**Factorize each of the following expressions:**

**Question 1. 16x**^{2} – 25y^{2}

^{2}– 25y

^{2}

**Solution:**

16x

^{2 }– 25y^{2}= (4x)

^{2 }– (5y)^{2}= (4x – 5y)(4x + 5y)

**Question 2. 27x**^{2 }– 12y^{2}

^{2 }– 12y

^{2}

**Solution:**

27x

^{2 }– 12y^{2}= 3(9x

^{2}– 4y^{2})= 3[(3x)

^{2 }– (2y)^{2}]= 3(3x – 2y)(3x + 2y)

**Question 3. 144a**^{2 }– 289b^{2}

^{2 }– 289b

^{2}

**Solution:**

144a

^{2 }– 289b^{2}= (12a)

^{2 }– (17b)^{2}= (12a – 17b)(12a + 17b)

**Question 4. 12m**^{2} – 27

^{2}– 27

**Solution:**

12m

^{2}– 27= 3(4m

^{2 }– 9)= 3[(2m)

^{2 }– 3^{2}]= 3(2m – 3)(2m + 3)

**Question 5. 125x**^{2 }– 45y^{2}

^{2 }– 45y

^{2}

**Solution:**

125x

^{2 }– 45y^{2}= 5(25x

^{2 }– 9y^{2})= 5[(5x)

^{2 }– (3y)^{2}]= 5(5x – 3y)(5x + 3y)

**Question 6. 144a**^{2 }– 169b^{2}

^{2 }– 169b

^{2}

**Solution:**

144a

^{2 }– 169b^{2}= (12a)

^{2}– (13b)^{2}= (12a – 13b)(12a + 13b)

**Question 7. (2a – b)**^{2} – 16c^{2}

^{2}– 16c

^{2}

**Solution:**

(2a – b)

^{2}– 16c^{2}= (2a – b)

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

= (2a – b + 4c)(2a – b – 4c)

**Question 8. (x + 2y)**^{2} – 4(2x – y)^{2}

^{2}– 4(2x – y)

^{2}

**Solution:**

(x + 2y)

^{2}– 4(2x – y)^{2}= (x + 2y)

^{2 }– [2(2x – y)]^{2}= [(x + 2y) – 2(2x – y)][(x + 2y) + 2(2x – y)]

= (x + 2y – 4x + 2y)(x + 2y + 4x – 2y)

= 5x(4y – 3x)

**Question 9. 3a**^{5 }– 48a^{3}

^{5 }– 48a

^{3}

**Solution:**

3a

^{5 }– 48a^{3}= 3a

^{3}(a^{2 }– 16)= 3a

^{3}(a^{2 }– (4)^{2})= 3a

^{3}(a – 4)(a + 4)

**Question 10. a**^{4 }– 16b^{4}

^{4 }– 16b

^{4}

**Solution:**

a

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

^{2 }– (2b)^{2}](a^{2 }+ 4b^{2})= (a – 2b)(a + 2b)(a

^{2 }+ 4b^{2})

**Question 11. x**^{8 }– 1

^{8 }– 1

**Solution:**

x

^{8 }– 1= (x

^{4})^{2}– 1^{2}= [(x

^{2})^{2}– 1^{2}](x^{4 }+ 1)= [(x

^{2 }– 1)(x^{2 }+ 1)](x^{4 }+ 1)= (x – 1)(x + 1)(x

^{2 }+ 1)(x^{4 }+ 1)

**Question 12. 64 – (a + 1)**^{2}

^{2}

**Solution:**

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)

**Question 13. 36L**^{2 }– (m + n)^{2}

^{2 }– (m + n)

^{2}

**Solution:**

36L

^{2 }– (m + n)^{2}= (6L)

^{2 }– (m + n)^{2}= [6L – (m + n)][6L + (m + n)]

= (6L – m – n)(6L + m + n)

**Question 14. 25x**^{4}y^{4 }– 1

^{4}y

^{4 }– 1

**Solution:**

25x

^{4}y^{4 }– 1= (5x

^{2}y^{2})^{2 }– 1^{2}= (5x

^{2}y^{2 }– 1)(5x^{2}y^{2 }+ 1)

**Question 15. a**^{4 }– 1/b^{4}

^{4 }– 1/b

^{4}

**Solution:**

a

^{4 }– 1/b^{4}= (a

^{2})^{2 }– (1/b^{2})^{2}= (a

^{2 }– 1/b^{2})(a^{2 }+ 1/b^{2})= (a – 1/b)(a + 1/b)(a

^{2 }+ 1/b^{2})

**Question 16. x**^{3 }– 144x

^{3 }– 144x

**Solution:**

x

^{3 }– 144x= x(x

^{2 }– 144)= x(x

^{2 }– 12^{2})= x(x – 12)(x + 12)

**Question 17. (x – 4y)**^{2 }– 625

^{2 }– 625

**Solution:**

(x – 4y)

^{2 }– 625= (x – 4y)

^{2 }– 25^{2}= [(x – 4y) – 25] [(x – 4y) + 25]

= (x – 4y – 25)(x – 4y + 25)

**Question 18. 9(a – b)**^{2 }– 100(x – y)^{2}

^{2 }– 100(x – y)

^{2}

**Solution:**

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)]

= (3a – 3b – 10x + 10y) (3a – 3b + 10x – 10y)

**Question 19. (3 + 2a)**^{2} – 25a^{2}

^{2}– 25a

^{2}

**Solution:**

(3 + 2a)

^{2 }– 25a^{2}= (3 + 2a)

^{2 }– (5a)^{2}= (3 + 2a + 5a)(3 + 2a – 5a)

= (3 + 7a)(3 – 3a)

**Question 20. (x + y)**^{2} – (a – b)^{2}

^{2}– (a – b)

^{2}

**Solution:**

(x + y)

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

= (x + y + a – b) (x + y – a + b)

**Question 21. 1/16 x**^{2}y^{2} – 4/49y^{2}z^{2}

^{2}y

^{2}– 4/49y

^{2}z

^{2}

**Solution:**

1/16 x

^{2}y^{2}– 4/49y^{2}z^{2}= (1/4xy)

^{2 }– (2/7 yz)^{2}= (1/4xy – 2/7yz) (1/4xy + 2/7yz)

= [y (1/4x – 2/7z)] [y(1/4x + 2/7z)]

= y

^{2}(1/4x – 2/7z) (1/4x + 2/7z)

**Question 22. 75a**^{3}b^{2} – 108ab^{4}

^{3}b

^{2}– 108ab

^{4}

**Solution:**

75a

^{3}b^{2}– 108ab^{4}= 3ab

^{2}(25a^{2 }– 36b^{2})= 3ab

^{2}[(5a)^{2 }– (6b)^{2}]= 3ab

^{2}[(5a – 6b)(5a + 6b)]= 3ab

^{2}(5a – 6b)(5a + 6b)

**Question 23. x**^{5 }– 16x^{3}

^{5 }– 16x

^{3}

**Solution:**

x

^{5 }– 16x^{3}= x

^{3}(x^{2 }– 16)= x

^{3}(x^{2 }– 4^{2})= x

^{3}(x – 4)(x + 4)

### Chapter 7 Factorization – Exercise 7.5 | Set 2

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