following expression:
x⁴ − 625

Answer:

$x^{4} - 625 = (x^{2})^{2} - 25^{2} = (x^{2} + 25) (x^{2} - 25) = (x^{2} + 25) (x^{2} - 5^{2}) = (x^{2} + 25) (x + 5) (x - 5)$

Page No 7.17:

Question 32:

Factorize each of the following expression:
x⁴ − 1

Answer:

$x^{4} - 1 = (x^{2})^{2} - 1 = (x^{2} + 1) (x^{2} - 1) = (x^{2} + 1) (x + 1) (x - 1)$

Page No 7.17:

Question 33:

Factorize each of the following expression:
49(a − b)² − 25(a + b)²

Answer:

$49 (a - b)^{2} - 25 (a + b)^{2} = [7 (a - b)]^{2} - [5 (a + b)]^{2} = [7 (a - b) - 5 (a + b)] [7 (a - b) + 5 (a + b)] = (7 a - 7 b - 5 a - 5 b) (7 a - 7 b + 5 a + 5 b) = (2 a - 12 b) (12 a - 2 b) = 2 (a - 6 b) 2 (6 a - b) = 4 (a - 6 b) (6 a - b)$

Page No 7.17:

Question 34:

Factorize each of the following expression:
x − y − x² + y²

Answer:

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

Page No 7.17:

Question 35:

Factorize each of the following expression:
16(2x − 1)² − 25y²

Answer:

$16 (2 x - 1)^{2} - 25 y^{2} = [4 (2 x - 1)]^{2} - (5 y)^{2} = [4 (2 x - 1) - 5 y] [4 (2 x - 1) + 5 y] = (8 x - 4 - 5 y) (8 x - 4 + 5 y) = (8 x - 5 y - 4) (8 x + 5 y - 4)$

Page No 7.17:

Question 36:

Factorize each of the following expression:
4(xy + 1)² − 9(x − 1)²

Answer:

$4 (x y + 1)^{2} - 9 (x - 1)^{2} = [2 (x y + 1)]^{2} - [3 (x - 1)]^{2} = [2 (x y + 1) - 3 (x - 1)] [2 (x y + 1) + 3 (x - 1)] = (2 x y + 2 - 3 x + 3) (2 x y + 2 + 3 x - 3) = (2 x y - 3 x + 5) (2 x y + 3 x - 1)$

Page No 7.17:

Question 37:

Factorize each of the following expression:
(2x + 1)² − 9x⁴

Answer:

$(2 x + 1)^{2} - 9 x^{4} = (2 x + 1)^{2} - (3 x^{2})^{2} = [(2 x + 1) - 3 x^{2}] [(2 x + 1) + 3 x^{2}] = (- 3 x^{2} + 2 x + 1) (3 x^{2} + 2 x + 1) We can factorise the quadratic expressions in the curved brackets as : (- 3 x^{2} + 3 x - x + 1) (3 x^{2} + 2 x + 1) = \{3 x (- x + 1) + 1 (- x + 1)\} (3 x^{2} + 2 x + 1) = (- x + 1) (3 x + 1) (3 x^{2} + 2 x + 1) = - (x - 1) (3 x + 1) (3 x^{2} + 2 x + 1)$

Page No 7.17:

Question 38:

Factorize each of the following expression:
x⁴ − (2y − 3z)²

Answer:

$x^{4} - (2 y - 3 z)^{2} = (x^{2})^{2} - (2 y - 3 z)^{2} = [x^{2} - (2 y - 3 z)] [x^{2} + (2 y - 3 z)] = (x^{2} - 2 y + 3 z) (x^{2} + 2 y - 3 z)$

Page No 7.17:

Question 39:

Factorize each of the following expression:
a² − b² + a − b

Answer:

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

Page No 7.17:

Question 40:

Factorize each of the following expression:
16a⁴ − b⁴

Answer:

$16 a^{4} - b^{4} = (4 a^{2})^{2} - (b^{2})^{2} = (4 a^{2} + b^{2}) (4 a^{2} - b^{2}) = (4 a^{2} + b^{2}) [(2 a)^{2} - b^{2}] = (4 a^{2} + b^{2}) (2 a + b) (2 a - b)$

Page No 7.17:

Question 41:

Factorize each of the following expression:
a⁴ − 16(b − c)⁴

Answer:

$a^{4} - 16 (b - c)^{4} = (a^{2})^{2} - [4 (b - c)^{2}]^{2} = [a^{2} + 4 (b - c)^{2}] [a^{2} - 4 (b - c)^{2}] = [a^{2} + 4 (b - c)^{2}] {a^{2} - [2 (b - c)]^{2}} = [a^{2} + 4 (b - c)^{2}] [a + 2 (b - c)] [a - 2 (b - c)] = [a^{2} + 4 (b - c)^{2}] (a + 2 b - 2 c) (a - 2 b + 2 c)$

Page No 7.17:

Question 42:

Factorize each of the following expression:
2a⁵ − 32a

Answer:

$2 a^{5} - 32 a = 2 a (a^{4} - 16) = 2 a [(a^{2})^{2} - 4^{2}] = 2 a (a^{2} + 4) (a^{2} - 4) = 2 a (a^{2} + 4) (a^{2} - 2^{2}) = 2 a (a^{2} + 4) (a + 2) (a - 2) = 2 a (a - 2) (a + 2) (a^{2} + 4)$

Page No 7.17:

Question 43:

Factorize each of the following expression:
a⁴b⁴ − 81c⁴

Answer:

$a^{4} b^{4} - 81 c^{4} = (a^{2} b^{2})^{2} - (9 c^{2})^{2} = (a^{2} b^{2} + 9 c^{2}) (a^{2} b^{2} - 9 c^{2}) = (a^{2} b^{2} + 9 c^{2}) [(a b)^{2} - (3 c)^{2}] = (a^{2} b^{2} + 9 c^{2}) (a b + 3 c) (a b - 3 c)$

Page No 7.17:

Question 44:

Factorize each of the following expression:
xy⁹ − yx⁹

Answer:

$x y^{9} - y x^{9} = x y (y^{8} - x^{8}) = x y [(y^{4})^{2} - (x^{4})^{2}] = x y (y^{4} + x^{4}) (y^{4} - x^{4}) = x y (y^{4} + x^{4}) [(y^{2})^{2} - (x^{2})^{2}] = x y (y^{4} + x^{4}) (y^{2} + x^{2}) (y^{2} - x^{2}) = x y (y^{4} + x^{4}) (y^{2} + x^{2}) (y + x) (y - x)$

Page No 7.17:

Question 45:

Factorize each of the following expression:
x³ − x

Answer:

$x^{3} - x = x (x^{2} - 1) = x (x - 1) (x + 1)$

Page No 7.17:

Question 46:

Factorize each of the following expression:
18a²x² − 32

Answer:

$18 a^{2} x^{2} - 32 = 2 (9 a^{2} x^{2} - 16) = 2 [(3 a x)^{2} - 4^{2}] = 2 (3 a x - 4) (3 a x + 4)$

Page No 7.22:

Question 1:

Factorize each of the following algebraic expression:
4x²+ 12xy +9y²

Answer:

$4 x^{2} + 12 x y + 9 y^{2} = (2 x)^{2} + 2 \times 2 x \times 3 y + (3 y)^{2} = (2 x + 3 y)^{2} = (2 x + 3 y) (2 x + 3 y)$

Page No 7.22:

Question 2:

Factorize each of the following algebraic expression:
9a² − 24ab + 16b²

Answer:

$9 a^{2} - 24 a b + 16 b^{2} = (3 a)^{2} - 2 \times 3 a \times 4 b + (4 b)^{2} = (3 a - 4 b)^{2} = (3 a - 4 b) (3 a - 4 b)$

Page No 7.22:

Question 3:

Factorize each of the following algebraic expression:
p²q² − 6pqr + 9r²

Answer:

$p^{2} q^{2} - 6 p q r + 9 r^{2} = (p q)^{2} - 2 \times p q \times 3 r + (3 r)^{2} = (p q - 3 r)^{2} = (p q - 3 r) (p q - 3 r)$

Page No 7.22:

Question 4:

Factorize each of the following algebraic expression:
36a² + 36a + 9

Answer:

$36 a^{2} + 36 a + 9 = 9 (4 a^{2} + 4 a + 1) = 9 \{(2 a)^{2} + 2 \times 2 a \times 1 + 1^{2}\} = 9 (2 a + 1)^{2} = 9 (2 a + 1) (2 a + 1)$

Page No 7.23:

Question 5:

Factorize each of the following algebraic expression:
a² + 2ab + b² − 16

Answer:

$a^{2} + 2 a b + b^{2} - 16 = a^{2} + 2 \times a \times b + b^{2} - 16 = (a + b)^{2} - 4^{2} = (a + b - 4) (a + b + 4)$

Page No 7.23:

Question 6:

Factorize each of the following algebraic expression:
9z² − x² + 4xy − 4y²

Answer:

$9 z^{2} - x^{2} + 4 x y - 4 y^{2} = 9 z^{2} - (x^{2} - 4 x y + 4 y^{2}) = 9 z^{2} - [x^{2} - 2 \times x \times 2 y + (2 y)^{2}] = (3 z)^{2} - (x - 2 y)^{2} = [3 z - (x - 2 y)] [3 z + (x - 2 y)] = (3 z - x + 2 y) (3 z + x - 2 y) = (x - 2 y + 3 z) (- x + 2 y + 3 z)$

Page No 7.23:

Question 7:

Factorize each of the following algebraic expression:
9a⁴ − 24a²b² + 16b⁴ − 256

Answer:

$9 a^{4} - 24 a^{2} b^{2} + 16 b^{4} - 256 = (9 a^{4} - 24 a^{2} b^{2} + 16 b^{4}) - 256 = [(3 a^{2})^{2} - 2 \times 3 a^{2} \times 4 b^{2} + (4 b^{2})^{2}] - 16^{2} = (3 a^{2} - 4 b^{2})^{2} - 16^{2} = [(3 a^{2} - 4 b^{2}) - 16] [(3 a^{2} - 4 b^{2}) + 16] = (3 a^{2} - 4 b^{2} - 16) (3 a^{2} - 4 b^{2} + 16)$

Page No 7.23:

Question 8:

Factorize each of the following algebraic expression:
16 − a⁶ + 4a³b³ − 4b⁶

Answer:

$16 - a^{6} + 4 a^{3} b^{3} - 4 b^{6} = 16 - (a^{6} - 4 a^{3} b^{3} + 4 b^{6}) = 4^{2} - [(a^{3})^{2} - 2 \times a^{3} \times 2 b^{3} + (2 b^{3})^{2}] = 4^{2} - (a^{3} - 2 b^{3})^{2} = [4 - (a^{3} - 2 b^{3})] [4 + (a^{3} - 2 b^{3})] = (4 - a^{3} + 2 b^{3}) (4 + a^{3} - 2 b^{3}) = (a^{3} - 2 b^{3} + 4) (- a^{3} + 2 b^{3} + 4)$

Page No 7.23:

Question 9:

Factorize each of the following algebraic expression:
a² − 2ab + b² − c²

Answer:

$a^{2} - 2 a b + b^{2} - c^{2} = (a^{2} - 2 a b + b^{2}) - c^{2} = (a^{2} - 2 \times a \times b + b^{2}) - c^{2} = (a - b)^{2} - c^{2} = [(a - b) - c] [(a - b) + c] = (a - b - c) (a - b + c)$

Page No 7.23:

Question 10:

Factorize each of the following algebraic expression:
x² + 2x + 1 − 9y²

Answer:

$x^{2} + 2 x + 1 - 9 y^{2} = (x^{2} + 2 x + 1) - 9 y^{2} = (x^{2} + 2 \times x \times 1 + 1) - 9 y^{2} = (x + 1)^{2} - (3 y)^{2} = [(x + 1) - 3 y] [(x + 1) + 3 y] = (x + 1 - 3 y) (x + 1 + 3 y) = (x + 3 y + 1) (x - 3 y + 1)$

Page No 7.23:

Question 11:

Factorize each of the following algebraic expression:
a² + 4ab + 3b²

Answer:

$a^{2} + 4 ab + 3 b^{2} = a^{2} + 4 ab + 4 b^{2} - b^{2} = [a^{2} + 2 \times a \times 2 b + (2 b)^{2}] - b^{2} = (a + 2 b)^{2} - b^{2} = [(a + 2 b) - b] [(a + 2 b) + b] = (a + 2 b - b) (a + 2 b + b) = (a + b) (a + 3 b)$

Page No 7.23:

Question 12:

Factorize each of the following algebraic expression:
96 − 4x − x²

Answer:

$96 - 4 x - x^{2} = 100 - 4 - 4 x - x^{2} = 100 - (x^{2} + 4 x + 4) = 100 - (x^{2} + 2 \times x \times 2 + 2^{2}) = 10^{2} - (x + 2)^{2} = [10 - (x + 2)] [10 + (x + 2)] = (10 - x - 2) (10 + x + 2) = (8 - x) (12 + x) = (x + 12) (- x + 8)$

Page No 7.23:

Question 13:

Factorize each of the following algebraic expression:
a⁴ + 3a² +4

Answer:

$a^{4} + 3 a^{2} + 4 = a^{4} + 4 a^{2} - a^{2} + 4 = (a^{4} + 4 a^{2} + 4) - a^{2} = [(a^{2})^{2} + 2 \times a^{2} \times 2 + 2^{2}] - a^{2} = (a^{2} + 2)^{2} - a^{2} = [(a^{2} + 2) - a] [(a^{2} + 2) + a] = (a^{2} - a + 2) ((a^{2} + a + 2)$

Page No 7.23:

Question 14:

Factorize each of the following algebraic expression:
4x⁴ + 1

Answer:

$4 x^{4} + 1 = 4 x^{4} + 4 x^{2} + 1 - 4 x^{2} = [(2 x^{2})^{2} + 2 \times 2 x^{2} \times 1 + 1] - 4 x^{2} = (2 x^{2} + 1)^{2} - (2 x)^{2} = [(2 x^{2} + 1) - 2 x] [(2 x^{2} + 1) + 2 x] = (2 x^{2} - 2 x + 1) (2 x^{2} + 2 x + 1)$

Page No 7.23:

Question 15:

Factorize each of the following algebraic expression:
4x⁴ + y⁴

Answer:

$4 x^{4} + y^{4} = 4 x^{4} + 4 x^{2} y^{2} + y^{4} - 4 x^{2} y^{2} = [(2 x^{2})^{2} + 2 \times 2 x^{2} \times y + (y^{2})^{2}] - (2 xy)^{2} = (2 x^{2} + y^{2})^{2} - (2 xy)^{2} = [(2 x^{2} + y^{2}) - 2 xy] [(2 x^{2} + y^{2}) + 2 xy] = (2 x^{2} - 2 xy + y^{2}) (2 x^{2} + 2 xy + y^{2})$

Page No 7.23:

Question 16:

Factorize each of the following algebraic expression:
(x + 2)² − 6(x + 2) + 9

Answer:

$(x + 2)^{2} - 6 (x + 2) + 9 = (x + 2)^{2} - 2 \times (x + 2) \times 3 + 3^{2} = [(x + 2) - 3]^{2} = (x + 2 - 3)^{2} = (x - 1)^{2} = (x - 1) (x - 1)$

Page No 7.23:

Question 17:

Factorize each of the following algebraic expression:
25 − p² − q² − 2pq

Answer:

$25 - p^{2} - q^{2} - 2 pq = 25 - (p^{2} + 2 pq + q^{2}) = 5^{2} - (p^{2} + 2 \times p \times q + q^{2}) = 5^{2} - (p + q)^{2} = [5 - (p + q)] [5 + (p + q)] = (5 - p - q) (5 + p + q) = - (p + q - 5) (p + q + 5)$

Page No 7.23:

Question 18:

Factorize each of the following algebraic expression:
x² + 9y² − 6xy − 25a²

Answer:

$x^{2} + 9 y^{2} - 6 xy - 25 a^{2} = (x^{2} - 6 xy + 9 y^{2}) - 25 a^{2} = [x^{2} - 2 \times x \times 3 y + (3 y)^{2}] - 25 a^{2} = (x - 3 y)^{2} - (5 a)^{2} = [(x - 3 y) - 5 a] [(x - 3 y) + 5 a] = (x - 3 y - 5 a) (x - 3 y + 5 a)$

Page No 7.23:

Question 19:

Factorize each of the following algebraic expression:
49 − a² + 8ab − 16b²

Answer:

$49 - a^{2} + 8 ab - 16 b^{2} = 49 - (a^{2} - 8 ab + 16 b^{2}) = 49 - [a^{2} - 2 \times a \times 4 b + (4 b)^{2}] = 7^{2} - (a - 4 b)^{2} = [7 - (a - 4 b)] [7 + (a - 4 b)] = (7 - a + 4 b) (7 + a - 4 b) = - (a - 4 b - 7) (a - 4 b + 7) = - (a - 4 b + 7) (a - 4 b - 7)$

Page No 7.23:

Question 20:

Factorize each of the following algebraic expression:
a² − 8ab + 16b² − 25c²

Answer:

$a^{2} - 8 a b + 16 b^{2} - 25 c^{2} = (a^{2} - 8 a b + 16 b^{2}) - 25 c^{2} = [a^{2} - 2 \times a \times 4 b + (4 b)^{2}] - 25 c^{2} = (a - 4 b)^{2} - (5 c)^{2} = [(a - 4 b) - 5 c] [(a - 4 b) + 5 c] = (a - 4 b - 5 c) (a - 4 b + 5 c)$

Page No 7.23:

Question 21:

Factorize each of the following algebraic expression:
x² − y² + 6y − 9

Answer:

$x^{2} - y^{2} + 6 y - 9 = x^{2} - (y^{2} - 6 y + 9) = x^{2} - (y^{2} - 2 \times y \times 3 + 3^{2}) = x^{2} - (y - 3)^{2} = [x - (y - 3)] [x + (y - 3)] = (x - y + 3) (x + y - 3)$

Page No 7.23:

Question 22:

Factorize each of the following algebraic expression:
25x² − 10x + 1 − 36y²

Answer:

$25 x^{2} - 10 x + 1 - 36 y^{2} = (25 x^{2} - 10 x + 1) - 36 y^{2} = [(5 x)^{2} - 2 \times 5 x \times 1 + 1] - 36 y^{2} = (5 x - 1)^{2} - (6 y)^{2} = [(5 x - 1) - 6 y] [(5 x - 1) + 6 y] = (5 x - 1 - 6 y) (5 x - 1 + 6 y) = (5 x - 6 y - 1) (5 x + 6 y - 1)$

Page No 7.23:

Question 23:

Factorize each of the following algebraic expression:
a² − b² + 2bc − c²

Answer:

$a^{2} - b^{2} + 2 b c - c^{2} = a^{2} - (b^{2} - 2 b c + c^{2}) = a^{2} - (b^{2} - 2 \times b \times c + c^{2}) = a^{2} - (b - c)^{2} = [(a - (b - c)] [(a + (b - c)] = (a - b + c) (a + b - c)$

Page No 7.23:

Question 24:

Factorize each of the following algebraic expression:
a² + 2ab + b² − c²

Answer:

$a^{2} + 2 a b + b^{2} - c^{2} = (a^{2} + 2 a b + b^{2}) - c^{2} = (a^{2} + 2 \times a \times b + b^{2}) - c^{2} = (a + b)^{2} - c^{2} = [(a + b) - c] [(a + b) + c] = (a + b - c) (a + b + c)$

Page No 7.23:

Question 25:

Factorize each of the following algebraic expression:
49 − x² − y² + 2xy

Answer:

$49 - x^{2} - y^{2} + 2 xy = 49 - (x^{2} - 2 xy + y^{2}) = 49 - (x^{2} - 2 \times x \times y + y^{2}) = 7^{2} - (x - y)^{2} = [7 - (x - y)] [7 + (x - y)] = (7 - x + y) (7 + x - y) = (x - y + 7) (y - x + 7)$

Page No 7.23:

Question 26:

Factorize each of the following algebraic expression:
a² + 4b² − 4ab − 4c²

Answer:

$a^{2} + 4 b^{2} - 4 a b - 4 c^{2} = (a^{2} - 4 a b + 4 b^{2}) - 4 c^{2} = [a^{2} - 2 \times a \times 2 b + (2 b)^{2}] - 4 c^{2} = (a - 2 b)^{2} - (2 c)^{2} = [(a - 2 b) - 2 c] [(a - 2 b) + 2 c] = (a - 2 b - 2 c) (a - 2 b + 2 c)$

Page No 7.23:

Question 27:

Factorize each of the following algebraic expression:
x² − y² − 4xz + 4z²

Answer:

$x^{2} - y^{2} - 4 xz + 4 z^{2} = (x^{2} - 4 xz + 4 z^{2}) - y^{2} = [x^{2} - 2 \times x \times 2 z + (2 z)^{2}] - y^{2} = (x - 2 z)^{2} - y^{2} = [(x - 2 z) - y] [(x - 2 z) + y] = (x - 2 z - y) (x - 2 z + y) = (x + y - 2 z) (x - y - 2 z)$

Page No 7.27:

Question 1:

Factorize each of the following algebraic expression:
x² + 12x − 45

Answer:

$To factorise x^{2} + 12 x - 45, we will find two numbers p and q such that p + q = 12 and pq = - 45 . Now, 15 + (- 3) = 12 and 15 \times (- 3) = - 45 Splitting the middle term 12 x in the given quadratic as - 3 x + 15 x, we get : x^{2} + 12 x - 45 = x^{2} - 3 x + 15 x - 45 = (x^{2} - 3 x) + (15 x - 45) = x (x - 3) + 15 (x - 3) = (x + 15) (x - 3)$

Page No 7.27:

Question 2:

Factorize each of the following algebraic expression:
40 + 3x − x²

Answer:

$We have : 40 + 3 x - x^{2} \Rightarrow - (x^{2} - 3 x - 40) To factorise (x^{2} - 3 x - 40), we will find two numbers p and q such that p + q = - 3 and pq = - 40 . Now, 5 + (- 8) = - 3 and 5 \times (- 8) = - 40 Splitting the middle term - 3 x in the given quadratic as 5 x - 8 x, we get : 40 + 3 x - x^{2} = - (x^{2} - 3 x - 40) = - (x^{2} + 5 x - 8 x - 40) = - [(x^{2} + 5 x) - (8 x + 40)] = - [x (x + 5) - 8 (x + 5)] = - (x - 8) (x + 5) = (x + 5) (- x + 8)$

Page No 7.27:

Question 3:

Factorize each of the following algebraic expression:
a² + 3a − 88

Answer:

$To factorise a^{2} + 3 a - 88, we will find two numbers p and q such that p + q = 3 and pq = - 88 . Now, 11 + (- 8) = 3 and 11 \times (- 8) = - 88 Splitting the middle term 3 a in the given quadratic as 11 a - 8 a, we get : a^{2} + 3 a - 88 = a^{2} + 11 a - 8 a - 88 = (a^{2} + 11 a) - (8 a + 88) = a (a + 11) - 8 (a + 11) = (a - 8) (a + 11)$

Page No 7.27:

Question 4:

Factorize each of the following algebraic expression:
a² − 14a − 51

Answer:

$To factorise a^{2} - 14 a - 51, we will find two numbers p and q such that p + q = - 14 and pq = - 51 . Now, 3 + (- 17) = - 14 and 3 \times (- 17) = - 51 Splitting the middle term - 14 a in the given quadratic as 3 a - 17 a, we get : a^{2} - 14 a - 51 = a^{2} + 3 a - 17 a - 51 = (a^{2} + 3 a) - (17 a + 51) = a (a + 3) - 17 (a + 3) = (a - 17) (a + 3)$

Page No 7.27:

Question 5:

Factorize each of the following algebraic expression:
x² + 14x + 45

Answer:

$To factorise x^{2} + 14 x + 45, we will find two numbers p and q such that p + q = 14 and pq = 45 . Now, 9 + 5 = 14 and 9 \times 5 = 45 Splitting the middle term 14 x in the given quadratic as 9 x + 5 x, we get : x^{2} + 14 x + 45 = x^{2} + 9 x + 5 x + 45 = (x^{2} + 9 x) + (5 x + 45) = x (x + 9) + 5 (x + 9) = (x + 5) (x + 9)$

Page No 7.27:

Question 6:

Factorize each of the following algebraic expression:
x² − 22x + 120

Answer:

$To factorise x^{2} - 22 x + 120, we will find two numbers p and q such that p + q = - 22 and pq = 120 . Now, (- 12) + (- 10) = - 22 and (- 12) \times (- 10) = 120 Splitting the middle term - 22 x in the given quadratic as - 12 x - 10 x, we get : x^{2} - 22 x + 12 = x^{2} - 12 x - 10 x + 120 = (x^{2} - 12 x) + (- 10 x + 120) = x (x - 12) - 10 (x - 12) = (x - 10) (x - 12)$

Page No 7.27:

Question 7:

Factorize each of the following algebraic expression:
x² − 11x − 42

Answer:

$To factorise x^{2} - 11 x - 42, we will find two numbers p and q such that p + q = - 11 and pq = - 42 . Now, 3 + (- 14) = - 22 and 3 \times (- 14) = 42 Splitting the middle term - 11 x in the given quadratic as - 14 x + 3 x, we get : x^{2} - 11 x - 42 = x^{2} - 14 x + 3 x - 42 = (x^{2} - 14 x) + (3 x - 42) = x (x - 14) + 3 (x - 14) = (x + 3) (x - 14)$

Page No 7.27:

Question 8:

Factorize each of the following algebraic expression:
a² + 2a − 3

Answer:

$To factorise a^{2} + 2 a - 3, we will find two numbers p and q such that p + q = 2 and pq = - 3 . Now, 3 + (- 1) = 2 and 3 \times (- 1) = - 3 Splitting the middle term 2 a in the given quadratic as - a + 3 a, we get : a^{2} + 2 a - 3 = a^{2} - a + 3 a - 3 = (a^{2} - a) + (3 a - 3) = a (a - 1) + 3 (a - 1) = (a + 3) (a - 1)$

Page No 7.27:

Question 9:

Factorize each of the following algebraic expression:
a² + 14a + 48

Answer:

$To factorise a^{2} + 14 a + 48, we will find two numbers p and q such that p + q = 14 and pq = 48 . Now, 8 + 6 = 14 and 8 \times 6 = 48 Splitting the middle term 14 a in the given quadratic as 8 a + 6 a, we get : a^{2} + 14 a + 48 = a^{2} + 8 a + 6 a + 48 = (a^{2} + 8 a) + (6 a + 48) = a (a + 8) + 6 (a + 8) = (a + 6) (a + 8)$

Page No 7.27:

Question 10:

Factorize each of the following algebraic expression:
x² − 4x − 21

Answer:

$To factorise x^{2} - 4 x - 21, we will find two numbers p and q such that p + q = - 4 and pq = - 21 . Now, 3 + (- 7) = - 4 and 3 \times (- 7) = - 21 Splitting the middle term - 4 x in the given quadratic as - 7 x + 3 x, we get : x^{2} - 4 x - 21 = x^{2} - 7 x + 3 x - 21 = (x^{2} - 7 x) + (3 x - 21) = x (x - 7) + 3 (x - 7) = (x + 3) (x - 7)$

Page No 7.27:

Question 11:

Factorize each of the following algebraic expression:
y² + 5y − 36

Answer:

$To factorise y^{2} + 5 y - 36, we will find two numbers p and q such that p + q = 5 and pq = - 36 . Now, 9 + (- 4) = 5 and 9 \times (- 4) = - 36 Splitting the middle term 5 y in the given quadratic as - 4 y + 9 y, we get : y^{2} + 5 y - 36 = y^{2} - 4 y + 9 y - 36 = (y^{2} - 4 y) + (9 y - 36) = y (y - 4) + 9 (y - 4) = (y + 9) (y - 4)$

Page No 7.27:

Question 12:

Factorize each of the following algebraic expression:
(a² − 5a)² − 36

Answer:

$(a^{2} - 5 a)^{2} - 36 = (a^{2} - 5 a)^{2} - 6^{2} = [(a^{2} - 5 a) - 6] [(a^{2} - 5 a) + 6] = (a^{2} - 5 a - 6) (a^{2} - 5 a + 6) I n o r d e r t o f a c t o r i s e a^{2} - 5 a - 6, w e w i l l f i n d t w o n u m b e r s p a n d q s u c h t h a t p + q = - 5 a n d p q = - 6 N o w, (- 6) + 1 = - 5 a n d (- 6) \times 1 = - 6 S p l i t t i n g t h e m i d d l e t e r m - 5 i n t h e g i v e n q u a d r a t i c a s - 6 a + a, w e g e t : a^{2} - 5 a - 6 = a^{2} - 6 a + a - 6 = (a^{2} - 6 a) + (a - 6) = a (a - 6) + (a - 6) = (a + 1) (a - 6) N o w, I n o r d e r t o f a c t o r i s e a^{2} - 5 a + 6, w e w i l l f i n d t w o n u m b e r s p a n d q s u c h t h a t p + q = - 5 a n d p q = 6 C l e a r l y, (- 2) + (- 3) = - 5 a n d (- 2) \times (- 3) = 6 S p l i t t i n g t h e m i d d l e t e r m - 5 i n t h e g i v e n q u a d r a t i c a s - 2 a - 3 a, w e g e t : a^{2} - 5 a + 6 = a^{2} - 2 a - 3 a + 6 = (a^{2} - 2 a) - (3 a - 6) = a (a - 2) - 3 (a - 2) = (a - 3) (a - 2) ∴ (a^{2} - 5 a - 6) (a^{2} - 5 a + 6) = (a - 6) (a + 1) (a - 3) (a - 2) = (a + 1) (a - 2) (a - 3) (a - 6)$

Page No 7.27:

Question 13:

Factorize each of the following algebraic expression:
(a + 7)(a − 10) + 16

Answer:

$(a + 7) (a - 10) + 16 = a^{2} - 10 a + 7 a - 70 + 16 = a^{2} - 3 a - 54 T o f a c t o r i s e a^{2} - 3 a - 54, w e w i l l f i n d t w o n u m b e r s p a n d q s u c h t h a t p + q = - 3 a n d p q = - 54 . N o w, 6 + (- 9) = - 3 a n d 6 \times (- 9) = - 54 S p l i t t i n g t h e m i d d l e t e r m - 3 a i n t h e g i v e n q u a d r a t i c a s - 9 a + 6 a, w e g e t : a^{2} - 3 a - 54 = a^{2} - 9 a + 6 a - 54 = (a^{2} - 9 a) + (6 a - 54) = a (a - 9) + 6 (a - 9) = (a + 6) (a - 9)$

Page No 7.30:

Question 1:

Resolve each of the following quadratic trinomial into factor:
2x² + 5x + 3

Answer:

$The given expression is 2 x^{2} + 5 x + 3 . (Coefficient of x^{2} = 2, coefficient of x = 5 and constant term = 3) We will split the coefficient of x into two parts such that their sum is 5 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 2 \times 3 = 6 . Now, 2 + 3 = 5 and 2 \times 3 = 6 Replacing the middle term 5 x by 2 x + 3 x, we have : 2 x^{2} + 5 x + 3 = 2 x^{2} + 2 x + 3 x + 3 = (2 x^{2} + 2 x) + (3 x + 3) = 2 x (x + 1) + 3 (x + 1) = (x + 1) (2 x + 3)$

Page No 7.30:

Question 2:

Resolve each of the following quadratic trinomial into factor:
2x² − 3x − 2

Answer:

$The given expression is 2 x^{2} - 3 x - 2 . (Coefficient of x^{2} = 2, coefficient of x = - 3 and constant term = - 2) We will split the coefficient of x into two parts such that their sum is - 3 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 2 \times (- 2) = - 4 . Now, (- 4) + 1 = - 3 and (- 4) \times 1 = - 4 Replacing the middle term 3 x by - 4 x + x, we have : 2 x^{2} - 3 x - 2 = 2 x^{2} - 4 x + x - 2 = (2 x^{2} - 4 x) + (x - 2) = 2 x (x - 2) + (x - 2) = (2 x + 1) (x - 2)$

Page No 7.30:

Question 3:

Resolve each of the following quadratic trinomial into factor:
3x² + 10x + 3

Answer:

$The given expression is 3 x^{2} + 10 x + 3 . (Coefficient of x^{2} = 3, coefficient of x = 10 and constant term = 3) We will split the coefficient of x into two parts such that their sum is 10 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 3 \times 3 = 9 . Now, 9 + 1 = 10 and 9 \times 1 = 9 Replacing the middle term 10 x by 9 x + x, we have : 3 x^{2} + 10 x + 3 = 3 x^{2} + 9 x + x + 3 = (3 x^{2} + 9 x) + (x + 3) = 3 x (x + 3) + (x + 3) = (3 x + 1) (x + 3)$

Page No 7.30:

Question 4:

Resolve each of the following quadratic trinomial into factor:
7x − 6 − 2x²

Answer:

$The given expression is 7 x - 6 - 2 x^{2} . (Coefficient of x^{2} = - 2, coefficient of x = 7 and constant term = - 6) We will split the coefficient of x into two parts such that their sum is 7 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., (- 2) \times (- 6) = 12 . Now, 4 + 3 = 7 and 4 \times 3 = 12 Replacing the middle term 7 x by 4 x + 3 x, we have : 7 x - 6 - 2 x^{2} = - 2 x^{2} + 4 x + 3 x - 6 = (- 2 x^{2} + 4 x) + (3 x - 6) = 2 x (2 - x) - 3 (2 - x) = (2 x - 3) (2 - x)$

Page No 7.30:

Question 5:

Resolve each of the following quadratic trinomial into factor:
7x² − 19x − 6

Answer:

$The given expression is 7 x^{2} - 19 x - 6 . (Coefficient of x^{2} = 7, coefficient of x = - 19 and constant term = - 6) We will split the coefficient of x into two parts such that their sum is - 19 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 7 \times (- 6) = - 42 . Now, (- 21) + 2 = - 19 and (- 21) \times 2 = - 42 Replacing the middle term - 19 x by - 21 x + 2 x, we have : 7 x^{2} - 19 x - 6 = 7 x^{2} - 21 x + 2 x - 6 = (7 x^{2} - 21 x) + (2 x - 6) = 7 x (x - 3) + 2 (x - 3) = (7 x + 2) (x - 3)$

Page No 7.30:

Question 6:

Resolve each of the following quadratic trinomial into factor:
28 − 31x − 5x²

Answer:

$The given expression is 28 - 31 x - 5 x^{2} . (Coefficient of x^{2} = - 5, coefficient of x = - 31 and constant term = 28) We will split the coefficient of x into two parts such that their sum is - 31 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., (- 5) \times (28) = - 140 . Now, (- 35) + 4 = - 31 and (- 35) \times 4 = - 140 Replacing the middle term - 31 x by - 35 x + 4 x, we have : - 5 x^{2} - 31 x + 28 = - 5 x^{2} - 35 x + 4 x + 28 = (- 5 x^{2} - 35 x) + (4 x + 28) = - 5 x (x + 7) + 4 (x + 7) = (4 - 5 x) (x + 7)$

Page No 7.30:

Question 7:

Resolve each of the following quadratic trinomial into factor:
3 + 23y − 8y²

Answer:

$The given expression is 3 + 23 y - 8 y^{2} . (Coefficient of y^{2} = - 8, coefficient of y = 23 and constant term = 3) We will split the coefficient of y into two parts such that their sum is 23 and their product equals the product of the coefficient of y^{2} and the constant term, i . e ., (- 8) \times 3 = - 24 . Now, (- 1) + 24 = 23 and (- 1) \times 24 = - 24 Replacing the middle term 23 y by - y + 24 y, we have : 3 + 23 y - 8 y^{2} = - 8 y^{2} + 23 y + 3 = - 8 y^{2} - y + 24 y + 3 = (- 8 y^{2} - y) + (24 y + 3) = - y (8 y + 1) + 3 (8 y + 1) = (3 - y) (8 y + 1)$

Page No 7.30:

Question 8:

Resolve each of the following quadratic trinomial into factor:
11x² − 54x + 63

Answer:

$The given expression is 11 x^{2} - 54 x + 63 . (Coefficient of x^{2} = 11, coefficient of x = - 54 and constant term = 63) We will split the coefficient of x into two parts such that their sum is - 54 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 11 \times 63 = 693 . Now, (- 33) + (- 21) = - 54 and (- 33) \times (- 21) = 693 Replacing the middle term - 54 x by - 33 x - 21 x, we have : 11 x^{2} - 54 x + 63 = 11 x^{2} - 33 x - 21 x + 63 = (11 x^{2} - 33 x) + (- 21 x + 63) = 11 x (x - 3) - 21 (x - 3) = (11 x - 21) (x - 3)$

Page No 7.30:

Question 9:

Resolve each of the following quadratic trinomial into factor:
7x − 6x² + 20

Answer:

$The given expression is 7 x - 6 x^{2} + 20 . (Coefficient of x^{2} = - 6, coefficient of x = 7 and constant term = 20) We will split the coefficient of x into two parts such that their sum is 7 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., (- 6) \times 20 = - 120 . Now, 15 + (- 8) = 7 and 15 \times (- 8) = - 120 Replacing the middle term 7 x by 15 x - 8 x, we get : 7 x - 6 x^{2} + 20 = - 6 x^{2} + 7 x + 20 = - 6 x^{2} + 15 x - 8 x + 20 = (- 6 x^{2} + 15 x) + (- 8 x + 20) = 3 x (- 2 x + 5) + 4 (- 2 x + 5) = (3 x + 4) (- 2 x + 5)$

Page No 7.30:

Question 10:

Resolve each of the following quadratic trinomial into factor:
3x² + 22x + 35

Answer:

$The given expression is 3 x^{2} + 22 x + 35 . (Coefficient of x^{2} = 3, coefficient of x = 22 and constant term = 35) We will split the coefficient of x into two parts such that their sum is 22 and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 3 \times 35 = 105 . Now, 15 + 7 = 22 and 15 \times 7 = 105 Replacing the middle term 22 x by 15 x + 7 x, we get : 3 x^{2} + 22 x + 35 = 3 x^{2} + 15 x + 7 x + 35 = (3 x^{2} + 15 x) + (7 x + 35) = 3 x (x + 5) + 7 (x + 5) = (3 x + 7) (x + 5)$

Page No 7.30:

Question 11:

Resolve each of the following quadratic trinomial into factor:
12x² − 17xy + 6y²

Answer:

$The given expression is 12 x^{2} - 17 xy + 6 y^{2} . (Coefficient of x^{2} = 12, coefficient of x = - 17 y and constant term = 6 y^{2}) We willsplit the coefficient of x into two parts such that their sum is - 17 y and their product equals the product of the coefficient of x^{2} and the constant term i . e ., 12 \times 6 y^{2} = 72 y^{2} . Now, (- 9 y) + (- 8 y) = - 17 y and (- 9 y) \times (- 8 y) = 72 y^{2} Replacing the middle term - 17 xy by - 9 xy - 8 xy, we get : 12 x^{2} - 17 xy + 6 y^{2} = 12 x^{2} - 9 xy - 8 xy + 6 y^{2} = (12 x^{2} - 9 xy) - (8 xy - 6 y^{2}) = 3 x (4 x - 3 y) - 2 y (4 x - 3 y) = (3 x - 2 y) (4 x - 3 y)$

Page No 7.30:

Question 12:

Resolve each of the following quadratic trinomial into factor:
6x² − 5xy − 6y²

Answer:

$The given expression is 6 x^{2} - 5 xy - 6 y^{2} . (Coefficient of x^{2} = 6, coefficient of x = - 5 y and constant term = - 6 y^{2}) We will split the coefficient of x into two parts such that their sum is - 5 y and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 6 \times (- 6 y^{2}) = - 36 y^{2} . Now, (- 9 y) + 4 y = - 5 y and (- 9 y) \times 4 y = - 36 y^{2} Replacing the middle term - 5 xy by - 9 xy + 4 xy, we get : 6 x^{2} - 5 xy - 6 y^{2} = 6 x^{2} - 9 xy + 4 xy - 6 y^{2} = (6 x^{2} - 9 xy) + (4 xy - 6 y^{2}) = 3 x (2 x - 3 y) + 2 y (2 x - 3 y) = (3 x + 2 y) (2 x - 3 y)$

Page No 7.30:

Question 13:

Resolve each of the following quadratic trinomial into factor:
6x² − 13xy + 2y²

Answer:

$The given expression is 6 x^{2} - 13 xy + 2 y^{2} . (Coefficient of x^{2} = 6, coefficient of x = - 13 y and constant term = 2 y^{2}) We will split the coefficient of x into two parts such that their sum is - 13 y and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 6 \times (2 y^{2}) = 12 y^{2} . Now, (- 12 y) + (- y) = - 13 y and (- 12 y) \times (- y) = 12 y^{2} Replacing the middle term - 13 xy by - 12 xy - xy, we get : 6 x^{2} - 13 xy + 2 y^{2} = 6 x^{2} - 12 xy - xy + 2 y^{2} = (6 x^{2} - 12 xy) - (xy - 2 y^{2}) = 6 x (x - 2 y) - y (x - 2 y) = (6 x - y) (x - 2 y)$

Page No 7.30:

Question 14:

Resolve each of the following quadratic trinomial into factor:
14x² + 11xy − 15y²

Answer:

$The given expression is 14 x^{2} + 11 xy - 15 y^{2} . (Coefficient of x^{2} = 14, coefficient of x = 11 y and constant term = - 15 y^{2}) Now, we will split the coefficient of x into two parts such that their sum is 11 y and their product equals the product of the coefficient of x^{2} and the constant term, i . e ., 14 \times (- 15 y^{2}) = - 210 y^{2} . Now, 21 y + (- 10 y) = 11 y and 21 y \times (- 10 y) = - 210 y^{2} Replacing the middle term - 11 xy by - 10 xy + 21 xy, we get : 14 x^{2} + 11 xy - 15 y^{2} = 14 x^{2} - 10 xy + 21 xy - 15 y^{2} = (14 x^{2} - 10 xy) + (21 xy - 15 y^{2}) = 2 x (7 x - 5 y) + 3 y (7 x - 5 y) = (2 x + 3 y) (7 x - 5 y)$

Page No 7.30:

Question 15:

Resolve each of the following quadratic trinomial into factor:
6a² + 17ab − 3b²

Answer:

$The given expression is 6 a^{2} + 17 ab - 3 b^{2} . (Coefficient of a^{2} = 6, coefficient of a = 17 b and constant term = - 3 b^{2}) Now, we will split the coefficient of a into two parts such that their sum is 17 b and their product equals the product of the coefficient of a^{2} and the constant term, i . e ., 6 \times (- 3 b^{2}) = - 18 b^{2} . Now, 18 b + (- b) = 17 b and 18 b \times (- b) = - 18 b^{2} Replacing the middle term 17 ab by - ab + 18 ab, we get : 16 a^{2} + 17 ab - 3 b^{2} = 6 a^{2} + - ab + 18 ab - 3 b^{2} = (6 a^{2} - ab) + (18 ab - 3 b^{2}) = a (6 a - b) + 3 b (6 a - b) = (a + 3 b) (6 a - b)$

Page No 7.30:

Question 16:

Resolve each of the following quadratic trinomial into factor:
36a² + 12abc − 15b²c²

Answer:

$The given expression is 36 a^{2} + 12 abc - 15 b^{2} c^{2} . (Coefficient of a^{2} = 36, coefficient of a = 12 bc and constant term = - 15 b^{2} c^{2}) Now, we will split the coefficient of a into two parts such that their sum is 12 bc and their product equals the product of the coefficient of a^{2} and the constant term, i . e ., 36 \times (- 15 b^{2} c^{2}) = - 540 b^{2} c^{2} . Now, (- 18 bc) + 30 bc = 12 bc and (- 18 bc) \times 30 bc = - 540 b^{2} c^{2} Replacing the middle term 12 abc by - 18 abc + 30 abc, we get : 36 a^{2} + 12 abc - 15 b^{2} c^{2} = 36 a^{2} - 18 abc + 30 abc - 15 b^{2} c^{2} = (36 a^{2} - 18 abc) + (30 abc - 15 b^{2} c^{2}) = 18 a (2 a - bc) + 15 bc (2 a - bc) = (18 a + 15 bc) (2 a - bc) = 3 (6 a + 5 bc) (2 a - bc)$

Page No 7.30:

Question 17:

Resolve each of the following quadratic trinomial into factor:
15x² − 16xyz − 15y²z²

Answer:

$T h e g i v e n e x p r e s s i o n i s 15 x^{2} - 16 x y z - 15 y^{2} z^{2} . (C o e f f i c i e n t o f x^{2} = 15, c o e f f i c i e n t o f x = - 16 y z a n d c o n s t a n t t e r m = - 15 y^{2} z^{2}) N o w, w e w i l l s p l i t t h e c o e f f i c i e n t o f x i n t o t w o p a r t s s u c h t h a t t h e i r s u m i s - 16 y z a n d t h e i r p r o d u c t e q u a l s t h e p r o d u c t o f t h e c o e f f i c i e n t o f x^{2} a n d t h e c o n s \tan t t e r m, i . e ., 15 \times (- 15 y^{2} z^{2}) = - 225 y^{2} z^{2} . N o w, (- 25 y z) + 9 y z = - 16 y x a n d (- 25 y z) \times 9 y z = - 225 y^{2} z^{2} R e p l a c i n g t h e m i d d l e t e r m - 16 x y z b y - 25 x y z + 9 x y z, w e h a v e : 15 x^{2} - 16 x y z - 15 y^{2} z^{2} = 15 x^{2} - 25 x y z + 9 x y z - 15 y^{2} z^{2} = (15 x^{2} - 25 x y z) + (9 x y z - 15 y^{2} z^{2}) = 5 x (3 x - 5 y z) + 3 y z (3 x - 5 y z) = (5 x + 3 y z) (3 x - 5 y z)$

Page No 7.30:

Question 18:

Resolve each of the following quadratic trinomial into factor:
(x − 2y)² − 5(x − 2y) + 6

Answer:

$The given expression is a^{2} - 5 a + 6 . Assuming a = x - 2 y, we have : (x - 2 y)^{2} - 5 (x - 2 y) + 6 = a^{2} - 5 a + 6 (Coefficient of a^{2} = 1, coefficient of a = - 5 and constant term = 6) Now, we will split the coefficient of a into two parts such that their sum is - 5 and their product equals the product of the coefficient of a^{2} and the constant term, i . e ., 1 \times 6 = 6 . Clearly, (- 2) + (- 3) = - 5 and (- 2) \times (- 3) = 6 Replacing the middle term - 5 a by - 2 a - 3 a, we have : a^{2} - 5 a + 6 = a^{2} - 2 a - 3 a + 6 = (a^{2} - 2 a) - (3 a - 6) = a (a - 2) - 3 (a - 2) = (a - 3) (a - 2) Replacing a by (x - 2 y), we get : (a - 3) (a - 2) = (x - 2 y - 3) (x - 2 y - 2)$

Page No 7.30:

Question 19:

Resolve each of the following quadratic trinomial into factor:
(2a − b)² + 2(2a − b) − 8

Answer:

$A s s u m i n g x = 2 a - b, w e h a v e : (2 a - b)^{2} + 2 (2 a - b) - 8 = x^{2} + 2 x - 8 T h e g i v e n e x p r e s s i o n b e c o m e s x^{2} + 2 x - 8 . (C o e f f i c i e n t o f x^{2} = 1 a n d t h a t o f x = 2; c o n s t a n t t e r m = - 8) N o w, w e w i l l s p l i t t h e c o e f f i c i e n t o f x i n t o t w o p a r t s s u c h t h a t t h e i r s u m i s 2 a n d t h e i r p r o d u c t e q u a l s t h e p r o d u c t o f t h e c o e f f i c i e n t o f x^{2} a n d t h e c o n s \tan t t e r m, i . e ., 1 \times (- 8) = - 8 . C l e a r l y, (- 2) + 4 = 2 a n d (- 2) \times 4 = - 8 R e p l a c i n g t h e m i d d l e t e r m 2 x b y - 2 x + 4 x, w e g e t : x^{2} + 2 x - 8 = x^{2} - 2 x + 4 x - 8 = (x^{2} - 2 x) + (4 x - 8) = x (x - 2) + 4 (x - 2) = (x + 4) (x - 2) R e l a c i n g x b y 2 a - b, w e g e t : (x + 4) (x - 2) = (2 a - b + 4) (2 a - b - 2)$

Page No 7.3:

Question 1:

Find the greatest common factor (GCF/HCF) of the following polynomial:
2x² and 12x²

Answer:

The numerical coefficients of the given monomials are 2 and 12. So, the greatest common factor of 2 and 12 is 2.
The common literal appearing in the given monomials is x.
The smallest power of x in the two monomials is 2.
The monomial of the common literals with the smallest powers is x².
Hence, the greatest common factor is 2x².

Page No 7.3:

Question 2:

Find the greatest common factor (GCF/HCF) of the following polynomial:
6x³y and 18x²y³

Answer:

The numerical coefficients of the given monomials are 6 and 18. The greatest common factor of 6 and 18 is 6.
The common literals appearing in the two monomials are x and y.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The monomial of the common literals with the smallest powers is x²y.
Hence, the greatest common factor is 6x²y.

Page No 7.3:

Question 3:

Find the greatest common factor (GCF/HCF) of the following polynomial:
7x, 21x² and 14xy²

Answer:

The numerical coefficients of the given monomials are 7, 21 and 14. The greatest common factor of 7, 21 and 14 is 7.
The common literal appearing in the three monomials is x.
The smallest power of x in the three monomials is 1.
The monomial of the common literals with the smallest powers is x.
Hence, the greatest common factor is 7x.

Page No 7.3:

Question 4:

Find the greatest common factor (GCF/HCF) of the following polynomial:
42x²yz and 63x³y²z³

Answer:

The numerical coefficients of the given monomials are 42 and 63. The greatest common factor of 42 and 63 is 21.
The common literals appearing in the two monomials are x, y and z.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 1.
The smallest power of z in the two monomials is 1.
The monomial of the common literals with the smallest powers is x²yz.
Hence, the greatest common factor is 21x²yz.

Page No 7.3:

Question 5:

Find the greatest common factor (GCF/HCF) of the following polynomial:
12ax², 6a²x³ and 2a³x⁵

Answer:

The numerical coefficients of the given monomials are 12, 6 and 2. The greatest common factor of 12, 6 and 2 is 2.
The common literals appearing in the three monomials are a and x.
The smallest power of a in the three monomials is 1.
The smallest power of x in the three monomials is 2.
The monomial of common literals with the smallest powers is ax².
Hence, the greatest common factor is 2ax².

Page No 7.3:

Question 6:

Find the greatest common factor (GCF/HCF) of the following polynomial:
9x², 15x²y³, 6xy² and 21x²y²

Answer:

The numerical coefficients of the given monomials are 9, 15, 6 and 21. The greatest common factor of 9, 15, 6 and 21 is 3.
The common literal appearing in the three monomials is x.
The smallest power of x in the four monomials is 1.
The monomial of common literals with the smallest powers is x.
Hence, the greatest common factor is 3x.

Page No 7.3:

Question 7:

Find the greatest common factor (GCF/HCF) of the following polynomial:
4a²b³, −12a³b, 18a⁴b³

Answer:

The numerical coefficients of the given monomials are 4, -12 and 18. The greatest common factor of 4, -12 and 18 is 2.
The common literals appearing in the three monomials are a and b.
The smallest power of a in the three monomials is 2.
The smallest power of b in the three monomials is 1.
The monomial of the common literals with the smallest powers is a²b.
Hence, the greatest common factor is 2a²b.

Page No 7.3:

Question 8:

Find the greatest common factor (GCF/HCF) of the following polynomial:
6x²y², 9xy³, 3x³y²

Answer:

The numerical coefficients of the given monomials are 6, 9 and 3. The greatest common factor of 6, 9 and 3 is 3.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 1.
The smallest power of y in the three monomials is 2.
The monomial of common literals with the smallest powers is xy².
Hence, the greatest common factor is 3xy².

Page No 7.32:

Question 1:

Factorize each of the following quadratic polynomials by using the method of completing the square:
p² + 6p + 8

Answer:

$p^{2} + 6 p + 8 = p^{2} + 6 p + {(\frac{6}{2})}^{2} - {(\frac{6}{2})}^{2} + 8 [Adding and subtracting {(\frac{6}{2})}^{2}, that is, 3^{2}] = p^{2} + 6 p + 3^{2} - 3^{2} + 8 = p^{2} + 2 \times p \times 3 + 3^{2} - 9 + 8 = p^{2} + 2 \times p \times 3 + 3^{2} - 1 = (p + 3)^{2} - 1^{2} [Completing the square] = [(p + 3) - 1] [(p + 3) + 1] = (p + 3 - 1) (p + 3 + 1) = (p + 2) (p + 4)$

Page No 7.32:

Question 2:

Factorize each of the following quadratic polynomials by using the method of completing the square:
q2 − 10q + 21

Answer:

$q^{2} - 10 q + 21 = q^{2} - 10 q + {(\frac{10}{2})}^{2} - {(\frac{10}{2})}^{2} + 21 [Adding and subtracting {(\frac{10}{2})}^{2}, that is, 5^{2}] = q^{2} - 2 \times q \times 5 + 5^{2} - 5^{2} + 21 = (q - 5)^{2} - 4 [Completing the square] = (q - 5)^{2} - 2^{2} = [(q - 5) - 2] [(q - 5) + 2] = (q - 5 - 2) (q - 5 + 2) = (q - 7) (q - 3)$

Page No 7.32:

Question 3:

Factorize each of the following quadratic polynomials by using the method of completing the square:
4y² + 12y + 5

Answer:

$4 y^{2} + 12 y + 5 = 4 (y^{2} + 3 y + \frac{5}{4}) [M a k i n g t h e c o e f f i c i e n t o f y^{2} = 1] = 4 [y^{2} + 3 y + {(\frac{3}{2})}^{2} - {(\frac{3}{2})}^{2} + \frac{5}{4}] [A d d i n g a n d s u b t r a c t i n g {(\frac{3}{2})}^{2}] = 4 [(y + \frac{3}{2})^{2} - \frac{9}{4} + \frac{5}{4}] = 4 [(y + \frac{3}{2})^{2} - 1^{2}] [C o m p l e t i n g t h e s q u a r e] = 4 [(y + \frac{3}{2}) - 1] [(y + \frac{3}{2}) + 1] = 4 (y + \frac{3}{2} - 1) (y + \frac{3}{2} + 1) = 4 (y + \frac{1}{2}) (y + \frac{5}{2}) = (2 y + 1) (2 y + 5)$

Page No 7.32:

Question 4:

Factorize each of the following quadratic polynomials by using the method of completing the square:
p² + 6p − 16

Answer:

$p^{2} + 6 p - 16 = p^{2} + 6 p + {(\frac{6}{2})}^{2} - {(\frac{6}{2})}^{2} - 16 [Adding and subtracting {(\frac{6}{2})}^{2}, that is, 3^{2}] = p^{2} + 6 p + 3^{2} - 9 - 16 = (p + 3)^{2} - 25 [Completing the square] = (p + 3)^{2} - 5^{2} = [(p + 3) - 5] [(p + 3) + 5] = (p + 3 - 5) (p + 3 + 5) = (p - 2) (p + 8)$

Page No 7.32:

Question 5:

Factorize each of the following quadratic polynomials by using the method of completing the square:
x² + 12x + 20

Answer:

$x^{2} + 12 x + 20 = x^{2} + 12 x + {(\frac{12}{2})}^{2} - {(\frac{12}{2})}^{2} + 20 [A d d i n g a n d s u b t r a c t i n g {(\frac{12}{2})}^{2}, t h a t i s, 6^{2}] = x^{2} + 12 x + 6^{2} - 6^{2} + 20 = (x + 6)^{2} - 16 [C o m p l e t i n g t h e s q u a r e] = (x + 6)^{2} - 4^{2} = [(x + 6) - 4] [(x + 6) + 4] = (x + 6 - 4) (x + 6 + 4) = (x + 2) (x + 10)$

Page No 7.32:

Question 6:

Factorize each of the following quadratic polynomials by using the method of completing the square:
a² − 14a − 51

Answer:

$a^{2} - 14 a - 51 = a^{2} - 14 a + {(\frac{14}{2})}^{2} - {(\frac{14}{2})}^{2} - 51 [A d d i n g a n d s u b t r a c t i n g {(\frac{14}{2})}^{2}, t h a t i s, 7^{2}] = a^{2} - 14 a + 7^{2} - 7^{2} - 51 = (a - 7)^{2} - 100 [C o m p l e t i n g t h e s q u a r e] = (a - 7)^{2} - 10^{2} = [(a - 7) - 10] [(a - 7) + 10] = (a - 7 - 10) (a - 7 + 10) = (a - 17) (a + 3)$

Page No 7.33:

Question 7:

Factorize each of the following quadratic polynomials by using the method of completing the square:
a² + 2a − 3

Answer:

$a^{2} + 2 a - 3 = a^{2} + 2 a + {(\frac{2}{2})}^{2} - {(\frac{2}{2})}^{2} - 3 [A d d i n g a n d s u b t r a c t i n g {(\frac{2}{2})}^{2}, t h a t i s, 1^{2}] = a^{2} + 2 a + 1^{2} - 1^{2} - 3 = (a + 1)^{2} - 4 [C o m p l e t i n g t h e s q u a r e] = (a + 1)^{2} - 2^{2} = [(a + 1) - 2] [(a + 1) + 2] = (a + 1 - 2) (a + 1 + 2) = (a - 1) (a + 3)$

Page No 7.33:

Question 8:

Factorize each of the following quadratic polynomials by using the method of completing the square:
4x² − 12x + 5

Answer:

$4 x^{2} - 12 x + 5 = 4 (x^{2} - 3 x + \frac{5}{4}) [M a k i n g t h e c o e f f i c i e n t o f x^{2} = 1] = 4 [x^{2} - 3 x + {(\frac{3}{2})}^{2} - {(\frac{3}{2})}^{2} + \frac{5}{4}] [A d d i n g a n d s u b t r a c t i n g {(\frac{3}{2})}^{2}] = 4 [(x - \frac{3}{2})^{2} - \frac{9}{4} + \frac{5}{4}] [C o m p l e t i n g t h e s q u a r e] = 4 [(x - \frac{3}{2})^{2} - 1^{2}] = 4 [(x - \frac{3}{2}) - 1] [(x - \frac{3}{2}) + 1] = 4 (x - \frac{3}{2} - 1) (x - \frac{3}{2} + 1) = 4 (x - \frac{5}{2}) (x - \frac{1}{2}) = (2 x - 5) (2 x - 1)$

Page No 7.33:

Question 9:

Factorize each of the following quadratic polynomials by using the method of completing the square:
y² − 7y + 12

Answer:

$y^{2} - 7 y + 12 = y^{2} - 7 y + {(\frac{7}{2})}^{2} - {(\frac{7}{2})}^{2} + 12 [A d d i n g a n d s u b t r a c t i n g {(\frac{7}{2})}^{2}] = (y - \frac{7}{2})^{2} - \frac{49}{4} + \frac{48}{4} [C o m p l e t i n g t h e s q u a r e] = (y - \frac{7}{2})^{2} - \frac{1}{4} = (y - \frac{7}{2})^{2} - {(\frac{1}{2})}^{2} = [(y - \frac{7}{2}) - \frac{1}{2}] [(y - \frac{7}{2}) + \frac{1}{2}] = (y - \frac{7}{2} - \frac{1}{2}) (y - \frac{7}{2} + \frac{1}{2}) = (y - 4) (y - 3)$

Page No 7.33:

Question 10:

Factorize each of the following quadratic polynomials by using the method of completing the square:
z² − 4z − 12

Answer:

$z^{2} - 4 z - 12 = z^{2} - 4 z + {(\frac{4}{2})}^{2} - {(\frac{4}{2})}^{2} - 12 [A d d i n g a n d s u b t r a c t i n g {(\frac{4}{2})}^{2}, t h a t i s, 2^{2}] = z^{2} - 4 z + 2^{2} - 2^{2} - 12 = (z - 2)^{2} - 16 [C o m p l e t i n g t h e s q u a r e] = (z - 2)^{2} - 4^{2} = [(z - 2) - 4] [(z - 2) + 4] = (z - 6) (z + 2)$

Page No 7.4:

Question 9:

Find the greatest common factor (GCF/HCF) of the following polynomial:
a²b³, a³b²

Answer:

The common literals appearing in the three monomials are a and b.
The smallest power of x in the two monomials is 2.
The smallest power of y in the two monomials is 2.
The monomial of common literals with the smallest powers is a²b².
Hence, the greatest common factor is a²b².

Page No 7.4:

Question 10:

Find the greatest common factor (GCF/HCF) of the following polynomial:
36a²b²c⁴, 54a⁵c², 90a⁴b²c²

Answer:

The numerical coefficients of the given monomials are 36, 54 and 90. The greatest common factor of 36, 54 and 90 is 18.
The common literals appearing in the three monomials are a and c.
The smallest power of a in the three monomials is 2.
The smallest power of c in the three monomials is 2.
The monomial of common literals with the smallest powers is a²c².
Hence, the greatest common factor is 18a²c².

Page No 7.4:

Question 11:

Find the greatest common factor (GCF/HCF) of the following polynomial:
x³, − yx²

Answer:

The common literal appearing in the two monomials is x.
The smallest power of x in both the monomials is 2.
Hence, the greatest common factor is x².

Page No 7.4:

Question 12:

Find the greatest common factor (GCF/HCF) of the following polynomial:
15a³, − 45a², − 150a

Answer:

The numerical coefficients of the given monomials are 15, -45 and -150. The greatest common factor of 15, -45 and -150 is 15.
The common literal appearing in the three monomials is a.
The smallest power of a in the three monomials is 1.
Hence, the greatest common factor is 15a.

Page No 7.4:

Question 13:

Find the greatest common factor (GCF/HCF) of the following polynomial:
2x³y², 10x²y³, 14xy

Answer:

The numerical coefficients of the given monomials are 2, 10 and 14. The greatest common factor of 2, 10 and 14 is 2.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 1.
The smallest power of y in the three monomials is 1.
The monomial of common literals with the smallest powers is xy.
Hence, the greatest common factor is 2xy.

Page No 7.4:

Question 14:

Find the greatest common factor (GCF/HCF) of the following polynomial:
14x³y⁵, 10x⁵y³, 2x²y²

Answer:

The numerical coefficients of the given monomials are 14, 10 and 2. The greatest common factor of 14, 10 and 2 is 2.
The common literals appearing in the three monomials are x and y.
The smallest power of x in the three monomials is 2.
The smallest power of y in the three monomials is 2.
The monomial of common literals with the smallest powers is x²y².
Hence, the greatest common factor is 2x²y².

Page No 7.4:

Question 15:

Find the greatest common factor of the terms in each of the following expression:
5a⁴+ 10a³ − 15a²

Answer:

Terms are expressions separated by plus or minus signs. Here, the terms are 5a⁴, 10a³ and 15a².
The numerical coefficients of the given monomials are 5, 10 and 15. The greatest common factor of 5, 10 and 15 is 5.
The common literal appearing in the three monomials is a.
The smallest power of a in the three monomials is 2.
The monomial of common literals with the smallest powers is a².
Hence, the greatest common factor is 5a².

Page No 7.4:

Question 16:

Find the greatest common factor of the terms in each of the following expression:
2xyz + 3x²y + 4y²

Answer:

The expression has three monomials: 2xyz, 3x²y and 4y².
The numerical coefficients of the given monomials are 2, 3 and 4. The greatest common factor of 2, 3 and 4 is 1.
The common literal appearing in the three monomials is y.
The smallest power of y in the three monomials is 1.
The monomial of common literals with the smallest powers is y.
Hence, the greatest common factor is y.

Page No 7.4:

Question 17:

Find the greatest common factor of the terms in each of the following expression:
3a²b² + 4b²c² + 12a²b²c²

Answer:

The expression has three monomials: 3a²b², 4b²c² and 12a²b²c².
The numerical coefficients of the given monomials are 3, 4 and 12. The greatest common factor of 3, 4 and 12 is 1.
The common literal appearing in the three monomials is b.
The smallest power of b in the three monomials is 2.
The monomial of common literals with the smallest powers is b².
Hence, the greatest common factor is b².

Page No 7.5:

Question 1:

Factorize the following:
3x − 9

Answer:

The greatest common factor of the terms 3x and -9 of the expression 3x - 9 is 3.
Now.
3x = 3x
and
-9 = 3.-3
Hence, the expression 3x - 9 can be factorised as 3(x - 3).

Page No 7.5:

Question 2:

Factorize the following:
5x − 15x²

Answer:

The greatest common factor of the terms 5x and 15x² of the expression 5x - 15x² is 5x.
Now,
5x = 5x $\times$ 1
and
-15x² = 5x $\times$ -3x
Hence, the expression 5x - 15x²can be factorised as 5x(1 - 3x).

Page No 7.5:

Question 3:

Factorize the following:
20a¹²b² − 15a⁸b⁴

Answer:

The greatest common factor of the terms 20a¹²b² and -15a⁸b⁴ of the expression 20a¹²b² - 15a⁸b⁴ is 5a⁸b².
20a¹²b² = 5×4×a⁸×a⁴×b² = 5a⁸×b² $\times$ 4a⁴ and -15a⁸b⁴= 5×-3×a⁸×b²×b² = 5a⁸b² $\times$ -3b²

Hence, the expression 20a¹²b² - 15a⁸b⁴can be factorised as 5a⁸b²(4a⁴-3b²)

Page No 7.5:

Question 4:

Factorize the following:
72x⁶y⁷ − 96x⁷y⁶

Answer:

The greatest common factor of the terms 72x⁶y⁷ and -96x⁷y⁶ of the expression 72x⁶y⁷ - 96x⁷y⁶⁴ is 24x⁶y⁶.

Now,
72x⁶y⁷ = 24x⁶y⁶ $\times$ 3y
and
-96x⁷y⁶ = 24x⁶y⁶ $\times$ -4x

Hence, the expression 72x⁶y⁷ - 96x⁷y⁶ can be factorised as 24x⁶y⁶(3y - 4x).

Page No 7.5:

Question 5:

Factorize the following:
20x³ − 40x² + 80x

Answer:

The greatest common factor of the terms 20x³, -40x² and 80x of the expression 20x³ - 40x² + 80x is 20x.
Now,
20x³ = 20x $\times$ x²
-40x² = 20x $\times$ -2x
and
80x = 20x $\times$ 4
Hence, the expression 20x³ - 40x² + 80x can be factorised as 20x(x² - 2x + 4).

Page No 7.5:

Question 6:

Factorize the following:
2x³y² − 4x²y³ + 8xy⁴

Answer:

The greatest common factor of the terms 2x³y², -4x²y³ and 8xy⁴of the expression 2x³y² - 4x²y³+ 8xy⁴y⁶⁴ is 2xy².

Now,
2x³y² = 2xy² $\times$ x²
-4x²y³= 2xy² $\times$ -2xy
8xy⁴ = 2xy² $\times$ 4y²

Hence, the expression 2x³y² - 4x²y³ + 8xy⁴ can be factorised as 2xy²(x² - 2xy + 4y²).

Page No 7.5:

Question 7:

Factorize the following:
10m³n² + 15m⁴n − 20m²n³

Answer:

The greatest common factor of the terms 10m³n², 15m⁴n and -20m²n³of the expression 10m³n² + 15m⁴n - 20m²n³ is 5m²n.

Now,
10m³n²= 5m²n $\times$ 2mn
15m⁴n = 5m²n $\times$ 3m²
-20m²n³= 5m²n $\times$ -4n²

Hence, 10m³n² + 15m²n - 20m²n³ can be factorised as 5m²n(2mn + 3m² - 4n²).

Page No 7.5:

Question 8:

Factorize the following:
2a⁴b⁴ − 3a³b⁵ + 4a²b⁵

Answer:

The greatest common factor of the terms 2a⁴b⁴, -3a³b⁵ and 4a²b⁵ of the expression 2a⁴b⁴ - 3a³b⁵ + 4a²b⁵ is a²b⁴.

Now,
2a⁴b⁴= a²b⁴ $\times$ 2a²
-3a³b⁵= a²b⁴ $\times$ -3ab
4a²b⁵= a²b⁴ $\times$ 4b

Hence, (2a⁴b⁴ - 3a³b⁵ + 4a²b⁵) can be factorised as [a²b⁴(2a² - 3ab + 4b)].

Page No 7.5:

Question 9:

Factorize the following:
28a² + 14a²b² − 21a⁴

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 28 a^{2}, 14 a^{2} b^{2} a n d 21 a^{4} o f t h e e x p r e s s i o n 28 a^{2} + 14 a^{2} b^{2} - 21 a^{4} i s 7 a^{2} . A l s o, w e c a n w r i t e 28 a^{2} = 7 a^{2} \times 4, 14 a^{2} b^{2} = 7 a^{2} \times 2 b^{2} a n d 21 a^{4} = 7 a^{2} \times 3 a^{2} . ∴ 28 a^{2} + 14 a^{2} b^{2} - 21 a^{4} = 7 a^{2} \times 4 + 7 a^{2} \times 2 b^{2} - 7 a^{2} \times 3 a^{2} = 7 a^{2} (4 + 2 b^{2} - 3 a^{2})$

Page No 7.5:

Question 10:

Factorize the following:
a⁴b − 3a²b² − 6ab³

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s a^{4} b, 3 a^{2} b^{2} a n d 6 a b^{3} o f t h e e x p r e s s i o n a^{4} b - 3 a^{2} b^{2} - 6 a b^{3} i s a b . A l s o, w e c a n w r i t e a^{4} b = a b \times a^{3}, 3 a^{2} b^{2} = a b \times 3 a b a n d 6 a b^{3} = a b \times 6 b^{2} . ∴ a^{4} b - 3 a^{2} b^{2} - 6 a b^{3} = a b \times a^{3} - a b \times 3 a b - a b \times 6 b^{2} = a b (a^{3} - 3 a b - 6 b^{2})$

Page No 7.5:

Question 11:

Factorize the following:
2l²mn - 3lm²n + 4lmn²

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 2 l^{2} m n, 3 l m^{2} n a n d 4 l m n^{2} o f t h e e x p r e s s i o n 2 l^{2} m n - 3 l m^{2} n + 4 l m n^{2} i s l m n . A l s o, w e c a n w r i t e 2 l^{2} m n = l m n \times 2 l, 3 l m^{2} n = l m n \times 3 m a n d 4 l m n^{2} = l m n \times 4 n . ∴ 2 l^{2} n m - 3 l m^{2} n + 4 l m n^{2} = l m n \times 2 l - l m n \times 3 m + l m n \times 4 n = l m n (2 l - 3 m + 4 n)$

Page No 7.5:

Question 12:

Factorize the following:
x⁴y² − x²y⁴ − x⁴y⁴

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s x^{4} y^{2}, x^{2} y^{4} a n d x^{4} y^{4} o f t h e e x p r e s s i o n x^{4} y^{2} - x^{2} y^{4} - x^{4} y^{4} i s x^{2} y^{2} . A l s o, w e c a n w r i t e x^{4} y^{2} = x^{2} y^{2} \times x^{2}, x^{2} y^{4} = x^{2} y^{2} \times y^{2} a n d x^{4} y^{4} = x^{2} y^{2} \times x^{2} y^{2} . ∴ x^{4} y^{2} - x^{2} y^{4} - x^{4} y^{4} = x^{2} y^{2} \times x^{2} - x^{2} y^{2} \times y^{2} - x^{2} y^{2} \times x^{2} y^{2} = x^{2} y^{2} (x^{2} - y^{2} - x^{2} y^{2})$

Page No 7.5:

Question 13:

Factorize the following:
9x²y + 3axy

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 9 x^{2} y a n d 3 a x y o f t h e e x p r e s s i o n 9 x^{2} y + 3 a x y i s 3 x y . A l s o, w e c a n w r i t e 9 x^{2} y = 3 x y \times 3 x a n d 3 a x y = 3 x y \times a . ∴ 9 x^{2} y + 3 a x y = 3 x y \times 3 x + 3 x y \times a = 3 x y (3 x + a)$

Page No 7.5:

Question 14:

Factorize the following:
16m − 4m²

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s 16 m a n d 4 m^{2} o f t h e e x p r e s s i o n 16 m - 4 m^{2} i s 4 m . A l s o, w e c a n w r i t e 16 m = 4 m \times 4 a n d 4 m^{2} = 4 m \times m . ∴ 16 m - 4 m^{2} = 4 m \times 4 - 4 m \times m = 4 m (4 - m)$

Page No 7.5:

Question 15:

Factorize the following:
−4a² + 4ab − 4ca

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s - 4 a^{2}, 4 a b a n d - 4 c a o f t h e e x p r e s s i o n - 4 a^{2} + 4 a b - 4 c a i s - 4 a . A l s o, w e c a n w r i t e - 4 a^{2} = - 4 a \times a, 4 a b = - 4 a \times (- b) a n d 4 c a = - 4 a \times c . ∴ - 4 a^{2} + 4 a b - 4 c a = - 4 a \times a + (- 4 a) \times (- b) - 4 a \times c = - 4 a (a - b + c)$

Page No 7.5:

Question 16:

Factorize the following:
x²yz + xy²z + xyz²

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s x^{2} y z, x y^{2} z a n d x y z^{2} o f t h e e x p r e s s i o n x^{2} y z + x y^{2} z + x y z^{2} i s x y z . A l s o, w e c a n w r i t e x^{2} y z = x y z \times x, x y^{2} z = x y z \times y a n d x y z^{2} = x y z \times z . ∴ x^{2} y z + x y^{2} z + x y z^{2} = x y z \times x + x y z \times y + x y z \times z = x y z (x + y + z)$

Page No 7.5:

Question 17:

Factorize the following:
ax²y + bxy² + cxyz

Answer:

$T h e g r e a t e s t c o m m o n f a c t o r o f t h e t e r m s a x^{2} y, b x y^{2} a n d c x y z o f t h e e x p r e s s i o n a x^{2} y + b x y^{2} + c x y z i s x y . A l s o, w e c a n w r i t e a x^{2} y = x y \times a x, b x y^{2} = x y \times b y a n d c x y z = x y \times c z . ∴ a x^{2} y + b x y^{2} + c x y z = x y \times a x + x y \times b y + x y \times c z = x y (a x + b y + c z)$

Page No 7.7:

Question 1:

Factorize each of the following algebraic expressions:
6x(2x − y) + 7y(2x − y)

Answer:

$6 x (2 x - y) + 7 y (2 x - y) = (6 x + 7 y) (2 x - y) [T a k i n g (2 x - y) a s t h e c o m m o n f a c t o r]$

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Question 2:

Factorize each of the following algebraic expressions:
2r(y − x) + s(x − y)

Answer:

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

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Question 3:

Factorize each of the following algebraic expressions:
7a(2x − 3) + 3b(2x − 3)

Answer:

$7 a (2 x - 3) + 3 b (2 x - 3) = (7 a + 3 b) (2 x - 3) [T a k i n g (2 x - 3) a s t h e c o m m o n f a c t o r]$

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Question 4:

Factorize each of the following algebraic expressions:
9a(6a − 5b) −12a²(6a − 5b)

Answer:

$9 a (6 a - 5 b) - 12 a^{2} (6 a - 5 b) = (9 a - 12 a^{2}) (6 a - 5 b) [T a k i n g (6 a - 5 b) a s t h e c o m m o n f a c t o r] = 3 a (3 - 4 a) (6 a - 5 b) [T a k i n g 3 a a s t h e c o m m o n f a c t o r o f t h e q u a d r a t i c (9 a - 12 a^{2})]$

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Question 5:

Factorize each of the following algebraic expressions:
5(x − 2y)² + 3(x − 2y)

Answer:

$5 (x - 2 y)^{2} + 3 (x - 2 y) = [5 (x - 2 y) + 3] (x - 2 y) [T a k i n g (x - 2 y) a s t h e c o m m o n f a c t o r] = (5 x - 10 y + 3) (x - 2 y)$

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Question 6:

Factorize each of the following algebraic expressions:
16(2l − 3m)² −12(3m − 2l)

Answer:

$16 (2 l - 3 m)^{2} - 12 (3 m - 2 l) = 16 (2 l - 3 m)^{2} + 12 (2 l - 3 m) [∵ (3 m - 2 l) = - (2 l - 3 m)] = [16 (2 l - 3 m) + 12] (2 l - 3 m) [T a k i n g (2 l - 3 m) a s t h e c o m m o n f a c t o r] = 4 [4 (2 l - 3 m) + 3] (2 l - 3 m) {T a k i n g 4 a s t h e c o m m o n f a c t o r o f [16 (2 l - 3 m) + 12]} = 4 (8 l - 12 m + 3) (2 l - 3 m)$

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Question 7:

Factorize each of the following algebraic expressions:
3a(x − 2y) −b(x − 2y)

Answer:

$3 a (x - 2 y) - b (x - 2 y) = (3 a - b) (x - 2 y) [Taking (x - 2 y) as the common factor]$

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Question 8:

Factorize each of the following algebraic expressions:
a²(x + y) +b²(x + y) +c²(x + y)

Answer:

$a^{2} (x + y) + b^{2} (x + y) + c^{2} (x + y) = (a^{2} + b^{2} + c^{2}) (x + y) [Taking (x + y) as the common factor]$

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Question 9:

Factorize each of the following algebraic expressions:
(x − y)² + (x − y)

Answer:

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

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Question 10:

Factorize each of the following algebraic expressions:
6(a + 2b) −4(a + 2b)²

Answer:

$6 (a + 2 b) - 4 (a + 2 b)^{2} = [6 - 4 (a + 2 b)] (a + 2 b) [T a k i n g (a + 2 b) a s t h e c o m m o n f a c t o r] = 2 [3 - 2 (a + 2 b)] (a + 2 b) {T a k i n g 2 a s t h e c o m m o n f a c t o r o f [6 - 4 (a + 2 b)]} = 2 (3 - 2 a - 4 b) (a + 2 b)$

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Question 11:

Factorize each of the following algebraic expressions:
a(x − y) + 2b(y − x) + c(x − y)²

Answer:

$a (x - y) + 2 b (y - x) + c (x - y)^{2} = a (x - y) - 2 b (x - y) + c (x - y)^{2} [∵ (y - x) = - (x - y)] = [a - 2 b + c (x - y)] (x - y) = (a - 2 b + c x - c y) (x - y)$

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Question 12:

Factorize each of the following algebraic expressions:
−4(x − 2y)² + 8(x −2y)

Answer:

$- 4 (x - 2 y)^{2} + 8 (x - 2 y) = [- 4 (x - 2 y) + 8] (x - 2 y) [T a k i n g (x - 2 y) a s t h e c o m m o n f a c t o r] = 4 [- (x - 2 y) + 2] (x - 2 y) {T a k i n g 4 a s t h e c o m m o n f a c t o r o f [- 4 (x - 2 y) + 8]} = 4 (2 y - x + 2) (x - 2 y)$

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Question 13:

Factorize each of the following algebraic expressions:
x³(a − 2b) + x²(a − 2b)

Answer:

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

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Question 14:

Factorize each of the following algebraic expressions:
(2x − 3y)(a + b) + (3x − 2y)(a + b)

Answer:

$(2 x - 3 y) (a + b) + (3 x - 2 y) (a + b) = (2 x - 3 y + 3 x - 2 y) (a + b) [T a k i n g (a + b) a s t h e c o m m o n f a c t o r] = (5 x - 5 y) (a + b) = 5 (x - y) (a + b) [T a k i n g 5 a s t h e c o m m o n f a c t o r o f (5 x - 5 y)]$

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Question 15:

Factorize each of the following algebraic expressions:
4(x + y) (3a − b) +6(x + y) (2b − 3a)

Answer:

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

View NCERT Solutions for all chapters of Class 8