Factorization of Algebraic Expressions- 1 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

 Page 1


Question:1
Factorize:
1. x
3
 + x - 3x
2
 - 3
Solution:
The given expression to be factorized is
Take common x from the first two terms and -3 from the last two terms. That is
Finally, take common x
2
 + 1from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization is .
Question:2
Factorize:
2. aa +b
3
 - 3a
2
b a +b
 
Solution:
The given expression to be factorized is
Take common  from the two terms. That is
Expand the term within the second braces.
We cannot further factorize the expression. 
So, the required factorization of is .
Question:3
Factorize:
3. x x
3
-y
3
 + 3xy x -y
Solution:
( ) ( )
Page 2


Question:1
Factorize:
1. x
3
 + x - 3x
2
 - 3
Solution:
The given expression to be factorized is
Take common x from the first two terms and -3 from the last two terms. That is
Finally, take common x
2
 + 1from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization is .
Question:2
Factorize:
2. aa +b
3
 - 3a
2
b a +b
 
Solution:
The given expression to be factorized is
Take common  from the two terms. That is
Expand the term within the second braces.
We cannot further factorize the expression. 
So, the required factorization of is .
Question:3
Factorize:
3. x x
3
-y
3
 + 3xy x -y
Solution:
( ) ( )
The given expression to be factorized is
We know that
The given expression then becomes
Take common  from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:4
Factorize:
a
2
x
2
 + (ax
2
 + 1)x + a
Solution:
The given expression to be factorized is
Simplify the middle term. That is
Take common  from the first two terms and 1 from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:5
Factorize:
x
2
 + y - xy - x
Solution:
The given expression to be factorized is
Rearrange the given expression as
Page 3


Question:1
Factorize:
1. x
3
 + x - 3x
2
 - 3
Solution:
The given expression to be factorized is
Take common x from the first two terms and -3 from the last two terms. That is
Finally, take common x
2
 + 1from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization is .
Question:2
Factorize:
2. aa +b
3
 - 3a
2
b a +b
 
Solution:
The given expression to be factorized is
Take common  from the two terms. That is
Expand the term within the second braces.
We cannot further factorize the expression. 
So, the required factorization of is .
Question:3
Factorize:
3. x x
3
-y
3
 + 3xy x -y
Solution:
( ) ( )
The given expression to be factorized is
We know that
The given expression then becomes
Take common  from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:4
Factorize:
a
2
x
2
 + (ax
2
 + 1)x + a
Solution:
The given expression to be factorized is
Simplify the middle term. That is
Take common  from the first two terms and 1 from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:5
Factorize:
x
2
 + y - xy - x
Solution:
The given expression to be factorized is
Rearrange the given expression as
Take common x from the first two terms and -1 from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:6
Factorize:
x
3
 - 2x
2
y + 3xy
2
 - 6y
3
Solution:
The given expression to be factorized is
Take common  from the first two terms and  from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:7
Factorize:
6ab - b
2
 + 12ac - 2bc
Solution:
The given expression to be factorized is
Take common b from the first two terms and  from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:8
Factorize:
Page 4


Question:1
Factorize:
1. x
3
 + x - 3x
2
 - 3
Solution:
The given expression to be factorized is
Take common x from the first two terms and -3 from the last two terms. That is
Finally, take common x
2
 + 1from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization is .
Question:2
Factorize:
2. aa +b
3
 - 3a
2
b a +b
 
Solution:
The given expression to be factorized is
Take common  from the two terms. That is
Expand the term within the second braces.
We cannot further factorize the expression. 
So, the required factorization of is .
Question:3
Factorize:
3. x x
3
-y
3
 + 3xy x -y
Solution:
( ) ( )
The given expression to be factorized is
We know that
The given expression then becomes
Take common  from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:4
Factorize:
a
2
x
2
 + (ax
2
 + 1)x + a
Solution:
The given expression to be factorized is
Simplify the middle term. That is
Take common  from the first two terms and 1 from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:5
Factorize:
x
2
 + y - xy - x
Solution:
The given expression to be factorized is
Rearrange the given expression as
Take common x from the first two terms and -1 from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:6
Factorize:
x
3
 - 2x
2
y + 3xy
2
 - 6y
3
Solution:
The given expression to be factorized is
Take common  from the first two terms and  from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:7
Factorize:
6ab - b
2
 + 12ac - 2bc
Solution:
The given expression to be factorized is
Take common b from the first two terms and  from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:8
Factorize:
x(x -2)(x -4) + 4x -8
Solution:
The given expression to be factorized is
Take common 4 from the last two terms. That is
Again take common from the two terms of the above expression.
                                    
We cannot further factorize the expression. 
So, the required factorization of is .
Question:9
Factorize:
a -b +c
2
 + b -c +a
2
 + 2a -b +c
b -c +a
Solution:
The given expression to be factorized is
This can be written as
We cannot further factorize the expression. 
So, the required factorization of is .
Question:10
Factorize:
a
2
 + 2ab +b
2
 - c
2
Solution:
The given expression to be factorized is
This can be arrange in the form
Page 5


Question:1
Factorize:
1. x
3
 + x - 3x
2
 - 3
Solution:
The given expression to be factorized is
Take common x from the first two terms and -3 from the last two terms. That is
Finally, take common x
2
 + 1from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization is .
Question:2
Factorize:
2. aa +b
3
 - 3a
2
b a +b
 
Solution:
The given expression to be factorized is
Take common  from the two terms. That is
Expand the term within the second braces.
We cannot further factorize the expression. 
So, the required factorization of is .
Question:3
Factorize:
3. x x
3
-y
3
 + 3xy x -y
Solution:
( ) ( )
The given expression to be factorized is
We know that
The given expression then becomes
Take common  from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:4
Factorize:
a
2
x
2
 + (ax
2
 + 1)x + a
Solution:
The given expression to be factorized is
Simplify the middle term. That is
Take common  from the first two terms and 1 from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:5
Factorize:
x
2
 + y - xy - x
Solution:
The given expression to be factorized is
Rearrange the given expression as
Take common x from the first two terms and -1 from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:6
Factorize:
x
3
 - 2x
2
y + 3xy
2
 - 6y
3
Solution:
The given expression to be factorized is
Take common  from the first two terms and  from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:7
Factorize:
6ab - b
2
 + 12ac - 2bc
Solution:
The given expression to be factorized is
Take common b from the first two terms and  from the last two terms. That is
Finally, take common from the two terms. That is
We cannot further factorize the expression. 
So, the required factorization of is .
Question:8
Factorize:
x(x -2)(x -4) + 4x -8
Solution:
The given expression to be factorized is
Take common 4 from the last two terms. That is
Again take common from the two terms of the above expression.
                                    
We cannot further factorize the expression. 
So, the required factorization of is .
Question:9
Factorize:
a -b +c
2
 + b -c +a
2
 + 2a -b +c
b -c +a
Solution:
The given expression to be factorized is
This can be written as
We cannot further factorize the expression. 
So, the required factorization of is .
Question:10
Factorize:
a
2
 + 2ab +b
2
 - c
2
Solution:
The given expression to be factorized is
This can be arrange in the form
Substituting in the above expression, we get.
Put .
We cannot further factorize the expression. 
So, the required factorization of is .
Question:11
Factorize:
a
2
 + 4b
2
 - 4ab - 4c
2
Solution:
The given expression to be factorized is
This can be arrange in the form
Substitute .
Put .
We cannot further factorize the expression. 
So, the required factorization of is .
Question:12
Factorize:
x
2
 - y
2
 - 4xz + 4z
2
Solution:
The given expression to be factorized is
Rearrange the terms as