RD Sharma 2020 solution class 9 chapter 5 Factorization of Algebraic Expressions Exercise 5.2

Exercise 5.2

Page-5.14

Question 1:

Factorize each of the following expressions:

 p³ + 27
 

Answer 1:

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 2:

y³ + 125

Answer 2:

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 3:

1 − 27a³

Answer 3:

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 4:

8x³y³ + 27a³

Answer 4:

The given expression to be factorized is

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 5:

64a³ − b³

Answer 5:

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 6:

$\frac{x^3}{216}-8y^3$

Answer 6:

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 7:

10x⁴y − 10xy⁴

Answer 7:

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 8:

54x⁶y + 2x³y⁴

Answer 8:

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 9:

32a³ + 108b³

Answer 9:

The given expression to be factorized is

Take common from the two terms,. Then we have

This can be written in the form

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 10:

(a − 2b)³ − 512b³

Answer 10:

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have 

We cannot further factorize the expression. 

So, the required factorization of is.

Question 11:

8x²y³ − x⁵

Answer 11:

The given expression to be factorized is

Take common. Then we have

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 12:

1029 − 3x³

Answer 12:

The given expression to be factorized is

Take common 3. Then we have from the above expression,

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 13:

x³y³ + 1

Answer 13:

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 14:

x⁴y⁴ − xy

Answer 14:

The given expression to be factorized is

Take common. Then we have from the above expression,

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 15:

a³ + b³ + a + b

Answer 15:

The given expression to be factorized is

This can be written as

=

Recall the formula for sum of two cubes

Using the above formula, we have

Take common. Then we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 16:

Simplify:

(i) $\frac{173\times 173\times 173+127\times 127\times 127}{173\times 173-173\times 127+127\times 127}$

(ii) $\frac{155\times 155\times 155-55\times 55\times 55}{155\times 155+155\times 55+55\times 55}$

(iii) $\frac{1.2\times 1.2\times 1.2-0.2\times 0.2\times 0.2}{1.2\times 1.2+1.2\times 0.2+0.2\times 0.2}$
 

Answer 16:

(i) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for sum of two cubes

Using the above formula, the expression becomes

Note that both and b are positive. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

(ii) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

(iii) The given expression is

Assumeand. Then the given expression can be rewritten as

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both, b is positive and unequal. So, neithernor any factor of it can be zero.

Therefore we can cancel the termfrom both numerator and denominator. Then the expression becomes

Question 17:

(a + b)³ − 8(a − b)³

Answer 17:

The given expression to be factorized is

This can be written in the form

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 18:

(x + 2)³ + (x − 2)³

Answer 18:

The given expression to be factorized is

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 19:

x⁶ + y⁶

Answer 19:

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 20:

a¹²+ b¹²

Answer 20:

The given expression to be factorized is

This can be written as

Recall the formula for difference of two cubes

Using the above formula, we have

We cannot further factorize the expression. 

So, the required factorization of is.

Question 21:

x³ + 6x² + 12x + 16

Answer 21:

The given expression to be factorized is

This can be written as

Take common x² from first two terms, 2x from the next two terms andfrom the last two terms. Then we have,

Finally, take common. Then we get,

We cannot further factorize the expression. 

So, the required factorization of is.

Question 22:

$a^3-\frac{1}{a^3} -2a+\frac{2}{a}$

Answer 22:

The given expression to be factorized is

This can be written as

Recall the formula for sum of two cubes

Using the above formula and taking common from the last two terms, we get

Take common. Then we have, 

We cannot further factorize the expression. 

So, the required factorization of is.

Question 23:

a³ + 3a²b + 3ab² + b³ − 8

Answer 23:

The given expression to be factorized is

Recall the well known formula

The given expression can be written as

Recall the formula for difference of two cubes

Using the above formula and taking common –2 from the last two terms, we get 

We cannot further factorize the expression. 

So, the required factorization of is.

Question 24:

8a³ − b³ − 4ax + 2bx

Answer 24:

The given expression to be factorized is

The given expression can be written as

Recall the formula for difference of two cubes

Using the above formula and taking common from the last two terms, we get 

Take common. Then we have,

We cannot further factorize the expression. 

So, the required factorization of is.