Exercise 3A RS Aggarwal Class 9 Factorisation of Polynomials Best Solutions

Exercise 3A RS Aggarwal Class 9 Mathematics Solutions

Q.1 Q.2 Q.3 Q.4 Q.5 Q.6 Q.7 Q.8 Q.9 Q.10 Q.11 Q.12 Q.13 Q.14 Q.15 Q.16 Q.17 Q.18 Q.19 Q.20 Q.21 Q.22 Q.23 Q.24 Q.25 Q.26 Q.27 Q.28 Q.29 Q.30 Q.31 Q.32 Q.33 Q.34

The question number 1 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

Factorise:

1. $9 x^{2} + 12 x y$

Answer

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

Method: Just think of each term and break it down into its prime factors. Then, find all the common factors among them.

The question number 2 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

2. $18 x^{2} y - 24 x y z$

Answer

We have
$18 x^{2} y - 24 x y z$
$= 2 \times 3 \times 3 \times x^{2} y - 2 \times 2 \times 2 \times 3 \times x y z$
$= 2 \times 3 \times x y (3 x - 4 z)$
$= 6 x y (3 x - 4 z)$

Method: Just think of each term and break it down into its prime factors. Then, find all the common factors among them.

The question number 3 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

3. $27 a^{3} b^{3} - 45 a^{4} b^{2}$

Answer

We have
$27 a^{3} b^{3} - 45 a^{4} b^{2}$
$= 3 \times 3 \times 3 \times a^{3} b^{3} - 3 \times 3 \times 5 \times a^{4} b^{2}$
$= 3 \times 3 \times a^{3} b^{2} (3 b - 5 a)$
$= 9 a^{3} b^{2} (3 b - 5 a)$

The question number 4 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

4. $2 a (x + y) - 3 b (x + y)$

Answer

We have
$2 a (x + y) - 3 b (x + y)$
$= (x + y) (2 a - 3 b)$

The question number 5 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

5. $2 x (p^{2} + q^{2}) + 4 y (p^{2} + q^{2})$

Answer

We have
$2 x (p^{2} + q^{2}) + 4 y (p^{2} + q^{2})$
$= (p^{2} + q^{2}) (2 x + 4 y)$
$= 2 (p^{2} + q^{2}) (x + 2 y)$

The question number 6 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

6. $x (a - 5) + y (5 - a)$

Answer

We have
$x (a - 5) + y (5 - a)$
$= x (a - 5) - y (a - 5)$
$= (a - 5) (x - y)$

The question number 7 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

7. $4 (a + b) - 6 {(a + b)}^{2}$

Answer

We have
$4 (a + b) - 6 {(a + b)}^{2}$
$= 2 (a + b) [2 - 3 (a + b)]$
$= 2 (a + b) (2 - 3 a - 3 b)$

The question number 8 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

8. $8 {(3 a - 2 b)}^{2} - 10 (3 a - 2 b)$

Answer

We have
$8 {(3 a - 2 b)}^{2} - 10 (3 a - 2 b)$
$= 2 (3 a - 2 b) [4 (3 a - 2 b) - 5]$
$= 2 (3 a - 2 b) (12 a - 8 b - 5)$

The question number 9 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

9. $x {(x + y)}^{3} - 3 x^{2} y (x + y)$

Answer

We have
$x {(x + y)}^{3} - 3 x^{2} y (x + y)$
$= x (x + y) [{(x + y)}^{2} - 3 x y]$
$= x (x + y) [x^{2} + y^{2} + 2 x y - 3 x y]$
$= x (x + y) (x^{2} + y^{2} - x y)$

The question number 10 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

10. $x^{3} + 2 x^{2} + 5 x + 10$

Answer

We have
$x^{3} + 2 x^{2} + 5 x + 10$
$= x^{2} (x + 2) + 5 (x + 2)$
$= (x + 2) (x^{2} + 5)$

The question number 11 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

11. $x^{2} + x y - 2 x z - 2 y z$

Answer

We have
$x^{2} + x y - 2 x z - 2 y z$
$= x (x + y) - 2 z (x + y)$
$= (x + y) (x - 2 z)$

The question number 12 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

12. $a^{3} b - a^{2} b + 5 a b - 5 b$

Answer

We have
$a^{3} b - a^{2} b + 5 a b - 5 b$
$= a^{2} b (a - 1) + 5 b (a - 1)$
$= (a - 1) (a^{2} b + 5 b)$
$= b (a - 1) (a^{2} + 5)$

The question number 13 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

13. $8 - 4 a - 2 a^{3} + a^{4}$

Answer

We have
$8 - 4 a - 2 a^{3} + a^{4}$
$= 4 (2 - a) - a^{3} (2 - a)$
$= (2 - a) (4 - a^{3})$

The question number 14 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

14. $x^{3} - 2 x^{2} y + 3 x y^{2} - 6 y^{3}$

Answer

We have
$x^{3} - 2 x^{2} y + 3 x y^{2} - 6 y^{3}$
$= x^{2} (x - 2 y) + 3 y^{2} (x - 2 y)$
$= (x - 2 y) (x^{2} + 3 y^{2})$

The question number 15 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

15. $p x - 5 q + p q - 5 x$

Answer

We have
$p x - 5 q + p q - 5 x$

Regrouping the terms having common coefficients, we get.

$p x + p q - 5 q - 5 x$
$= p (x + q) - 5 (q + x)$
$= p (x + q) - 5 (x + q)$
$= (p - 5) (x + q)$

The question number 16 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

16. $x^{2} + y - x y - x$

Answer

We have
$x^{2} + y - x y - x$

Rearranging the terms having common coefficients, we get.

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

The question number 17 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

17. ${(3 a - 1)}^{2} - 6 a + 2$

Answer

We have
${(3 a - 1)}^{2} - 6 a + 2$
$= {(3 a - 1)}^{2} - 2 (3 a - 1)$
$= (3 a - 1) [(3 a - 1) - 2]$
$= (3 a - 1) [3 a - 1 - 2]$
$= (3 a - 1) (3 a - 3)$
$= (3 a - 1) \times 3 (a - 1)$
$= 3 (3 a - 1) (a - 1)$

The question number 18 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

18. ${(2 x - 3)}^{2} - 8 x + 12$

Answer

We have
${(2 x - 3)}^{2} - 8 x + 12$
$= {(2 x - 3)}^{2} - 4 (2 x - 3)$
$= (2 x - 3) [(2 x - 3) - 4]$
$= (2 x - 3) [2 x - 3 - 4]$
$= (2 x - 3) (2 x - 7)$

The question number 19 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

19. $a^{3} + a - 3 a^{2} - 3$

Answer

We have
$a^{3} + a - 3 a^{2} - 3$
$= a (a^{2} + 1) - 3 (a^{2} + 1)$
$= (a - 3) (a^{2} + 1)$

The question number 20 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

20. $3 a x - 6 a y - 8 b y + 4 b x$

Answer

We have
$3 a x - 6 a y - 8 b y + 4 b x$
$= 3 a (x - 2 y) - 4 b (2 y - x)$
$= 3 a (x - 2 y) + 4 b (x - 2 y)$
$= (3 a + 4 b) (x - 2 y)$

The question number 21 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

21. $a b x^{2} + a^{2} x + b^{2} x + a b$

Answer

We have
$a b x^{2} + a^{2} x + b^{2} x + a b$
$= a x (b x + a) + b (b x + a)$
$= (a x + b) (b x + a)$

The question number 22 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

22. $x^{3} - x^{2} + a x + x - a - 1$

Answer

We have
$x^{3} - x^{2} + a x + x - a - 1$

Rearranging the terms having common coefficients, we get.

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

The question number 23 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

23. $2 x + 4 y - 8 x y - 1$

Answer

We have
$2 x + 4 y - 8 x y - 1$

Rearranging the terms having common coefficients, we get.

$2 x - 8 x y - 1 + 4 y$
$= 2 x (1 - 4 y) - (1 - 4 y)$
$= (2 x - 1) (1 - 4 y)$

The question number 24 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

24. $a b (x^{2} + y^{2}) - x y (a^{2} + b^{2})$

Answer

We have
$a b (x^{2} + y^{2}) - x y (a^{2} + b^{2})$
$= a b x^{2} + a b y^{2} - x y a^{2} - x y b^{2}$

Rearranging the terms having common coefficients, we get.

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

The question number 25 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

25. $a^{2} + a b (b + 1) + b^{3}$

Answer

We have
$a^{2} + a b (b + 1) + b^{3}$
$= a^{2} + a b^{2} + a b + b^{3}$
$= a (a + b^{2}) + b (a + b^{2})$
$= (a + b) (a + b^{2})$

The question number 26 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

26. $a^{3} + a b (1 - 2 a) - 2 b^{2}$

Answer

We have
$a^{3} + a b (1 - 2 a) - 2 b^{2}$
$= a^{3} + a b - 2 a^{2} b - 2 b^{2}$
$= a (a^{2} + b) - 2 b (a^{2} + b)$
$= (a - 2 b) (a^{2} + b)$

The question number 27 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

27. $2 a^{2} + b c - 2 a b - a c$

Answer

We have
$2 a^{2} + b c - 2 a b - a c$

Rearranging the terms having common coefficients, we get.

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

The question number 28 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

28. ${(a x + b y)}^{2} + {(b x - a y)}^{2}$

Answer

We have
${(a x + b y)}^{2} + {(b x - a y)}^{2}$
$= {(a x)}^{2} + {(b y)}^{2} + 2 \times a x \times b y + {(b x)}^{2} + {(a y)}^{2} - 2 \times b x \times a y$
$= a^{2} x^{2} + b^{2} y^{2} + 2 a b x y + b^{2} x^{2} + a^{2} y^{2} - 2 a b x y$
$= a^{2} x^{2} + b^{2} y^{2} + b^{2} x^{2} + a^{2} y^{2}$
$= a^{2} x^{2} + b^{2} x^{2} + a^{2} y^{2} + b^{2} y^{2}$
$= x^{2} (a^{2} + b^{2}) + y^{2} (a^{2} + b^{2})$
$= (x^{2} + y^{2}) (a^{2} + b^{2})$

The question number 29 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

29. $a (a + b - c) - b c$

Answer

We have
$a (a + b - c) - b c$
$= a^{2} + a b - a c - b c$
$= a (a + b) - c (a + b)$
$= (a - c) (a + b)$

The question number 30 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

30. $a (a - 2 b - c) + 2 b c$

Answer

We have
$a (a - 2 b - c) + 2 b c$
$= a^{2} - 2 a b - a c + 2 b c$
$= a (a - 2 b) - c (a - 2 b)$
$= (a - c) (a - 2 b)$

The question number 31 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

31. $a^{2} x^{2} + (a x^{2} + 1) x + a$

Answer

We have
$a^{2} x^{2} + (a x^{2} + 1) x + a$
$= a^{2} x^{2} + a x^{3} + x + a$
$= a x^{2} (a + x) + 1 (a + x)$
$= (a x^{2} + 1) (a + x)$

The question number 32 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

32. $a b (x^{2} + 1) + x (a^{2} + b^{2})$

Answer

We have
$a b (x^{2} + 1) + x (a^{2} + b^{2})$
$= a b x^{2} + a b + a^{2} x + b^{2} x$

Rearranging the terms having common coefficients, we get.

$= a b x^{2} + a^{2} x + b^{2} x + a b$
$= a x (b x + a) + b (b x + a)$
$= (a x + b) (b x + a)$

The question number 33 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

33. $x^{2} - (a + b) x + a b$

Answer

We have
$x^{2} - (a + b) x + a b$
$= x^{2} - a x - b x + a b$
$= x (x - a) - b (x - a)$
$= (x - a) (x - b)$

The question number 34 of Exercise 3A RS Aggarwal Class 9 asks us to factorise the given algebraic expression.

34. $x^{2} + \frac{1}{x^{2}} - 2 - 3 x + \frac{3}{x}$

Answer

We have
$x^{2} + \frac{1}{x^{2}} - 2 - 3 x + \frac{3}{x}$ .

$= x^{2} + \frac{1}{x^{2}} - 2 \times x \times \frac{1}{x} - 3 x + \frac{3}{x}$ ... (∵ $x \times \frac{1}{x} = 1$ ).

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

$= (x - \frac{1}{x}) [(x - \frac{1}{x}) - 3]$ .

$= (x - \frac{1}{x}) (x - \frac{1}{x} - 3)$ .

Exercise 3A RS Aggarwal Class 9 Mathematics Solutions

Factorise:

1. 9x2+12xy

Answer

2. 18x2y−24xyz

Answer

3. 27a3b3−45a4b2

Answer

4. 2a(x+y)−3b(x+y)

Answer

5. 2x(p2+q2)+4y(p2+q2)

Answer

6. x(a−5)+y(5−a)

Answer

7. 4(a+b)−6(a+b)2

Answer

8. 8(3a−2b)2−10(3a−2b)

Answer

9. x(x+y)3−3x2y(x+y)

Answer

10. x3+2x2+5x+10

Answer

11. x2+xy−2xz−2yz

Answer

12. a3b−a2b+5ab−5b

Answer

13. 8−4a−2a3+a4

Answer

14. x3−2x2y+3xy2−6y3

Answer

15. px−5q+pq−5x

Answer

16. x2+y−xy−x

Answer

17. (3a−1)2−6a+2

Answer

18. (2x−3)2−8x+12

Answer

19. a3+a−3a2−3

Answer

20. 3ax−6ay−8by+4bx

Answer

21. abx2+a2x+b2x+ab

Answer

22. x3−x2+ax+x−a−1

Answer

23. 2x+4y−8xy−1

Answer

24. ab(x2+y2)−xy(a2+b2)

Answer

25. a2+ab(b+1)+b3

Answer

26. a3+ab(1−2a)−2b2

Answer

27. 2a2+bc−2ab−ac

Answer

28. (ax+by)2+(bx−ay)2

Answer

29. a(a+b−c)−bc

Answer

30. a(a−2b−c)+2bc

Answer

31. a2x2+(ax2+1)x+a

Answer

32. ab(x2+1)+x(a2+b2)

Answer

33. x2−(a+b)x+ab

Answer

34. x2+1x2−2−3x+3x

Answer

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1. $9 x^{2} + 12 x y$

2. $18 x^{2} y - 24 x y z$

3. $27 a^{3} b^{3} - 45 a^{4} b^{2}$

4. $2 a (x + y) - 3 b (x + y)$

5. $2 x (p^{2} + q^{2}) + 4 y (p^{2} + q^{2})$

6. $x (a - 5) + y (5 - a)$

7. $4 (a + b) - 6 {(a + b)}^{2}$

8. $8 {(3 a - 2 b)}^{2} - 10 (3 a - 2 b)$

9. $x {(x + y)}^{3} - 3 x^{2} y (x + y)$

10. $x^{3} + 2 x^{2} + 5 x + 10$

11. $x^{2} + x y - 2 x z - 2 y z$

12. $a^{3} b - a^{2} b + 5 a b - 5 b$

13. $8 - 4 a - 2 a^{3} + a^{4}$

14. $x^{3} - 2 x^{2} y + 3 x y^{2} - 6 y^{3}$

15. $p x - 5 q + p q - 5 x$

16. $x^{2} + y - x y - x$

17. ${(3 a - 1)}^{2} - 6 a + 2$

18. ${(2 x - 3)}^{2} - 8 x + 12$

19. $a^{3} + a - 3 a^{2} - 3$

20. $3 a x - 6 a y - 8 b y + 4 b x$

21. $a b x^{2} + a^{2} x + b^{2} x + a b$

22. $x^{3} - x^{2} + a x + x - a - 1$

23. $2 x + 4 y - 8 x y - 1$

24. $a b (x^{2} + y^{2}) - x y (a^{2} + b^{2})$

25. $a^{2} + a b (b + 1) + b^{3}$

26. $a^{3} + a b (1 - 2 a) - 2 b^{2}$

27. $2 a^{2} + b c - 2 a b - a c$

28. ${(a x + b y)}^{2} + {(b x - a y)}^{2}$

29. $a (a + b - c) - b c$

30. $a (a - 2 b - c) + 2 b c$

31. $a^{2} x^{2} + (a x^{2} + 1) x + a$

32. $a b (x^{2} + 1) + x (a^{2} + b^{2})$

33. $x^{2} - (a + b) x + a b$

34. $x^{2} + \frac{1}{x^{2}} - 2 - 3 x + \frac{3}{x}$