RD Sharma Solutions for Class 8 Math Chapter 7 - Factorization (Part-5) | RD Sharma Solutions for Class 8 Mathematics PDF Download

Question 1: Factorize each of the following expression:
16x² − 25y²
Answer 1: 16x2−25y2 
=(4x)2−(5y)2=(4x−5y)(4x+5y)Question 2: Factorize each of the following expression:
27x² − 12y²
Answer 2: 27x2−12y2 
=3(9x2−4y2) 
=3[(3x)2−(2y)2] 
=3(3x−2y)(3x+2y) 
Question 3: Factorize each of the following expression:
144a² − 289b²
Answer 3: 144a2−289b2 
=(12a)2−(17b)2 
=(12a−17b)(12a+17b) 
Question 4: Factorize each of the following expression:
12m² − 27
Answer 4: 12m2−27 
=3(4m2−9) 
=3[(2m)2−32] 
=3(2m−3)(2m+3)
Question 5: Factorize each of the following expression:
125x² − 45y² 
Answer 5: 125x2−45y2 
=5(25x2−9y2) 
=5[(5x)2−(3y)2] 
=5(5x−3y)(5x+3y) 
Question 6: Factorize each of the following expression:
144a² − 169b²
Answer 6: 144a2−169b2 
=(12a)2−(13b)2 
=(12a−13b)(12a+13b) 
Question 7: Factorize each of the following expression:
(2a − b)² − 16c²
Answer 7: (2a−b)2−16c2 
=(2a−b)2−(4c)2 
=[(2a−b)−4c][(2a−b)+4c] 
=(2a−b−4c)(2a−b+4c) 
Question 8: Factorize each of the following expression:
(x + 2y)² − 4(2x − y)² 
Answer 8: (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: Factorize each of the following expression:
3a⁵ − 48a³
Answer 9: 3a5−48a3
=3a3(a2−16)=3a3(a2−42) =3a3(a−4)(a+4)
Question 10: Factorize each of the following expression:
a⁴ − 16b⁴ 
Answer 10: a4−16b4=a4−24b4=(a2)2−(22b2)2 
=(a2−22b2)(a2+22b2) 
=[a2−(2b)2](a2+4b2) 
=(a−2b)(a+2b)(a2+4b2) 
Question 11: Factorize each of the following expression:
x⁸ − 1
Answer 11: x8−1 
=(x4)2−12 
=(x4−1)(x4+1) 
=[(x2)2−12](x4+1) 
=(x2−1)(x2+1)(x4+1) 
=(x2−12)(x2+1)(x4+1) 
=(x−1)(x+1)(x2+1)(x4+1) 
Question 12: Factorize each of the following expression:
64 − (a + 1)²
Answer 12: 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: Factorize each of the following expression:
36l² − (m + n)²
Answer 13: 36l2−(m+n)2 
=(6l)2−(m+n)2 
=[6l−(m+n)][6l+(m+n)] 
=(6l−m−n)(6l+m+n) 
Question 14: Factorize each of the following expression:
25x⁴y⁴ − 1
Answer 14: 25x4y4−1 
=(5x2y2)2−1 
=(5x2y2−1)(5x2y2+1) 
Question 15: Factorize each of the following expression:
a4− 1/b⁴ 
Answer 15:  a   a⁴ - 1/b⁴ 
= (a²)² - 1/(b²)² 
= a²- 1/b²a² + 1/b² 
= a - 1/ba + 1/ba² + 1/b² 
Question 16: Factorize each of the following expression:
x³ − 144x
Answer 16: x3−144x 
=x(x2−144) 
=x(x2−122) 
=x(x−12)(x+12) 
Question 17: Factorize each of the following expression:
(x - 4y)² − 625
Answer 17: (x−4y)2−625 
=(x−4y)2−252 
=[(x−4y)−25][(x−4y)+25] 
=(x−4y−25)(x−4y+25) 
Question 18: Factorize each of the following expression:
9(a − b)² − 100(x − y)²
Answer 18: 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: Factorize each of the following expression:
(3 + 2a)² − 25a²
Answer 19: (3+2a)2−25a2 
=(3+2a)2−(5a)2 
=[(3+2a)−5a][(3+2a)+5a] 
=(3+2a−5a)(3+2a+5a) 
=(3−3a)(3+7a) 
=3(1−a)(3+7a) 
Question 20: Factorize each of the following expression:
(x + y)² − (a − b)²
Answer 20: (x+y)2−(a−b)2 
=[(x+y)−(a−b)][(x+y)+(a−b)] 
=(x+y−a+b)(x+y+a−b) 
Question 21: Factorize each of the following expression:

Answer 21:

 



Question 22: Factorize each of the following expression:
75a³b² - 108ab⁴
Answer 22: 75a3b2−108ab4 
=3ab2(25a2−36b2) 
=3ab2[(5a)2−(6b)2] 
=3ab2(5a−6b)(5a+6b) 
Question 23: Factorize each of the following expression:
x⁵ − 16x³
Answer 23: x5−16x3
=x3(x2−16) 
=x3(x2−42) 
=x3(x−4)(x+4) 
Question 24: Factorize each of the following expression:

Answer 24:
 

Question 25: Factorize each of the following expression:
256x⁵ − 81x
Answer 25: 256x5−81x 
=x(256x4−81) 
=x[(16x2)2−92] 
=x(16x2+9)(16x2−9) 
=x(16x2+9)[(4x)2−32] 
=x(16x2+9)(4x+3)(4x−3) 
Question 26: Factorize each of the following expression:
a⁴ − (2b + c)⁴ 
Answer 26: a4−(2b+c)4 
=(a2)2−[(2b+c)2]2 
=[a2+(2b+c)2][a2−(2b+c)2] 
=[a2+(2b+c)2]{[a+(2b+c)][a−(2b+c)]} 
=[a2+(2b+c)2](a+2b+c)(a−2b−c) 
Question 27: Factorize each of the following expression:
(3x + 4y)⁴ − x⁴
Answer 27: (3x+4y)4−x4 
=[(3x+4y)2]2−(x2)2 
=[(3x+4y)2+x2][(3x+4y)2−x2] 
=[(3x+4y)2+x2][(3x+4y)+x][(3x+4y)−x] 
={(3x+4y)2+x2}(3x+4y+x)(3x+4y−x) 
={(3x+4y)2+x2}(4x+4y)(2x+4y) 
={(3x+4y)2+x2}4(x+y)2(x+2y) 
=8{(3x+4y)2+x2}(x+y)(x+2y) 
Question 28: Factorize each of the following expression:
p²q² − p⁴q⁴
Answer 28: p2q2−p4q4 
=p2q2(1−p2q2) 
=p2q2[1−(pq)2] 
=p2q2(1−pq)(1+pq) 
Question 29: Factorize each of the following expression:
3x³y − 243xy³
Answer 29: 3x3y−243xy3 
=3xy(x2−81y2) 
=3xy[x2−(9y)2]
=3xy(x−9y)(x+9y)
Question 30: Factorize each of the following expression:
a⁴b⁴ − 16c⁴ 
Answer 30: a4b4−16c4 
=[(a²b²)²−(4c²)²]
=(a2b2+4c2)(a2b2−4c2) 
=(a2b2+4c2)[(ab)2−(2c)2] 
=(a2b2+4c2)(ab+2c)(ab−2c) 
Question 31: Factorize each of the following expression:
x⁴ − 625
Answer 31: x4−625 
=(x2)2−252 
=(x2+25)(x2−25) 
=(x2+25)(x2−52) 
=(x2+25)(x+5)(x−5)
Question 32: Factorize each of the following expression:
x⁴ − 1
Answer 32: x4−1 
=(x²)²−1
=(x²+1)(x²−1)
=(x²+1)(x+1)(x−1)
Question 33: Factorize each of the following expression:
49(a − b)² − 25(a + b)²
Answer 33: 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) 
=2(a−6b)2(6a−b) 
=4(a−6b)(6a−b) 
Question 34: Factorize each of the following expression:
x − y − x² + y²
Answer 34: x−y−x2+y2 
=(x−y)+(y2−x2)               [Regrouping the terms] 
=(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) 
Question 35: Factorize each of the following expression:
16(2x − 1)² − 25y²
Answer 35: 16(2x−1)2−25y2 
=[4(2x−1)]2−(5y)2 
=[4(2x−1)−5y][4(2x−1)+5y] 
=(8x−4−5y)(8x−4+5y) 
=(8x−5y−4)(8x+5y−4) 

Question 36: Factorize each of the following expression:
4(xy + 1)² − 9(x − 1)²
Answer 36: 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: Factorize each of the following expression:
(2x + 1)² − 9x⁴
Answer 37: (2x+1)2−9x4 
=(2x+1)2−(3x2)2 
=[(2x+1)−3x2][(2x+1)+3x2] 
=(−3x2+2x+1)(3x2+2x+1) 
We can factorise the quadratic expressions in the curved brackets as: (−3x2+3x−x+1)(3x2+2x+1)
={3x(−x+1)+1(−x+1)}(3x2+2x+1) 
=(−x+1)(3x+1)(3x2+2x+1) 
=−(x−1)(3x+1)(3x2+2x+1) 
Question 38: Factorize each of the following expression:
x⁴ − (2y − 3z)²
Answer 38: x4−(2y−3z)2 
=(x2)2−(2y−3z)2 
=[x2−(2y−3z)][x2+(2y−3z)] 
=(x2−2y+3z)(x2+2y−3z)
Question 39: Factorize each of the following expression:
a² − b² + a − b
Answer 39: a2−b2+a−b=(a2−b2)+(a−b)            [Grouping the terms] 
=(a+b)(a−b)+(a−b) 
=(a−b)(a+b+1)             [Taking out the common factor (a−b)]
Question 40: Factorize each of the following expression:
16a⁴ − b⁴
Answer 40: 16a4−b4 
=(4a2)2−(b2)2 
=(4a2+b2)(4a2−b2) 
=(4a2+b2)[(2a)2−b2] 
=(4a2+b2)(2a+b)(2a−b)
Question 41: Factorize each of the following expression:
a⁴ − 16(b − c)⁴
Answer 41: a4−16(b−c)4 
=(a2)2−[4(b−c)2]2 
=[a2+4(b−c)2][a2−4(b−c)2] 
=[a2+4(b−c)2]{a2−[2(b−c)]2} 
=[a2+4(b−c)2][a+2(b−c)][a−2(b−c)] 
=[a2+4(b−c)2](a+2b−2c)(a−2b+2c) 
Question 42: Factorize each of the following expression:
2a⁵ − 32a
Answer 42: 2a5−32a 
=2a(a4−16) 
=2a[(a2)2−42] 
=2a(a2+4)(a2−4) 
=2a(a2+4)(a2−22) 
=2a(a2+4)(a+2)(a−2) 
=2a(a−2)(a+2)(a2+4) 
Question 43: Factorize each of the following expression:
a⁴b⁴ − 81c⁴
Answer 43: a4b4−81c4 
=(a2b2)2−(9c2)2 
=(a2b2+9c2)(a2b2−9c2) 
=(a2b2+9c2)[(ab)2−(3c)2] 
=(a2b2+9c2)(ab+3c)(ab−3c) 
Question 44: Factorize each of the following expression:
xy⁹ − yx⁹
Answer 44: xy9−yx9 
=xy(y8−x8) 
=xy[(y4)2−(x4)2] 
=xy(y4+x4)(y4−x4)=xy(y⁴+x⁴)[(y²)²−(x²)²]
=xy(y4+x4)(y2+x2)(y2−x2) 
=xy(y4+x4)(y2+x2)(y+x)(y−x) 
Question 45: Factorize each of the following expression:
x³ − x
Answer 45: x3−x 
=x(x2−1) 
=x(x−1)(x+1) 
Question 46: Factorize each of the following expression:
18a²x² − 32
Answer 46: 18a2x2−32 
=2(9a2x2−16) 
=2[(3ax)2−42] 
=2(3ax−4)(3ax+4)