myhelper: RS AGGARWAL CLASS 9 Chapter 3 FACTORISATION OF POLYNOMIAL EXERCISE 3B

EXERCISE 3B

PAGE NO-105

Question 1:

Factorise:
9x² – 16y²

Answer 1:

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

Question 2:

Factorise:
$(\frac{25}{4} x^{2} - \frac{1}{9} y^{2})$ $(\frac{25}{4} x^{2} - \frac{1}{9} y^{2})$

Answer 2:

$(\frac{25}{4} x^{2} - \frac{1}{9} y^{2}) = {(\frac{5}{2} x)}^{2} - {(\frac{1}{3} y)}^{2} = (\frac{5}{2} x + \frac{1}{3} y) (\frac{5}{2} x - \frac{1}{3} y) [a^{2} - b^{2} = (a + b) (a - b)]$ $(\frac{25}{4} x^{2} - \frac{1}{9} y^{2}) = {(\frac{5}{2} x)}^{2} - {(\frac{1}{3} y)}^{2} = (\frac{5}{2} x + \frac{1}{3} y) (\frac{5}{2} x - \frac{1}{3} y) [a^{2} - b^{2} = (a + b) (a - b)]$

Question 3:

Factorise:
81 – 16x²

Answer 3:

$81 - 16 x^{2} = 9^{2} - {(4 x)}^{2} = (9 + 4 x) (9 - 4 x) [a^{2} - b^{2} = (a + b) (a - b)]$ $81 - 16 x^{2} = 9^{2} - {(4 x)}^{2} = (9 + 4 x) (9 - 4 x) [a^{2} - b^{2} = (a + b) (a - b)]$

Question 4:

Factorise:
5 – 20x²

Answer 4:

$5 - 20 x^{2} = 5 (1 - 4 x^{2}) = 5 [1^{2} - {(2 x)}^{2}] = 5 (1 + 2 x) (1 - 2 x) [a^{2} - b^{2} = (a + b) (a - b)]$ $5 - 20 x^{2} = 5 (1 - 4 x^{2}) = 5 [1^{2} - {(2 x)}^{2}] = 5 (1 + 2 x) (1 - 2 x) [a^{2} - b^{2} = (a + b) (a - b)]$

Question 5:

Factorise:
2x⁴ – 32

Answer 5:

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

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

Question 6:

Factorize:
3a³b − 243ab³

Answer 6:

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

Question 7:

Factorize:
3x³ − 48x

Answer 7:

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

Question 8:

Factorize:
27a² − 48b²

Answer 8:

$27 a^{2} - 48 b^{2} = 3 (9 a^{2} - 16 b^{2}) = 3 [{(3 a)}^{2} - {(4 b)}^{2}] = 3 (3 a - 4 b) (3 a + 4 b)$ $27 a^{2} - 48 b^{2} = 3 (9 a^{2} - 16 b^{2}) = 3 [{(3 a)}^{2} - {(4 b)}^{2}] = 3 (3 a - 4 b) (3 a + 4 b)$

Question 9:

Factorize:
x − 64x³

Answer 9:

$x - 64 x^{3} = x (1 - 64 x^{2}) = x [1 - {(8 x)}^{2}] = x (1 - 8 x) (1 + 8 x)$ $x - 64 x^{3} = x (1 - 64 x^{2}) = x [1 - {(8 x)}^{2}] = x (1 - 8 x) (1 + 8 x)$

Question 10:

Factorize:
8ab² − 18a³

Answer 10:

$8 a b^{2} - 18 a^{3} = 2 a (4 b^{2} - 9 a^{2}) = 2 a [{(2 b)}^{2} - {(3 a)}^{2}] = 2 a (2 b - 3 a) (2 b + 3 a)$ $8 a b^{2} - 18 a^{3} = 2 a (4 b^{2} - 9 a^{2}) = 2 a [{(2 b)}^{2} - {(3 a)}^{2}] = 2 a (2 b - 3 a) (2 b + 3 a)$

Question 11:

Factorize:
150 − 6x²

Answer 11:

$150 - 6 x^{2} = 6 (25 - x^{2}) = 6 (5^{2} - x^{2}) = 6 (5 - x) (5 + x)$ $150 - 6 x^{2} = 6 (25 - x^{2}) = 6 (5^{2} - x^{2}) = 6 (5 - x) (5 + x)$

Question 12:

Factorise:
2 – 50x²

Answer 12:

$2 - 50 x^{2} = 2 (1 - 25 x^{2}) = 2 [1^{2} - {(5 x)}^{2}] = 2 (1 + 5 x) (1 - 5 x) [a^{2} - b^{2} = (a + b) (a - b)]$

Question 13:

Factorise:
20x²– 45

Answer 13:

$20 x^{2} - 45 = 5 (4 x^{2} - 9) = 5 [{(2 x)}^{2} - 3^{2}] = 5 (2 x + 3) (2 x - 3) [a^{2} - b^{2} = (a + b) (a - b)]$

Question 14:

Factorise:
(3a + 5b)² – 4c²

Answer 14:

${(3 a + 5 b)}^{2} - 4 c^{2} = {(3 a + 5 b)}^{2} - {(2 c)}^{2} = (3 a + 5 b + 2 c) (3 a + 5 b - 2 c) [a^{2} - b^{2} = (a + b) (a - b)]$

Question 15:

Factorise:
a²– b² – a – b

Answer 15:

$a^{2} - b^{2} - a - b = (a + b) (a - b) - 1 (a + b) [a^{2} - b^{2} = (a + b) (a - b)] = (a + b) [(a - b) - 1] = (a + b) (a - b - 1)$

Question 16:

Factorise:
4a²– 9b²– 2a – 3b

Answer 16:

$4 a^{2} - 9 b^{2} - 2 a - 3 b = {(2 a)}^{2} - {(3 b)}^{2} - 1 (2 a + 3 b) = (2 a + 3 b) (2 a - 3 b) - 1 (2 a + 3 b) [a^{2} - b^{2} = (a + b) (a - b)] = (2 a + 3 b) [(2 a - 3 b) - 1] = (2 a + 3 b) (2 a - 3 b - 1)$

Question 17:

Factorise:
a²– b²+ 2bc – c²

Answer 17:

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

Question 18:

Factorise:
4a²– 4b²+ 4a + 1

Answer 18:

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

$= [(2 a + 1) + 2 b] [(2 a + 1) - 2 b] [a^{2} - b^{2} = (a + b) (a - b)] = (2 a + 1 + 2 b) (2 a + 1 - 2 b) = (2 a + 2 b + 1) (2 a - 2 b + 1)$

Question 19:

Factorize:
a² + 2ab + b² − 9c²

Answer 19:

$a^{2} + 2 a b + b^{2} - 9 c^{2} = {(a + b)}^{2} - {(3 c)}^{2} = (a + b - 3 c) (a + b + 3 c)$

Question 20:

Factorize:
108a² − 3(b − c)²

Answer 20:

$108 a^{2} - 3 {(b - c)}^{2} = 3 [36 a^{2} - {(b - c)}^{2}] = 3 [{(6 a)}^{2} - {(b - c)}^{2}] = 3 (6 a - b + c) (6 a + b - c)$

Question 21:

Factorize:
(a + b)³ − a − b

Answer 21:

${(a + b)}^{3} - a - b = {(a + b)}^{3} - (a + b) = (a + b) [{(a + b)}^{2} - 1] = (a + b) [{(a + b)}^{2} - 1^{2}] = (a + b) (a + b - 1) (a + b + 1)$

Question 22:

Factorise:
x²+ y² – z²– 2xy

Answer 22:

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

Question 23:

Factorise:
x²+ 2xy + y²– a² + 2ab – b²

Answer 23:

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

Question 24:

Factorise:
25x²– 10x + 1 – 36y²

Answer 24:

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

Question 25:

Factorize:
a − b − a² + b²

Answer 25:

$a - b - a^{2} + b^{2} = (a - b) - (a^{2} - b^{2}) = (a - b) - (a - b) (a + b) = (a - b) [1 - (a + b)] = (a - b) (1 - a - b)$

Question 26:

Factorize:
a² − b² − 4ac + 4c²

Answer 26:

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

Question 27:

Factorize:
9 − a² + 2ab − b²

Answer 27:

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

Question 28:

Factorize:
x³ − 5x² − x + 5

Answer 28:

$x^{3} - 5 x^{2} - x + 5 = x^{2} (x - 5) - 1 (x - 5) = (x - 5) (x^{2} - 1) = (x - 5) (x^{2} - 1^{2}) = (x - 5) (x - 1) (x + 1)$

Question 29:

Factorise:
1 + 2ab – (a²+ b²)

Answer 29:

$1 + 2 a b - (a^{2} + b^{2}) = 1 + 2 a b - a^{2} - b^{2} = 1 - a^{2} + 2 a b - b^{2} = 1^{2} - (a^{2} - 2 a b + b^{2})$

$= 1^{2} - {(a - b)}^{2} [a^{2} - 2 a b + b^{2} = {(a - b)}^{2}] = [1 + (a - b)] [1 - (a - b)] [a^{2} - b^{2} = (a + b) (a - b)] = (1 + a - b) (1 - a + b)$

Question 30:

Factorise:
9a² + 6a + 1 – 36b²

Answer 30:

$9 a^{2} + 6 a + 1 - 36 b^{2} = [{(3 a)}^{2} + 2 \times 3 a \times 1 + 1^{2}] - {(6 b)}^{2} = {(3 a + 1)}^{2} - {(6 b)}^{2} [a^{2} + 2 a b + b^{2} = {(a + b)}^{2}] = (3 a + 1 - 6 b) (3 a + 1 + 6 b) [a^{2} - b^{2} = (a - b) (a + b)] = (3 a - 6 b + 1) (3 a + 6 b + 1)$

Question 31:

Factorize:
x² − y² + 6y − 9

Answer 31:

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

Question 32:

Factorize:
4x² − 9y² − 2x − 3y

Answer 32:

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

Question 33:

Factorize:
9a² + 3a − 8b − 64b²

Answer 33:

$9 a^{2} + 3 a - 8 b - 64 b^{2} = 9 a^{2} - 64 b^{2} + 3 a - 8 b = {(3 a)}^{2} - {(8 b)}^{2} + (3 a - 8 b) = (3 a - 8 b) (3 a + 8 b) + 1 (3 a - 8 b) = (3 a - 8 b) (3 a + 8 b + 1)$

Question 34:

Factorise:
$x^{2} + \frac{1}{x^{2}} - 3$

Answer 34:

$x^{2} + \frac{1}{x^{2}} - 3 = x^{2} + \frac{1}{x^{2}} - 2 - 1 = [x^{2} + {(\frac{1}{x})}^{2} - 2 \times x \times \frac{1}{x}] - 1 = {(x - \frac{1}{x})}^{2} - 1^{2} [a^{2} - 2 a b + b^{2} = {(a - b)}^{2}] = (x - \frac{1}{x} + 1) (x - \frac{1}{x} - 1) [a^{2} - b^{2} = (a - b) (a + b)]$

Question 35:

Factorise:
$x^{2} - 2 + \frac{1}{x^{2}} y^{2}$

Answer 35:

$x^{2} - 2 + \frac{1}{x^{2}} - y^{2} = [x^{2} - 2 \times x \times \frac{1}{x} + {(\frac{1}{x})}^{2}] - y^{2} = {(x - \frac{1}{x})}^{2} - y^{2} [a^{2} - 2 a b + b^{2} = {(a - b)}^{2}] = (x - \frac{1}{x} + y) (x - \frac{1}{x} - y) [a^{2} - b^{2} = (a - b) (a + b)]$

Disclaimer: The expression of the question should be $x^{2} - 2 + \frac{1}{x^{2}} - y^{2}$ . The same has been done before solving the question.

Question 36:

Factorise:
$x^{4} + \frac{4}{x^{4}}$

Answer 36:

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

Question 37:

Factorise:
x⁸ – 1

Answer 37:

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

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

Question 38:

Factorise:
16x⁴ – 1

Answer 38:

$16 x^{4} - 1 = {(4 x^{2})}^{2} - 1^{2} = (4 x^{2} + 1) (4 x^{2} - 1) [a^{2} - b^{2} = (a + b) (a - b)] = (4 x^{2} + 1) [{(2 x)}^{2} - 1^{2}] = (4 x^{2} + 1) (2 x + 1) (2 x - 1) [a^{2} - b^{2} = (a + b) (a - b)]$

Question 39:

81x⁴ – y⁴

Answer 39:

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

Question 40:

x⁴ – 625

Answer 40:

$x^{4} - 625 = {(x^{2})}^{2} - 25^{2} = (x^{2} + 25) (x^{2} - 25) [a^{2} - b^{2} = (a + b) (a - b)] = (x^{2} + 25) (x^{2} - 5^{2}) = (x^{2} + 25) (x + 5) (x - 5) [a^{2} - b^{2} = (a + b) (a - b)]$