RD Sharma 2020 solution class 9 chapter 5 Factorization of Algebraic Expressions Exercise 5.1

Exercise 5.1

Page-5.9

Question 1:

Factorize:

1. $x^3 + x - 3x^2 - 3$

Answer 1:

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 x² + 1from the two terms. That is

We cannot further factorize the expression. 

So, the required factorization is.

Question 2:

Factorize:

2. a(a+b)³ − 3a²b (a + b)
 

Answer 2:

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. $x\left(x^3-y^3\right) + 3xy (x-y)$

Answer 3:

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:

a²x² + (ax² + 1)x + a

Answer 4:

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 commonfrom the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Question 5:

Factorize:
x² + y − xy − x

Answer 5:

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 commonfrom the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Question 6:

Factorize:

x³ − 2x²y + 3xy² − 6y³

Answer 6:

The given expression to be factorized is

Take common from the first two terms and from the last two terms. That is

Finally, take commonfrom the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Question 7:

Factorize:

6ab − b² + 12ac − 2bc

Answer 7:

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 commonfrom the two terms. That is

We cannot further factorize the expression. 

So, the required factorization of is.

Question 8:

Factorize:

$x(x-2)(x-4) + 4x -8$

Answer 8:

The given expression to be factorized is

Take common 4 from the last two terms. That is

Again take commonfrom 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 + 2(a − b + c) (b − c + a)

Answer 9:

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:

a² + 2ab +b² − c²

Answer 10:

The given expression to be factorized is

This can be arrange in the form

Substitutingin the above expression, we get.

Put.

We cannot further factorize the expression. 

So, the required factorization of is.

Question 11:

Factorize:

a² + 4b² − 4ab − 4c²

Answer 11:

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:

x² − y² − 4xz + 4z²

Answer 12:

The given expression to be factorized is

Rearrange the terms as

Substitutingin the avove expression, 

Put.

We cannot further factorize the expression. 

So, the required factorization ofis.

Question 13:

Factorize:

$2x^2-\frac{5}{6}x+\frac{1}{12}$

Answer 13:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 14:

Factorize:

$x^2+\frac{12}{35}x+\frac{1}{35}$

Answer 14:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 15:

Factorize:

$21x^2-2x+\frac{1}{21}$

Answer 15:

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression. 

So, the required factorization of is.

Question 16:

Give possible expressions for the length and breadth of the rectangle having 35y² + 13y − 12 as its area.

Answer 16:

The area of the rectangle is

First we will factorize the above expression. This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

The area of a rectangle having length a and breadth bis ab.

Here we don’t know the bigger or the smaller factor. So, the two possibilities are

(i) Length isand breadth is

(ii) Length is and breadth is

Question 17:

What are the possible expressions for the dimensions of the cuboid whose volume is 3x²− 12x.

Answer 17:

The volume of the cuboid is

First we will factorize the above expression. 

Take commonfrom the two terms of the above expression,

The volume of a cuboid having length, breadth b and height is.

Here the word ‘dimensions’ stands for the length, breadth and height of the cuboid. So, the three possibilities are

(i) Length is, breadth is x and height is

(ii) Length is x , breadth isand height is

(iii) Length is, breadth isand height is x

There are many other possibilities also, because we can consider the product of two simple factors as a single factor.

Question 18:

Factorize:

$\left(x^2+\frac{1}{x^2}\right) - 4 \left(x + \frac{1}{x}\right)+6$

Answer 18:

The given expression to be factorized is

We have

Use the above result in the original expression to get

Substituting in the above , we get

Put.

We cannot further factorize the expression. 

So, the required factorization of is.

Question 19:

Factorize:

(x+2) (x²+25) − 10x² − 20x

Answer 19:

The given expression to be factorized is

Take common from the last two terms. That is

Again take commonfrom the two terms of the above expression. Then

We cannot further factorize the expression. 

So, the required factorization of is.

Question 20:

Factorize:

$2a^2 + 2\sqrt{6}ab+3b^2$
 

Answer 20:

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression. 

So, the required factorization of is.

Question 21:

Factorize:

a² + b² + 2(ab + bc + ca)

Answer 21:

The given expression to be factorized is

This can be written as

Take commonfrom the last two terms.

Finally, take common from the two terms of the above expression.

We cannot further factorize the expression. 

So, the required factorization of is.

Question 22:

Factorize:

4(x − y)² − 12(x − y) (x + y) + 9(x + y)²

Answer 22:

The given expression to be factorized is

Substituting andin the above expression, we get

=

This can be arrange in the form

Putand.

Take common -1 from the expression within the braces.

We cannot further factorize the expression. 

So, the required factorization of is.

Question 23:

Factorize:
a² + b² + 2bc − c²

Answer 23:

The given expression to be factorized is

This can be arrange in the form

Substitutingin the above expression, we get.

Put.

We cannot further factorize the expression. 

So, the required factorization of is.

Question 24:

Factorize:

xy⁹ − yx⁹

Answer 24:

The given expression to be factorized is

This can be written in the form

Take commonfrom the two terms of the above expression

We cannot further factorize the expression. 

So, the required factorization of is

Question 25:

Factorize:

x⁴ + x²y² + y⁴

Answer 25:

The given expression to be factorized is

Add and subtract the termin the given expression.

Substitutingin the above expression, we get

Putin the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 26:

Factorize:

$x^2 + 6\sqrt{2}x+10$

Answer 26:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms.

Finally, take commonfrom the above expression. Then we have

We cannot further factorize the expression.

So, the required factorization is.

Question 27:

Factorize:

$x^2-2\sqrt{2}x-30$

Answer 27:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 28:

Factorize:

$x^2-\sqrt{3}x-6$

Answer 28:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms. Then we have

Finally take commonfrom the above expression. Then we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 29:

Factorize:

$x^2+5\sqrt{5}x + 30$

Answer 29:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms. Then we have

Finally take commonfrom the above expression,

We cannot further factorize the expression.

So, the required factorization of is.

Question 30:

Factorize:

$x^2+2\sqrt{3}x-24$

Answer 30:

The given expression to be factorized is

This can be written in the form

Take common x from the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 31:

Factorize:

$5\sqrt{5}{x}^2+20x+3\sqrt{5}$

Answer 31:

The given expression to be factorized is

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 32:

Factorize:

$2x^2+3\sqrt{5}x+5$

Answer 32:

The given expression to be factorized is

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 33:

Factorize:

9(2a − b)² − 4(2a − b) − 13

Answer 33:

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take common x from the first two terms and 1 from the last two terms,

Finally take commonfrom the above expression,

Put,

We cannot further factorize the expression. 

So, the required factorization ofis.

Question 34:

Factorize:

7(x − 2y)² − 25(x − 2y) + 12

Answer 34:

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

Put in the above expression,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 35:

Factorize:

2(x + y)² − 9(x + y) − 5

Answer 35:

The given expression to be factorized is

Substitutingin the above expression, we get

This can be written in the form

Take commonfrom the first two terms andfrom the last two terms,

Finally take commonfrom the above expression,

Put. Then we have

We cannot further factorize the expression. 

So, the required factorization of is.