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Factorization - Exercise 7.4 | Mathematics (Maths) Class 8 PDF Download

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Q u e s t i o n : 5 0
Factorize each of the following expressions:
qr - pr + qs - ps
S o l u t i o n :
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]
Q u e s t i o n : 5 1
Factorize each of the following expressions:
p
2
q - pr
2
 - pq + r
2
S o l u t i o n :
p
2
q -pr
2
-pq +r
2
= p
2
q -pq + r
2
-pr
2
   Grouping the expressions = pq p -1 +r
2
1 -p = pq p -1 -r
2
p -1      ? 1 -p = - p -1 = pq -r
2
p -1              Taking p -1 as the commo
Q u e s t i o n : 5 2
Factorize each of the following expressions:
1 + x + xy + x
2
y
S o l u t i o n :
1 +x +xy +x
2
y = 1 +x + xy +x
2
y   Grouping the expressions = 1 +x +xy 1 +x = 1 +xy 1 +x           Taking 1 +x as the common factor
Q u e s t i o n : 5 3
Factorize each of the following expressions:
ax + ay - bx - by
S o l u t i o n :
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]
Q u e s t i o n : 5 4
Factorize each of the following expressions:
xa
2
 + xb
2
 - ya
2
 - yb
2
S o l u t i o n :
xa
2
+xb
2
-ya
2
-yb
2
= xa
2
+xb
2
- ya
2
+yb
2
   Grouping the expressions = x a
2
+b
2
-y a
2
+b
2
= x -y a
2
+b
2
                   Taking a
2
+b
2
 as the common factor
Q u e s t i o n : 5 5
Factorize each of the following expressions:
x
2
 + xy + xz + yz
S o l u t i o n :
x
2
+xy +xz +yz = x
2
+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
Q u e s t i o n : 5 6
Factorize each of the following expressions:
2ax + bx + 2ay + by
S o l u t i o n :
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]
Q u e s t i o n : 5 7
Factorize each of the following expressions:
ab - by - ay + y
2
S o l u t i o n :
ab -by -ay +y
2
= ab -ay + y
2
-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
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Q u e s t i o n : 5 0
Factorize each of the following expressions:
qr - pr + qs - ps
S o l u t i o n :
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]
Q u e s t i o n : 5 1
Factorize each of the following expressions:
p
2
q - pr
2
 - pq + r
2
S o l u t i o n :
p
2
q -pr
2
-pq +r
2
= p
2
q -pq + r
2
-pr
2
   Grouping the expressions = pq p -1 +r
2
1 -p = pq p -1 -r
2
p -1      ? 1 -p = - p -1 = pq -r
2
p -1              Taking p -1 as the commo
Q u e s t i o n : 5 2
Factorize each of the following expressions:
1 + x + xy + x
2
y
S o l u t i o n :
1 +x +xy +x
2
y = 1 +x + xy +x
2
y   Grouping the expressions = 1 +x +xy 1 +x = 1 +xy 1 +x           Taking 1 +x as the common factor
Q u e s t i o n : 5 3
Factorize each of the following expressions:
ax + ay - bx - by
S o l u t i o n :
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]
Q u e s t i o n : 5 4
Factorize each of the following expressions:
xa
2
 + xb
2
 - ya
2
 - yb
2
S o l u t i o n :
xa
2
+xb
2
-ya
2
-yb
2
= xa
2
+xb
2
- ya
2
+yb
2
   Grouping the expressions = x a
2
+b
2
-y a
2
+b
2
= x -y a
2
+b
2
                   Taking a
2
+b
2
 as the common factor
Q u e s t i o n : 5 5
Factorize each of the following expressions:
x
2
 + xy + xz + yz
S o l u t i o n :
x
2
+xy +xz +yz = x
2
+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
Q u e s t i o n : 5 6
Factorize each of the following expressions:
2ax + bx + 2ay + by
S o l u t i o n :
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]
Q u e s t i o n : 5 7
Factorize each of the following expressions:
ab - by - ay + y
2
S o l u t i o n :
ab -by -ay +y
2
= ab -ay + y
2
-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
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( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ]
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ] ( ) ( )
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ( )] ( ) ( ) [ (
Q u e s t i o n : 5 8
Factorize each of the following expressions:
axy + bcxy - az - bcz
S o l u t i o n :
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]
Q u e s t i o n : 5 9
Factorize each of the following expressions:
lm
2
 - mn
2
 - lm + n
2
S o l u t i o n :
lm
2
-mn
2
-lm +n
2
= lm
2
-lm + n
2
-mn
2
   Regrouping the expressions                                  = lm m -1 +n
2
1 -m                                  = lm m -1 -n
2
m -1      ? 1 -m = - m -1 
Q u e s t i o n : 6 0
Factorize each of the following expressions:
x
3
 - y
2
 + x - x
2
y
2
S o l u t i o n :
x
3
-y
2
+x -x
2
y
2
= x
3
+x - x
2
y
2
+y
2
         Regrouping the expressions = x x
2
+1 -y
2
x
2
+1 = x -y
2
x
2
+1                  Taking x
2
+1 as the common factor
Q u e s t i o n : 6 1
Factorize each of the following expressions:
6xy + 6 - 9y - 4x
S o l u t i o n :
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)]                             = (2
Q u e s t i o n : 6 2
Factorize each of the following expressions:
x
2
 - 2ax - 2ab + bx
S o l u t i o n :
x
2
-2ax -2ab +bx = x
2
-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
Q u e s t i o n : 6 3
Factorize each of the following expressions:
x
3
 - 2x
2
y + 3xy
2
 - 6y
3
S o l u t i o n :
x
3
- 2x
2
y + 3xy
2
- 6y
3
= x
3
-2x
2
y + 3xy
2
-6y
3
           Grouping the expressions = x
2
x -2y +3y
2
x -2y = x
2
+3y
2
x -2y                         Taking x -2y as the common factor
Q u e s t i o n : 6 4
Factorize each of the following expression:
abx
2
 + (ay - b) x - y
S o l u t i o n :
abx
2
+ ay -b x -y = abx
2
+axy -bx -y                                  = abx
2
-bx + axy -y    Regrouping the expressions                                  = bx ax -1 +y ax -1                                  = bx
Q u e s t i o n : 6 5
Factorize each of the following expression:
(ax + by)
2
 + (bx - ay)
2
S o l u t i o n :
ax +by)
2
+ bx -ay)
2
= a
2
x
2
+2abxy +b
2
y
2
+b
2
x
2
-2abxy +a
2
y
2
                                      = a
2
x
2
+b
2
y
2
+b
2
x
2
+a
2
y
2
                                      = a
2
x
2
+a
2
y
2
+ b
2
x
2
+b
2
y
2
   Regrouping the ex
Q u e s t i o n : 6 6
Factorize each of the following expression:
16(a - b)
3
 - 24 (a - b)
2
S o l u t i o n :
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
Q u e s t i o n : 6 7
Factorize each of the following expression:
ab(x
2
 + 1) + x(a
2
 + b
2
)
S o l u t i o n :
ab x
2
+1 +x a
2
+b
2
= abx
2
+ab +a
2
x +b
2
x                                      = abx
2
+a
2
x + b
2
x +ab   Regrouping the expressions                                      = ax bx +a +b bx +a                  
Q u e s t i o n : 6 8
Factorize each of the following expression:
a
2
x
2
 + (ax
2
 + 1)x + a
S o l u t i o n :
a
2
x
2
+ ax
2
+1 x +a = a
2
x
2
+ax
3
+x +a                                   = ax
3
+a
2
x
2
+ x +a   Regrouping the expressions                                   = ax
2
x +a + x +a                                   = ax
Q u e s t i o n : 6 9
Factorize each of the following expression:
a(a - 2b - c) + 2bc
S o l u t i o n :
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( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ]
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ] ( ) ( )
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ]
( ) ( ) ( ) [ ] ( ) ( ) (
( ( ( ) ( ) [
( ( ( [ ( ) ] { [ ( ] } ( ( )
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( ) ( ) ( ) [ ] ( ) ( ) (
( ) ( ) ( ) [ ] ( ) ( ) ( ) (
Page 3


     
      
     
                     
      
      
                          
      
          
                                     
      
           
                       
Q u e s t i o n : 5 0
Factorize each of the following expressions:
qr - pr + qs - ps
S o l u t i o n :
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]
Q u e s t i o n : 5 1
Factorize each of the following expressions:
p
2
q - pr
2
 - pq + r
2
S o l u t i o n :
p
2
q -pr
2
-pq +r
2
= p
2
q -pq + r
2
-pr
2
   Grouping the expressions = pq p -1 +r
2
1 -p = pq p -1 -r
2
p -1      ? 1 -p = - p -1 = pq -r
2
p -1              Taking p -1 as the commo
Q u e s t i o n : 5 2
Factorize each of the following expressions:
1 + x + xy + x
2
y
S o l u t i o n :
1 +x +xy +x
2
y = 1 +x + xy +x
2
y   Grouping the expressions = 1 +x +xy 1 +x = 1 +xy 1 +x           Taking 1 +x as the common factor
Q u e s t i o n : 5 3
Factorize each of the following expressions:
ax + ay - bx - by
S o l u t i o n :
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]
Q u e s t i o n : 5 4
Factorize each of the following expressions:
xa
2
 + xb
2
 - ya
2
 - yb
2
S o l u t i o n :
xa
2
+xb
2
-ya
2
-yb
2
= xa
2
+xb
2
- ya
2
+yb
2
   Grouping the expressions = x a
2
+b
2
-y a
2
+b
2
= x -y a
2
+b
2
                   Taking a
2
+b
2
 as the common factor
Q u e s t i o n : 5 5
Factorize each of the following expressions:
x
2
 + xy + xz + yz
S o l u t i o n :
x
2
+xy +xz +yz = x
2
+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
Q u e s t i o n : 5 6
Factorize each of the following expressions:
2ax + bx + 2ay + by
S o l u t i o n :
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]
Q u e s t i o n : 5 7
Factorize each of the following expressions:
ab - by - ay + y
2
S o l u t i o n :
ab -by -ay +y
2
= ab -ay + y
2
-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
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( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ]
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ]
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ] ( ) ( )
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ( )] ( ) ( ) [ (
Q u e s t i o n : 5 8
Factorize each of the following expressions:
axy + bcxy - az - bcz
S o l u t i o n :
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]
Q u e s t i o n : 5 9
Factorize each of the following expressions:
lm
2
 - mn
2
 - lm + n
2
S o l u t i o n :
lm
2
-mn
2
-lm +n
2
= lm
2
-lm + n
2
-mn
2
   Regrouping the expressions                                  = lm m -1 +n
2
1 -m                                  = lm m -1 -n
2
m -1      ? 1 -m = - m -1 
Q u e s t i o n : 6 0
Factorize each of the following expressions:
x
3
 - y
2
 + x - x
2
y
2
S o l u t i o n :
x
3
-y
2
+x -x
2
y
2
= x
3
+x - x
2
y
2
+y
2
         Regrouping the expressions = x x
2
+1 -y
2
x
2
+1 = x -y
2
x
2
+1                  Taking x
2
+1 as the common factor
Q u e s t i o n : 6 1
Factorize each of the following expressions:
6xy + 6 - 9y - 4x
S o l u t i o n :
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)]                             = (2
Q u e s t i o n : 6 2
Factorize each of the following expressions:
x
2
 - 2ax - 2ab + bx
S o l u t i o n :
x
2
-2ax -2ab +bx = x
2
-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
Q u e s t i o n : 6 3
Factorize each of the following expressions:
x
3
 - 2x
2
y + 3xy
2
 - 6y
3
S o l u t i o n :
x
3
- 2x
2
y + 3xy
2
- 6y
3
= x
3
-2x
2
y + 3xy
2
-6y
3
           Grouping the expressions = x
2
x -2y +3y
2
x -2y = x
2
+3y
2
x -2y                         Taking x -2y as the common factor
Q u e s t i o n : 6 4
Factorize each of the following expression:
abx
2
 + (ay - b) x - y
S o l u t i o n :
abx
2
+ ay -b x -y = abx
2
+axy -bx -y                                  = abx
2
-bx + axy -y    Regrouping the expressions                                  = bx ax -1 +y ax -1                                  = bx
Q u e s t i o n : 6 5
Factorize each of the following expression:
(ax + by)
2
 + (bx - ay)
2
S o l u t i o n :
ax +by)
2
+ bx -ay)
2
= a
2
x
2
+2abxy +b
2
y
2
+b
2
x
2
-2abxy +a
2
y
2
                                      = a
2
x
2
+b
2
y
2
+b
2
x
2
+a
2
y
2
                                      = a
2
x
2
+a
2
y
2
+ b
2
x
2
+b
2
y
2
   Regrouping the ex
Q u e s t i o n : 6 6
Factorize each of the following expression:
16(a - b)
3
 - 24 (a - b)
2
S o l u t i o n :
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
Q u e s t i o n : 6 7
Factorize each of the following expression:
ab(x
2
 + 1) + x(a
2
 + b
2
)
S o l u t i o n :
ab x
2
+1 +x a
2
+b
2
= abx
2
+ab +a
2
x +b
2
x                                      = abx
2
+a
2
x + b
2
x +ab   Regrouping the expressions                                      = ax bx +a +b bx +a                  
Q u e s t i o n : 6 8
Factorize each of the following expression:
a
2
x
2
 + (ax
2
 + 1)x + a
S o l u t i o n :
a
2
x
2
+ ax
2
+1 x +a = a
2
x
2
+ax
3
+x +a                                   = ax
3
+a
2
x
2
+ x +a   Regrouping the expressions                                   = ax
2
x +a + x +a                                   = ax
Q u e s t i o n : 6 9
Factorize each of the following expression:
a(a - 2b - c) + 2bc
S o l u t i o n :
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ( )]
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ]
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ] ( ) ( )
( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ( ) ]
( ) ( ) ( ) [ ] ( ) ( ) (
( ( ( ) ( ) [
( ( ( [ ( ) ] { [ ( ] } ( ( )
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( ) ( ) ( ) [ ] ( ) ( ) (
( ) ( ) ( ) [ ] ( ) ( ) ( ) (
a a -2b -c +2bc = a
2
-2ab -ac +2bc                                = a
2
-ac + 2bc -2ab             Regrouping the terms                                = a a -c +2b c -a                                = a a -c -2b
Q u e s t i o n : 7 0
Factorize each of the following expression:
a(a + b - c) - bc
S o l u t i o n :
a a +b -c -bc = a
2
+ab -ac -bc                            = a
2
-ac + ab -bc   Regrouping the expressions                            = a a -c +b a -c                            = a +b a -c               Taki
Q u e s t i o n : 7 1
Factorize each of the following expression:
x
2
 - 11xy - x + 11y
S o l u t i o n :
x
2
-11xy -x +11y = x
2
-x + 11y -11xy   Regrouping the expressions                                 = x x -1 +11y 1 -x                                 = x x -1 -11y x -1       ? 1 -x = - x -1        
Q u e s t i o n : 7 2
Factorize each of the following expression:
ab - a - b + 1
S o l u t i o n :
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 c
Q u e s t i o n : 7 3
Factorize each of the following expression:
x
2
 + y - xy - x
S o l u t i o n :
x
2
+y -xy -x = x
2
-xy + y -x   Regrouping the expressions                        = x x -y + y -x                        = x x -y - x -y     ? y -x = - x -y                        = x -1 x -y        
     
  
     
  
     
  
     
  
     
  
     
  
     
    
     
      
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