Factorization - Exercise 7.2 | Mathematics (Maths) Class 8 PDF Download

 Page 1


                      
          
          
          
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Q u e s t i o n : 1 8
Factorize the following:
3x - 9
S o l u t i o n :
The greatest common factor of the terms 3x and -9 of the expression 3x - 9 is 3. 
Now.
3x = 3x
and
-9 = 3.-3
Hence, the expression 3x - 9 can be factorised as 3x -3.
Q u e s t i o n : 1 9
Factorize the following:
5x - 15x
2
S o l u t i o n :
The greatest common factor of the terms 5x and 15x
2
 of the expression 5x - 15x
2
 is 5x. 
Now,
5x = 5x × 1 
and
-15x
2
 = 5x × -3x
Hence, the expression 5x - 15x
2 
can be factorised as 5x1 -3x ?.
Q u e s t i o n : 2 0
Factorize the following:
20a
12
b
2
 - 15a
8
b
4
S o l u t i o n :
The greatest common factor of the terms 20a
12
b
2
 and -15a
8
b
4
 of the expression 20a
12
b
2
 - 15a
8
b
4
 is 5a
8
b
2
.
20a
12
b
2
 = 5×4×a
8
×a
4
×b
2
 = 5a
8
×b
2
×4a
4
 and -15a
8
b
4 
= 5×-3×a
8
×b
2
×b
2
 = 5a
8
b
2 
× -3b
2
Hence, the expression 20a
12
b
2
 - 15a
8
b
4 
can be factorised as 5a
8
b
2
(4a
4
-3b
2
) ?? Q u e s t i o n : 2 1
Factorize the following:
72x
6
y
7
 - 96x
7
y
6
S o l u t i o n :
The greatest common factor of the terms 72x
6
y
7
 and -96x
7
y
6
 of the expression 72x
6
y
7
 - 96x
7
y
64
 is 24x
6
y
6
.
Now,
72x
6
y
7
 = 24x
6
y
6
? × 3y
and 
-96x
7
y
6
? = 24x
6
y
6
 × -4x
Hence, the expression 72x
6
y
7
 - 96x
7
y
6
 can be factorised as 24x
6
y
6
3y -4x ?.
Q u e s t i o n : 2 2
Factorize the following:
20x
3
 - 40x
2
 + 80x
S o l u t i o n :
The greatest common factor of the terms 20x
3
?, -40x
2
? and 80x ? of the expression 20x
3
 - 40x
2
 + 80x ? is 20x.
Now,
20x
3
? = 20x × x
2
-40x
2
? = 20x × -2x
and
80x ?= 20x × 4
Hence, the expression 20x
3
 - 40x
2
 + 80x ? ?can be factorised as 20x(x
2
 - 2x + 4) ?.
Q u e s t i o n : 2 3
Factorize the following:
2x
3
y
2
 - 4x
2
y
3
 + 8xy
4
S o l u t i o n :
The greatest common factor of the terms 2x
3
y
2
, -4x
2
y
3
 and 8xy
4 
of the expression 2x
3
y
2
 - 4x
2
y
3
+ 8xy
4
y
64
 is 2xy
2
.
 
Now,
2x
3
y
2
 = 2xy
2  
× x
2  
-4x
2
y
3 
= 2xy
2
 × -2xy
Page 2


                      
          
          
          
?      
             
    
         
                      
          
          
          
?      
Q u e s t i o n : 1 8
Factorize the following:
3x - 9
S o l u t i o n :
The greatest common factor of the terms 3x and -9 of the expression 3x - 9 is 3. 
Now.
3x = 3x
and
-9 = 3.-3
Hence, the expression 3x - 9 can be factorised as 3x -3.
Q u e s t i o n : 1 9
Factorize the following:
5x - 15x
2
S o l u t i o n :
The greatest common factor of the terms 5x and 15x
2
 of the expression 5x - 15x
2
 is 5x. 
Now,
5x = 5x × 1 
and
-15x
2
 = 5x × -3x
Hence, the expression 5x - 15x
2 
can be factorised as 5x1 -3x ?.
Q u e s t i o n : 2 0
Factorize the following:
20a
12
b
2
 - 15a
8
b
4
S o l u t i o n :
The greatest common factor of the terms 20a
12
b
2
 and -15a
8
b
4
 of the expression 20a
12
b
2
 - 15a
8
b
4
 is 5a
8
b
2
.
20a
12
b
2
 = 5×4×a
8
×a
4
×b
2
 = 5a
8
×b
2
×4a
4
 and -15a
8
b
4 
= 5×-3×a
8
×b
2
×b
2
 = 5a
8
b
2 
× -3b
2
Hence, the expression 20a
12
b
2
 - 15a
8
b
4 
can be factorised as 5a
8
b
2
(4a
4
-3b
2
) ?? Q u e s t i o n : 2 1
Factorize the following:
72x
6
y
7
 - 96x
7
y
6
S o l u t i o n :
The greatest common factor of the terms 72x
6
y
7
 and -96x
7
y
6
 of the expression 72x
6
y
7
 - 96x
7
y
64
 is 24x
6
y
6
.
Now,
72x
6
y
7
 = 24x
6
y
6
? × 3y
and 
-96x
7
y
6
? = 24x
6
y
6
 × -4x
Hence, the expression 72x
6
y
7
 - 96x
7
y
6
 can be factorised as 24x
6
y
6
3y -4x ?.
Q u e s t i o n : 2 2
Factorize the following:
20x
3
 - 40x
2
 + 80x
S o l u t i o n :
The greatest common factor of the terms 20x
3
?, -40x
2
? and 80x ? of the expression 20x
3
 - 40x
2
 + 80x ? is 20x.
Now,
20x
3
? = 20x × x
2
-40x
2
? = 20x × -2x
and
80x ?= 20x × 4
Hence, the expression 20x
3
 - 40x
2
 + 80x ? ?can be factorised as 20x(x
2
 - 2x + 4) ?.
Q u e s t i o n : 2 3
Factorize the following:
2x
3
y
2
 - 4x
2
y
3
 + 8xy
4
S o l u t i o n :
The greatest common factor of the terms 2x
3
y
2
, -4x
2
y
3
 and 8xy
4 
of the expression 2x
3
y
2
 - 4x
2
y
3
+ 8xy
4
y
64
 is 2xy
2
.
 
Now,
2x
3
y
2
 = 2xy
2  
× x
2  
-4x
2
y
3 
= 2xy
2
 × -2xy
8xy
4
? = 2xy
2 
× 4y
2
Hence, the expression 2x
3
y
2
 - 4x
2
y
3
 + 8xy
4
 ?can be factorised as 2xy
2
(x
2
 - 2xy + 4y
2
) ?.
Q u e s t i o n : 2 4
Factorize the following:
10m
3
n
2
 + 15m
4
n - 20m
2
n
3
S o l u t i o n :
The greatest common factor of the terms 10m
3
n
2
, 15m
4
n and -20m
2
n
3
 of the expression 10m
3
n
2
 + 15m
4
n - 20m
2
n
3
 is 5m
2
n ?.
Now,
10m
3
n
2 
= 5m
2
n ? × 2mn
15m
4
n = 5m
2
n × 3m
2
-20m
2
n
3 
?= 5m
2
n × -4n
2
Hence, 10m
3
n
2
 + 15m
2
n - 20m
2
n
3
 ?can be factorised as 5m
2
n(2mn + 3m
2
 - 4n
2
). ? Q u e s t i o n : 2 5
Factorize the following:
2a
4
b
4
 - 3a
3
b
5
 + 4a
2
b
5
S o l u t i o n :
The greatest common factor of the terms 2a
4
b
4
, -3a
3
b
5
? and 4a
2
b
5
 of the expression 2a
4
b
4
 - 3a
3
b
5
 + 4a
2
b
5
 is a
2
b
4
?.
Now,
2a
4
b
4 
= a
2
b
4 
× 2a
2
-3a
3
b
5 
= a
2
?b
4
 × -3ab 
4a
2
b
5 
= a
2
?b
4
 × 4b
Hence, (2a
4
b
4
 - 3a
3
b
5
 + 4a
2
b
5
) ?can be factorised as [a
2
b
4
(2a
2
 - 3ab + 4b)] ?.
Q u e s t i o n : 2 6
Factorize the following:
28a
2
 + 14a
2
b
2
 - 21a
4
S o l u t i o n :
The greatest common factor of the terms 28a
2
, 14a
2
b
2
 and 21a
4
 of the expression 28a
2
+14a
2
b
2
-21a
4
 is 7a
2
. Also, we can write 28a
2
= 7a
2
×4, 14a
2
b
2
= 7a
2
×2b
2
 and 21a
4
= 7a
2
×3a
2
. ?
Q u e s t i o n : 2 7
Factorize the following:
a
4
b - 3a
2
b
2
 - 6ab
3
S o l u t i o n :
The greatest common factor of the terms a
4
b, 3a
2
b
2
 and 6ab
3
 of the expression a
4
b -3a
2
b
2
-6ab
3
 is ab. Also, we can write a
4
b = ab ×a
3
, 3a
2
b
2
= ab ×3ab and 6ab
3
= ab ×6b
2
. ? a
4
b -3a
2
b
2
Q u e s t i o n : 2 8
Factorize the following:
2l
2
mn - 3lm
2
n + 4lmn
2
S o l u t i o n :
The greatest common factor of the terms 2l
2
mn, 3lm
2
n and 4lmn
2
 of the expression 2l
2
mn -3lm
2
n +4lmn
2
 is lmn. Also, we can write 2l
2
mn = lmn ×2l, 3lm
2
n = lmn ×3m and 4lmn
2
= lmn ×4
Q u e s t i o n : 2 9
Factorize the following:
x
4
y
2
 - x
2
y
4
 - x
4
y
4
S o l u t i o n :
The greatest common factor of the terms x
4
y
2
, x
2
y
4
 and x
4
y
4
 of the expression x
4
y
2
-x
2
y
4
-x
4
y
4
 is x
2
y
2
. Also, we can write x
4
y
2
= x
2
y
2
×x
2
, x
2
y
4
= x
2
y
2
×y
2
 and x
4
y
4
= x
2
y
2
×x
2
y
2
. ? x
4
y
2
-
Q u e s t i o n : 3 0
Factorize the following:
9x
2
y + 3axy
S o l u t i o n :
The greatest common factor of the terms 9x
2
y and 3axy of the expression 9x
2
y +3axy is 3xy. Also, we can write 9x
2
y = 3xy ×3x and 3axy = 3xy ×a. ? 9x
2
y +3axy = 3xy ×3x +3xy ×a     
Q u e s t i o n : 3 1
Factorize the following:
16m - 4m
2
S o l u t i o n :
The greatest common factor of the terms 16m and 4m
2 
of the expression 16m -4m
2 
is 4m. Also, we can write 16m = 4m ×4 and 4m
2
= 4m ×m. ? 16m -4m
2
= 4m ×4 -4m ×m                        
Q u e s t i o n : 3 2
Factorize the following:
-4a
2
 + 4ab - 4ca
S o l u t i o n :
The greatest common factor of the terms -4a
2
, 4ab and - 4ca of the expression -4a
2
+4ab -4ca is -4a. Also, we can write -4a
2
= -4a ×a, 4ab = -4a × -b and 4ca = -4a ×c. ? -4a
2
+4ab
Q u e s t i o n : 3 3
Factorize the following:
( )
Page 3


                      
          
          
          
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Q u e s t i o n : 1 8
Factorize the following:
3x - 9
S o l u t i o n :
The greatest common factor of the terms 3x and -9 of the expression 3x - 9 is 3. 
Now.
3x = 3x
and
-9 = 3.-3
Hence, the expression 3x - 9 can be factorised as 3x -3.
Q u e s t i o n : 1 9
Factorize the following:
5x - 15x
2
S o l u t i o n :
The greatest common factor of the terms 5x and 15x
2
 of the expression 5x - 15x
2
 is 5x. 
Now,
5x = 5x × 1 
and
-15x
2
 = 5x × -3x
Hence, the expression 5x - 15x
2 
can be factorised as 5x1 -3x ?.
Q u e s t i o n : 2 0
Factorize the following:
20a
12
b
2
 - 15a
8
b
4
S o l u t i o n :
The greatest common factor of the terms 20a
12
b
2
 and -15a
8
b
4
 of the expression 20a
12
b
2
 - 15a
8
b
4
 is 5a
8
b
2
.
20a
12
b
2
 = 5×4×a
8
×a
4
×b
2
 = 5a
8
×b
2
×4a
4
 and -15a
8
b
4 
= 5×-3×a
8
×b
2
×b
2
 = 5a
8
b
2 
× -3b
2
Hence, the expression 20a
12
b
2
 - 15a
8
b
4 
can be factorised as 5a
8
b
2
(4a
4
-3b
2
) ?? Q u e s t i o n : 2 1
Factorize the following:
72x
6
y
7
 - 96x
7
y
6
S o l u t i o n :
The greatest common factor of the terms 72x
6
y
7
 and -96x
7
y
6
 of the expression 72x
6
y
7
 - 96x
7
y
64
 is 24x
6
y
6
.
Now,
72x
6
y
7
 = 24x
6
y
6
? × 3y
and 
-96x
7
y
6
? = 24x
6
y
6
 × -4x
Hence, the expression 72x
6
y
7
 - 96x
7
y
6
 can be factorised as 24x
6
y
6
3y -4x ?.
Q u e s t i o n : 2 2
Factorize the following:
20x
3
 - 40x
2
 + 80x
S o l u t i o n :
The greatest common factor of the terms 20x
3
?, -40x
2
? and 80x ? of the expression 20x
3
 - 40x
2
 + 80x ? is 20x.
Now,
20x
3
? = 20x × x
2
-40x
2
? = 20x × -2x
and
80x ?= 20x × 4
Hence, the expression 20x
3
 - 40x
2
 + 80x ? ?can be factorised as 20x(x
2
 - 2x + 4) ?.
Q u e s t i o n : 2 3
Factorize the following:
2x
3
y
2
 - 4x
2
y
3
 + 8xy
4
S o l u t i o n :
The greatest common factor of the terms 2x
3
y
2
, -4x
2
y
3
 and 8xy
4 
of the expression 2x
3
y
2
 - 4x
2
y
3
+ 8xy
4
y
64
 is 2xy
2
.
 
Now,
2x
3
y
2
 = 2xy
2  
× x
2  
-4x
2
y
3 
= 2xy
2
 × -2xy
8xy
4
? = 2xy
2 
× 4y
2
Hence, the expression 2x
3
y
2
 - 4x
2
y
3
 + 8xy
4
 ?can be factorised as 2xy
2
(x
2
 - 2xy + 4y
2
) ?.
Q u e s t i o n : 2 4
Factorize the following:
10m
3
n
2
 + 15m
4
n - 20m
2
n
3
S o l u t i o n :
The greatest common factor of the terms 10m
3
n
2
, 15m
4
n and -20m
2
n
3
 of the expression 10m
3
n
2
 + 15m
4
n - 20m
2
n
3
 is 5m
2
n ?.
Now,
10m
3
n
2 
= 5m
2
n ? × 2mn
15m
4
n = 5m
2
n × 3m
2
-20m
2
n
3 
?= 5m
2
n × -4n
2
Hence, 10m
3
n
2
 + 15m
2
n - 20m
2
n
3
 ?can be factorised as 5m
2
n(2mn + 3m
2
 - 4n
2
). ? Q u e s t i o n : 2 5
Factorize the following:
2a
4
b
4
 - 3a
3
b
5
 + 4a
2
b
5
S o l u t i o n :
The greatest common factor of the terms 2a
4
b
4
, -3a
3
b
5
? and 4a
2
b
5
 of the expression 2a
4
b
4
 - 3a
3
b
5
 + 4a
2
b
5
 is a
2
b
4
?.
Now,
2a
4
b
4 
= a
2
b
4 
× 2a
2
-3a
3
b
5 
= a
2
?b
4
 × -3ab 
4a
2
b
5 
= a
2
?b
4
 × 4b
Hence, (2a
4
b
4
 - 3a
3
b
5
 + 4a
2
b
5
) ?can be factorised as [a
2
b
4
(2a
2
 - 3ab + 4b)] ?.
Q u e s t i o n : 2 6
Factorize the following:
28a
2
 + 14a
2
b
2
 - 21a
4
S o l u t i o n :
The greatest common factor of the terms 28a
2
, 14a
2
b
2
 and 21a
4
 of the expression 28a
2
+14a
2
b
2
-21a
4
 is 7a
2
. Also, we can write 28a
2
= 7a
2
×4, 14a
2
b
2
= 7a
2
×2b
2
 and 21a
4
= 7a
2
×3a
2
. ?
Q u e s t i o n : 2 7
Factorize the following:
a
4
b - 3a
2
b
2
 - 6ab
3
S o l u t i o n :
The greatest common factor of the terms a
4
b, 3a
2
b
2
 and 6ab
3
 of the expression a
4
b -3a
2
b
2
-6ab
3
 is ab. Also, we can write a
4
b = ab ×a
3
, 3a
2
b
2
= ab ×3ab and 6ab
3
= ab ×6b
2
. ? a
4
b -3a
2
b
2
Q u e s t i o n : 2 8
Factorize the following:
2l
2
mn - 3lm
2
n + 4lmn
2
S o l u t i o n :
The greatest common factor of the terms 2l
2
mn, 3lm
2
n and 4lmn
2
 of the expression 2l
2
mn -3lm
2
n +4lmn
2
 is lmn. Also, we can write 2l
2
mn = lmn ×2l, 3lm
2
n = lmn ×3m and 4lmn
2
= lmn ×4
Q u e s t i o n : 2 9
Factorize the following:
x
4
y
2
 - x
2
y
4
 - x
4
y
4
S o l u t i o n :
The greatest common factor of the terms x
4
y
2
, x
2
y
4
 and x
4
y
4
 of the expression x
4
y
2
-x
2
y
4
-x
4
y
4
 is x
2
y
2
. Also, we can write x
4
y
2
= x
2
y
2
×x
2
, x
2
y
4
= x
2
y
2
×y
2
 and x
4
y
4
= x
2
y
2
×x
2
y
2
. ? x
4
y
2
-
Q u e s t i o n : 3 0
Factorize the following:
9x
2
y + 3axy
S o l u t i o n :
The greatest common factor of the terms 9x
2
y and 3axy of the expression 9x
2
y +3axy is 3xy. Also, we can write 9x
2
y = 3xy ×3x and 3axy = 3xy ×a. ? 9x
2
y +3axy = 3xy ×3x +3xy ×a     
Q u e s t i o n : 3 1
Factorize the following:
16m - 4m
2
S o l u t i o n :
The greatest common factor of the terms 16m and 4m
2 
of the expression 16m -4m
2 
is 4m. Also, we can write 16m = 4m ×4 and 4m
2
= 4m ×m. ? 16m -4m
2
= 4m ×4 -4m ×m                        
Q u e s t i o n : 3 2
Factorize the following:
-4a
2
 + 4ab - 4ca
S o l u t i o n :
The greatest common factor of the terms -4a
2
, 4ab and - 4ca of the expression -4a
2
+4ab -4ca is -4a. Also, we can write -4a
2
= -4a ×a, 4ab = -4a × -b and 4ca = -4a ×c. ? -4a
2
+4ab
Q u e s t i o n : 3 3
Factorize the following:
( )
x
2
yz + xy
2
z + xyz
2
S o l u t i o n :
The greatest common factor of the terms x
2
yz, xy
2
z and xyz
2
 of the expression x
2
yz +xy
2
z +xyz
2
 is xyz. Also, we can write x
2
yz = xyz ×x, xy
2
z = xyz ×y and xyz
2
= xyz ×z. ? x
2
yz +xy
Q u e s t i o n : 3 4
Factorize the following:
ax
2
y + bxy
2
 + cxyz
S o l u t i o n :
The greatest common factor of the terms ax
2
y, bxy
2
 and cxyz of the expression ax
2
y +bxy
2
+cxyz is xy. Also, we can write ax
2
y = xy ×ax, bxy
2
= xy ×by and cxyz = xy ×cz. ? ax
2
y +bx