RD Sharma Solutions for Class 8 Math Chapter 7 - Factorization (Part-4) | RD Sharma Solutions for Class 8 Mathematics PDF Download

Question 1: Factorize each of the following expressions:
qr − pr + qs − ps
Answer 1: qr−pr+qs−ps  
=(qr−pr)+(qs−ps)  [Grouping the expressions] 
=r(q−p)+s(q−p) 
=(r+s)(q−p)              [Taking (q−p) as the common factor] 
Question 2: Factorize each of the following expressions: 
p²q − pr² − pq + r² 
Answer 2: p2q−pr2−pq+r2 
=(p2q−pq)+(r2−pr2)   [Grouping the expressions] 
=pq(p−1)+r2(1−p) 
=pq(p−1)−r2(p−1)      [∵(1−p)=−(p−1)]=(pq−r2)(p−1)              [Taking (p−1) as the common factor] 
Question 3: Factorize each of the following expressions:
1 + x + xy + x²y 
Answer 3: 1+x+xy+x2y 
=(1+x)+(xy+x2y)   [Grouping the expressions] 
=(1+x)+xy(1+x) 
=(1+xy)(1+x)           [Taking (1+x) as the common factor] 
Question 4: Factorize each of the following expressions: 
ax + ay − bx − by 
Answer 4: ax+ay−bx−by 
=(ax+ay)−(bx+by)   [Grouping the expressions] 
= a(x+y)−b(x+y) 
= (a−b)(x+y)              [Taking (x+y) as the common factor] 
Question 5: Factorize each of the following expressions:
xa² + xb² − ya² − yb² 
Answer 5:  xa2+xb2−ya2−yb2 
=(xa2+xb2)−(ya2+yb2)   [Grouping the expressions] 
=x(a2+b2)−y(a2+b2) 
=(x−y)(a2+b2)                   [Taking (a2+b2) as the common factor] 
Question 6: Factorize each of the following expressions:
x² + xy + xz + yz 
Answer 6: x2+xy+xz+yz 
=(x2+xy)+(xz+yz)   [Grouping the expressions] 
=x(x+y)+z(x+y) 
=(x+z)(x+y)              [Taking (x+y) as the common factor] 
= (x+y)(x+z) 
Question 7: Factorize each of the following expressions:
2ax + bx + 2ay + by
Answer 7: 2ax+bx+2ay+by 
=(2ax+bx)+(2ay+by)   [Grouping the expressions] 
=x(2a+b)+y(2a+b) 
=(x+y)(2a+b)                 [Taking (2a+b) as the common factor] 
Question 8: Factorize each of the following expressions: 
ab − by − ay + y² 
Answer 8: ab−by−ay+y2 
=(ab−ay)+(y2−by)              [Grouping the expressions] 
= a(b−y)+y(y−b) 
=a(b−y)−y(b−y)                  [∵(y−b)=−(b−y)] 
=(a−y)(b−y)                          [Taking (b−y) as the common factor]
Question 9: Factorize each of the following expressions:
axy + bcxy − az − bcz
Answer 9: axy+bcxy−az−bcz 
=(axy+bcxy)−(az+bcz)       [Grouping the expressions] 
=xy(a+bc)−z(a+bc) 
=(xy−z)(a+bc)                       [Taking (a+bc) as the common factor] 
Question 10: Factorize each of the following expressions:
lm² − mn² − lm + n²
Answer 10: lm2−mn2−lm+n2=(lm2−lm)+(n2−mn2)   [Regrouping the expressions] 
= lm(m−1)+n2(1−m) 
=lm(m−1)−n2(m−1)      [∵(1−m)=−(m−1)] 
=(lm−n2)(m−1)               [Taking (m−1) as the common factor]
Question 11: Factorize each of the following expressions:
x³ − y² + x − x²y²
Answer 11: x3−y2+x−x2y2 
=(x3+x)−(x2y2+y2)         [Regrouping the expressions] 
=x(x2+1)−y2(x2+1) 
=(x−y2)(x2+1)                  [Taking (x2+1) as the common factor] 
Question 12: Factorize each of the following expressions:
6xy + 6 − 9y − 4x
Answer 12: 6xy+6−9y−4x=(6xy−4x)+(6−9y)     [Regrouping the expressions] 
=2x(3y−2)+3(2−3y) 
=2x(3y−2)−3(3y−2)   [∵(2−3y)=−(3y−2)] 
=(2x−3)(3y−2)             [Taking (3y−2) as the common factor] 
Question 13: Factorize each of the following expressions:
x² − 2ax − 2ab + bx
Answer 13: x2−2ax−2ab+bx 
=(x2−2ax)+(bx−2ab)    [Regrouping the expressions] 
=x(x−2a)+b(x−2a) 
=(x+b)(x−2a)                  [Taking (x−2a) as the common factor] 
=(x−2a)(x+b) 
Question 14: Factorize each of the following expressions:
x³ − 2x²y + 3xy² − 6y³ 
Answer 14: x3− 2x2y + 3xy2− 6y3 
=(x3−2x2y)+(3xy2−6y3)           [Grouping the expressions] 
=x2(x−2y)+3y2(x−2y) 
=(x2+3y2)(x−2y)                         [Taking (x−2y) as the common factor] 
Question 15: Factorize each of the following expression:
abx² + (ay − b) x − y
Answer 15: abx2+(ay−b)x−y=abx2+axy−bx−y
=(abx2−bx)+(axy−y)    [Regrouping the expressions] 
=bx(ax−1)+y(ax−1) 
=(bx+y)(ax−1)                [Taking (ax−1) as the common factor]
Question 16: Factorize each of the following expression:
(ax + by)² + (bx − ay)²
Answer 16: (ax+by)2+(bx−ay)2=a2x2+2abxy+b2y2+b2x2−2abxy+a2y2 
=a2x2+b2y2+b2x2+a2y2 
=(a2x2+a2y2)+(b2x2+b2y2)   [Regrouping the expressions] 
=a2(x2+y2)+b2(x2+y2) 
=(a2+b2)(x2+y2)                        [Taking (x2+y2) as the common factor]
Question 17: Factorize each of the following expression:
16(a − b)³ − 24 (a − b)²
Answer 17: 16(a−b)3−24(a−b)2 
=8(a−b)2[2(a−b)−3]         {Taking [8(a−b)2] as the common factor} 
=8(a−b)2(2a−2b−3) 
Question 18: Factorize each of the following expression:
ab(x² + 1) + x(a² + b²)
Answer 18: ab(x2+1)+x(a2+b2)=abx2+ab+a2x+b2x 
=(abx2+a2x)+(b2x+ab)   [Regrouping the expressions] 
=ax(bx+a)+b(bx+a) 
=(ax+b)(bx+a)                   [Taking (bx+a) as the common factor] 
Question 19: Factorize each of the following expression:
a²x² + (ax² + 1)x + a
Answer 19:  a2x2+(ax2+1)x+a=a2x2+ax3+x+a
=(ax3+a2x2)+(x+a)   [Regrouping the expressions] 
=ax2(x+a)+(x+a) 
=(ax2+1)(x+a)             [Taking (x+a) as the common factor] 
Question 20: Factorize each of the following expression:
a(a − 2b − c) + 2bc
Answer 20: a(a−2b−c)+2bc=a2−2ab−ac+2bc 
=(a2−ac)+(2bc−2ab)             [Regrouping the terms] 
=a(a−c)+2b(c−a) 
=a(a−c)−2b(a−c)                   [∵(c−a)=−(a−c)] 
=(a−2b)(a−c)                            [Taking (a−c)  as the common factor]
Question 21: Factorize each of the following expression:
a(a + b − c) − bc
Answer 21: a(a+b−c)−bc=a2+ab−ac−bc 
=(a2−ac)+(ab−bc)   [Regrouping the expressions] 
=a(a−c)+b(a−c) 
=(a+b)(a−c)               [Taking (a−c) as the common factor] 
Question 22: Factorize each of the following expression:
x² − 11xy − x + 11y
Answer 22: x2−11xy−x+11y=(x2−x)+(11y−11xy)   [Regrouping the expressions] 
=x(x−1)+11y(1−x) 
=x(x−1)−11y(x−1)       [∵(1−x)=−(x−1)] 
=(x−11y)(x−1)               [Taking out the common factor (x−1)] 
Question 23: Factorize each of the following expression:
ab − a − b + 1
Answer 23: ab−a−b+1=(ab−b)+(1−a)   [Regrouping the expressions
]
=b(a−1)+(1−a)  
=b(a−1)−(a−1)   [∵(1−a)=−(a−1)] 
=(a−1)(b−1)         [Taking out the common factor (a−1)] 
Question 24: Factorize each of the following expression:
x² + y − xy − x 
Answer 24: x2+y−xy−x=(x2−xy)+(y−x)   
[Regrouping the expressions]
=x(x−y)+(y−x)
=x(x−y)−(x−y)     [∵(y−x)=−(x−y)] 
=(x−1)(x−y)           [Taking (x−y) as the common expression]