**Factorize each of the following expressions:**

**Question 24. 50/x**^{2} – 2x^{2}/81

^{2}– 2x

^{2}/81

**Solution:**

50/x

^{2}– 2x^{2}/81= 2(25/x

^{2}– x^{2}/81)= 2[(5/x)

^{2 }– (x/9)^{2}]= 2(5/x – x/9) (5/x + x/9)

**Question 25. 256x**^{5 }– 81x

^{5 }– 81x

**Solution:**

256x

^{5 }– 81x= x(256x

^{4}– 81)= x[(16x

^{2})^{2}– (9)^{2}]= x(16x

^{2}– 9) (16x^{2 }+ 9)

**Question 26. a**^{4} – (2b + c)^{4}

^{4}– (2b + c)

^{4}

**Solution:**

a

^{4}– (2b + c)^{4}= (a

^{2})^{2}– [(2b + c)^{2}]^{2}

= [a

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

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

^{2}+ (2b + c)^{2}]

**Question 27. (3x + 4y)**^{4} – x^{4}

^{4}– x

^{4}

**Solution:**

(3x + 4y)

^{4}– x^{4}= [(3x + 4y)

^{2}]^{2}– (x^{2})^{2}= [(3x + 4y)

^{2}– x^{2}] [(3x + 4y)^{2}+ x^{2}]= [(3x + 4y) – x] [(3x + 4y) + x] [(3x + 4y)

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

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

^{2}+ x^{2}]

**Question 28. p**^{2}q^{2} – p^{4}q^{4}

^{2}q

^{2}– p

^{4}q

^{4}

**Solution:**

p

^{2}q^{2}– p^{4}q^{4}= p

^{2}q^{2}(1 – p^{2}q^{2})= p

^{2}q^{2}(1 + pq) (1 – pq)

**Question 29. 3x**^{3}y – 243xy^{3}

^{3}y – 243xy

^{3}

**Solution:**

3x

^{3}y – 243xy^{3}= 3xy (x

^{2}– 81y^{2})= 3xy [x

^{2}– (9y)^{2}]= 3xy (x – 9y) (x + 9y)

**Question 30. a**^{4}b^{4} – 16c^{4}

^{4}b

^{4}– 16c

^{4}

**Solution:**

a

^{4}b^{4}– 16c^{4}= (a

^{2}b^{2})^{2}– (4c^{2})^{2}= (a

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

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

^{2}b^{2}+ c^{2})

**Question 31. x**^{4} – 625

^{4}– 625

**Solution:**

x

^{4}– 625= (x

^{2})^{2 }– (25)^{2}= (x

^{2}– 25) (x^{2}+ 25)= (x

^{2}– 5^{2}) (x^{2}+ 25)= (x – 5) (x + 5) (x

^{2}+ 25)

**Question 32. x**^{4} – 1

^{4}– 1

**Solution:**

x

^{4}– 1= (x

^{2})^{2}– 1^{2}= (x

^{2}– 1)(x^{2}+1)= (x + 1)(x – 1) (x

^{2 }+ 1)

**Question 33. 49(a – b)**^{2} – 25(a + b)^{2}

^{2}– 25(a + b)

^{2}

**Solution:**

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)]

= (7a – 7b -5a – 5b) (7a – 7b + 5a + 5b)

= (2a – 12b) (12a – 2b)

= 4(a – 6b)(6a – b)

**Question 34. x – y – x**^{2} + y^{2}

^{2}+ y

^{2}

**Solution:**

x – y – x

^{2}+ y^{2}= (x – y) – (x

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

= (x – y) [1 – (x + y)]

= (x – y) (1 – x – y)

**Question 35. 16(2x – 1)**^{2} – 25y^{2}

^{2}– 25y

^{2}

**Solution:**

16(2x – 1)

^{2}– 25y^{2}= [4(2x – 1)]

^{2}– (5y)^{2}= [4(2x – 1) – 5y] [4(2x – 1) + 5y]

= (8x – 4 – 5y) ( 8x – 4 + 5y)

**Question 36. 4(xy + 1)**^{2} – 9(x – 1)^{2}

^{2}– 9(x – 1)

^{2}

**Solution:**

4(xy + 1)

^{2}– 9( x- 1)^{2}= [2(xy + 1)]

^{2}– [3(x – 1)]^{2}= [2(xy + 1) – 3(x – 1)] [2(xy + 1) + 3(x – 1)]

= (2xy + 2 – 3x + 3) (2xy + 2 + 3x – 3)

= (2xy – 3x + 5) (2xy + 3x – 1)

**Question 37. (2x + 1)**^{2} – 9x^{4}

^{2}– 9x

^{4}

**Solution:**

(2x + 1)

^{2}– 9x^{4}= (2x + 1)

^{2}– (3x^{2})^{2}= [(2x +1) – 3x

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

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

^{2}+ 3x – x + 1) (3x^{2}+ 2x + 1)= [3x(1 – x) + (1 – x)] (3x

^{2}+ 2x + 1)= (3x +1) (1 – x) (3x

^{2}+ 2x + 1)

**Question 38. x**^{4} – (2y – 3z)^{2}

^{4}– (2y – 3z)

^{2}

**Solution:**

x

^{4}– (2y – 3z)^{2}= (x

^{2})^{2}– (2y – 3z)^{2}= [x

^{2}– (2y – 3z)] [x^{2}+ (2y – 3z)]= (x

^{2}– 2y + 3z) (x^{2 }+ 2y – 3z)

**Question 39. a**^{2} – b^{2} + a – b

^{2}– b

^{2}+ a – b

**Solution:**

a

^{2}– b^{2}+ a – b= (a

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

= (a – b) (a + b + 1)

**Question 40. 16a**^{4} – b^{4}

^{4}– b

^{4}

**Solution:**

16a

^{4}– b^{4}= (4a

^{2})^{2 }– (b^{2})^{2}= (4a

^{2}– b^{2}) (4a^{2}+ b^{2})= [(2a)

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

^{2}+ b^{2})

**Question 41. a**^{4} – 16(b – c)^{4}

^{4}– 16(b – c)

^{4}

**Solution:**

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}– [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 – 2b + 2c) (a + 2b – 2c) [a

^{2}+ 4(b – c)^{2}]

**Question 42. 2a**^{5} – 32a

^{5}– 32a

**Solution:**

2a

^{5}– 32a= 2a(a

^{5}– 1 )= 2a [ (a

^{2})^{2}– 1^{2}]= 2a ( a2 – 1) (a2+ 1)

= 2a (a+1) (a-1) (a

^{2}+1)

**Question 43. a**^{4}b^{4} – 81c^{4}

^{4}b

^{4}– 81c

^{4}

**Solution:**

Answer:a

^{4}b^{4}– 81c^{4}= (a

^{2}b^{2})^{2}– (9c^{2})^{2}= (a

^{2}b^{2}– 9c^{2}) (a^{2}b^{2}+ 9c^{2})= [(ab)

^{2}– (3c)^{2}] (a^{2}b^{2}+ 9c^{2})= (ab – 3c) (ab + 3c) (a

^{2}b^{2}+ 9c^{2})

**Question 44. xy**^{9} – x^{9}y

^{9}– x

^{9}y

**Solution:**

xy

^{9}– x^{9}y= xy(y

^{8}– x^{8})= xy [(y

^{4})^{2}– (x^{4})^{2}]= xy (y

^{4}– x^{4})(y^{4}+ x^{4})= xy [(y

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

^{2}– x^{2})(y^{2 }+ x^{2})(y^{4 }+ x^{4})= xy (y – x) (y + x) (y

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

**Question 45. x**^{3} – x

^{3}– x

**Solution:**

x

^{3}– x= x(x

^{2}– 1)= x (x + 1)(x – 1)

**Question 46. 18a**^{2}x^{2} – 32

^{2}x

^{2}– 32

**Solution:**

18a

^{2}x^{2}– 32= 2(9a

^{2}x^{2}– 16)= 2[(3ax)

^{2}– 42]= 2(3ax – 4) (3ax + 4)