# Algebraic Identities

### Page no. 4.11 Ex. 4.2

Q1.

**Answer :**

In the given problem, we have to find expended form

(i) Given

We shall use the identity

By applying in identity we get

Hence the expended form of

is

(ii) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(iii) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(iv) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(v) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(vi) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(vii) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(viii) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(ix) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(x) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(xi) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

(xii) Given

We shall use the identity

Here

By applying in identity we get

Hence the expended form of

Q2.

**Answer :**

In the given problem, we have to simplify the expressions

(i) Given

By using identity

Hence the equation becomes

Taking 2 as common factor we get

Hence the simplified value of

(ii) Given

By using identity

Hence the equation becomes

Taking 4 as common factor we get

Hence the simplified value of

(iii) Given

By using identity

Taking 3 as a common factor we get

Hence the value of

(iv) Given

By using identity

By canceling the opposite terms, we get

Taking

Hence the value of

(v) We have

Using formula

By canceling the opposite terms, we get

Taking

Hence the value of

### Page no. 4.12 Ex. 4.2

Q3.

**Answer :**

**Answer :**

In the given problem, we have to find value of

Given

Squaring the equation

Now putting the value of

Taking 2 as common factor we get

Hence the value of

Q4.

**Answer :**

In the given problem, we have to find value of

Given

Multiply equation

Now adding both equation

We shall use the identity

Hence the value of

Q5.

**Answer :**

In the given problem, we have to find value of

Given

Squaring both sides of

Substituting

Hence the value of

Q6.

**Answer :**

In the given problem, we have to find value of

Given

We have

This equation can also be written as

Using the identity

Hence the value of

Q7.

**Answer :**

In the given problem, we have to simplify the value of each expression

(i) Given

We shall use the identity

By arranging the like terms we get

Now adding or subtracting like terms,

Hence the value of

(ii) Given

We shall use the identity

Now arranging liked terms we get,

Hence the value of

(iii) Given

We shall use the identity

Canceling the opposite term and simplifies

Hence the value of

### Page no. 4.19 Ex. 4.3

Q1.

**Answer :**

In the given problem, we have to find cube of the binomial expressions

(i) Given

We shall use the identity

Here

By applying the identity we get

Hence cube of the binomial expression

(ii) Given

We shall use the identity

Here

By applying the identity we get

Hence cube of the binomial expression of

(iii) Given

We shall use the identity

Here

By applying identity we get

Hence cube of the binomial expression of

(iv) Given

We shall use the identity

Here

By applying in identity we get

Hence cube of the binomial expression of

Q2.

**Answer :**

In the given problem, we have to simplify equation

(i) Given

We shall use the identity

Here

By applying identity we get

Hence simplified form of expression

(ii) Given

We shall use the identity

Here

By applying identity we get

By rearranging the variable we get

Hence the simplified value of

(iii) Given

We shall use the identity

Here

By applying identity we get

By rearranging the variable we get,

Hence the simplified value of

(iv) Given

We shall use the identity

Here

By applying the identity we get

By rearranging the variable we get,

Hence the simplified value of

Q3.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

Hence the value of

Q4.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

Hence the value of

Q5.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

Hence the value of

Q6.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

Hence the value of

Q7.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

Hence the value of

Q8.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

In order to find

Here

Hence the value of

Q9.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

In order to find

Hence the value of

Q10.

**Answer :**

In the given problem, we have to find the value of

Given

In order to find

Here putting,

Hence the value of

Q11.

**Answer :**

In the given problem, we have to find the value of

Given

In order to find

Here putting,

Hence the value of

### Page no. 4.20 Ex. 4.3

Q12.

**Answer :**

Given

We shall use the identity

Here putting

In order to find

In order to find

Hence the value of

Q13.

**Answer :**

In the given problem, we have to find the value of numbers

(i) Given

In order to find

We can write

Hence where

The value of

(ii) Given

In order to find

We can write

Hence where

The value of

(iii) Given

In order to find

We can write

Hence where

The value of

(iv) Given

In order to find

We can write

Hence where

The value of

(v) Given

In order to find

We can write

Hence where

The value of

(vi) Given

In order to find

We can write

Hence where

The value of

Q14.

**Answer :**

In the given problem, we have to find the value of numbers

(i) Given

We can write

We shall use the identity

Here

Hence the value of

(ii) Given

We can write

We shall use the identity

Here

Hence the value of

(iii) Given

We can write

We shall use the identity

Here

Hence the value of

(iv) Given

We can write

We shall use the identity

Here

Hence the value of

Q15.

**Answer :**

In the given problem, we have to find the value of

Given

We shall use the identity

Here putting

Again squaring on both sides we get,

We shall use the identity

Again cubing on both sides we get,

We shall use identity

Hence the value of

Q16.

**Answer :**

In the given problem, we have to find the value of

Given

By adding and subtracting

Again by adding and subtracting

Now cubing on both sides of we get

we shall use identity

Hence the value of

Q17.

**Answer :**

In the given problem, we have to find the value of

(i) Given

On cubing both sides we get,

We shall use identity

Hence the value of

(ii) Given

On cubing both sides we get,

We shall use identity

Hence the value of

Q18.

**Answer :**

From given problem we have to find the value of

Given

On cubing both sides of

We shall use identity

Hence the value of

Q19.

**Answer :**

In the given problem, we have to find the value of

Given

Cubing on both sides of

We shall use identity

Hence the value of

# Factorization Of Algebraic Expressions

### Page no. 5.13 Ex. 5.2

Q1.

**Answer :**

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

Q2.

**Answer :**

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

Q3.

**Answer :**

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

Q4.

**Answer :**

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

Q5.

**Answer :**

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

Q6.

**Answer :**

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

Q7.

**Answer :**

The given expression to be factorized is

Take common

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

Q8.

**Answer :**

The given expression to be factorized is

Take common

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

Q9.

**Answer :**

The given expression to be factorized is

Take common

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

### Page no. 5.14 Ex. 5.2

Q10.

**Answer :**

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

Q11.

**Answer :**

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

Q12.

**Answer :**

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

Q13.

**Answer :**

The given expression to be factorized is

Take common

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

Q14.

**Answer :**

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

Q15.

**Answer :**

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

Q16.

**Answer :**

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

Q17.

**Answer :**

The given expression to be factorized is

Take common

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

Q18.

**Answer :**

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

Q19.

**Answer :**

The given expression to be factorized is

This can be written as

Take common *x*^{2} from first two terms, 2*x* from the next two terms and

Finally, take common

We cannot further factorize the expression.

So, the required factorization of

Q20.

**Answer :**

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

We cannot further factorize the expression.

So, the required factorization of

Q21.

**Answer :**

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

Take common

We cannot further factorize the expression.

So, the required factorization of

Q22.

**Answer :**

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

Q23.

**Answer :**

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

Take common

We cannot further factorize the expression.

So, the required factorization of

Q24.

**Answer :**

(i) The given expression is

Assume

Recall the formula for sum of two cubes

Using the above formula, the expression becomes

Note that both *b* are positive. So, neither

Therefore we can cancel the term

(ii) The given expression is

Assume

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both*b* is positive and unequal. So, neither

Therefore we can cancel the term

(iii) The given expression is

Assume

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both*b* is positive and unequal. So, neither

Therefore we can cancel the term

### Page no. 5.17 Ex. 5.3

Q1.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of

Q2.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q3.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q4.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q5.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

This can be written in the following form

Recall the formula for the sum of two cubes

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q6.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q7.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common 6*xy* from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q8.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the sum of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q9.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common – 18*ab* from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q10.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common – 12*x* from the last two terms,. Then we get

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q11.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common

This can be written in the following form

Recall the formula for the cube of the difference of two numbers

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

### Page no. 5.22 Ex. 5.4

Q1.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization of

Q2.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q3.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q4.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q5.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q6.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q7.

**Answer :**

The given expression to be factorized is

Let

Note that

Recall the formula

When

So, we have the new formula

when

Using the above formula, the given expression can be written as

Put

We cannot further factorize the expression.

So, the required factorization is of

Q8.

**Answer :**

The given expression to be factorized is

Let

Note that

Recall the formula

When

So, we have the new formula

Using the above formula, the given expression can be written as

Put

We cannot further factorize the expression.

So, the required factorization is of

Q9.

**Answer :**

The given expression to be factorized is

Let

Note that

Recall the formula

When

So, we have the new formula

Using the above formula, the given expression can be written as

Put

Then we have

We cannot further factorize the expression.

So, the required factorization is of

Q10.

**Answer :**

The given expression to be factorized is

Let

Note that

Recall the formula

When

So, we have the new formula

Using the above formula, the given expression can be written as

Put

We cannot further factorize the expression.

So, the required factorization is of

Q11.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

### Page no. 5.23 Ex. 5.4

Q12.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q13.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q14.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

We cannot further factorize the expression.

So, the required factorization is of

Q15.

**Answer :**

The given expression is

It is given that

The given expression can be written in the form

Recall the formula

Using the above formula, we have

Q16.

**Answer :**

(i) The given expression is

We have to multiply the above expression by

The required product is

Recall the formula

Using the above formula, we have

(ii) The given expression is

We have to multiply the above expression by

The required product is

Recall the formula

Using the above formula, we have

(iii) The given expression is

We have to multiply the above expression by

The required product is

Recall the formula

Using the above formula, we have

(iii) The given expression is

We have to multiply the above expression by

The required product is

Recall the formula

Using the above formula, we have

### Page no. 5.24 Formative Assessment_VSA

Q1.

**Answer :**

The given expression to be factorized is

This can be written in the form

We cannot further factorize the expression.

So, the required factorization is

Q2.

**Answer :**

The given expression to be factorized is

Take common

We cannot further factorize the expression.

So, the required factorization is

Q3.

**Answer :**

Recall the formula

When

Q4.

**Answer :**

Recall the formula

Given that

Then we have

Q5.

**Answer :**

Recall the formula

Given that

Then we have

Q6.

**Answer :**

Recall the formula

Given that

Then we have

Q7.

**Answer :**

The given expression is

Let

Note that

Recall the formula

When

So, we have the new formula

Using the above formula, the value of the given expression is

Q8.

**Answer :**

The given expression is

Let

Note that

Recall the formula

When

So, we have the new formula

Using the above formula, the value of the given expression is

Q9.

**Answer :**

The given expression is

Let

Note that:

Recall the formula

When

So, we have the new formula

Using the above formula, the value of the given expression is

Q10.

**Answer :**

The given expression is

Let

Note that

Recall the formula

When

So, we have the new formula

Using the above formula, the value of the given expression is

### Page no. 5.24 Formative Assessment_MCQ

Q1.

**Answer :**

The given expression to be factorized is

Take common

So, the correct choice is (c).

Q2.

**Answer :**

The given expression to be factorized is

This can be written in the form

So, the correct choice is (a).

Q3.

**Answer :**

The given expression to be factorized is

This can be arrange in the form

Let

Put

So, the correct choice is (a).

Q4.

**Answer :**

The given expression to be factorized is

Take common

Finally, take common

So, the correct choice is (d).

### Page no. 5.25 Formative Assessment_MCQ

Q5.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

So, the correct choice is (a).

Q6.

**Answer :**

The given expression to be factorized is

This can be written in the form

Recall the formula

Using the above formula, we have

So, the correct choice is (c).

Q7.

**Answer :**

The given expression to be factorized is

Recall the formula for difference of two cubes

Using the above formula, we have,

So, the correct choice is (a).

Q8.

**Answer :**

The given expression to be factorized is

This can be written in the form

Take common *x* from the first two terms and

Finally, take common

So, the correct choice is (d).

Q9.

**Answer :**

The given expression is

Let

Note that:

Recall the formula

When

So, we have the new formula

Using the above formula, the value of the given expression is

So, the correct choice is (b).

Q10.

**Answer :**

The given expression to be factorized is

This can be written in the form

So, the correct choice is (a).

Q11.

**Answer :**

The given expression is

Recall the formula

Using the above formula the given expression becomes

Given that

Therefore the value of the given expression is

So, the correct choice is (c).

Q12.

**Answer :**

The given equation is

Recall the formula

Using the above formula, we have

So, the correct choice is (d).

Q13.

**Answer :**

The given equation is

This can be written as

Comparing the coefficients on both sides of the equation.

We get,

Putting the value of *a* from (1) in (2)

We get,

So the value of a, b and c is 1, – 4 and 7 respectively.

Therefore,

*a* + *b* + *c* =1-4 + 7 = 4

So, the correct choice is (a).

Q14.

**Answer :**

The given expression is

This can be written in the form

Assume

Recall the formula for difference of two cubes

Using the above formula, the expression becomes

Note that both *a* and *b* are positive, unequal. So, neither

Therefore we can cancel the term

So, the correct choice is (a).

### Page no. 5.25 Formative Assessment_MCQ

Q15.

**Answer :**

The given expression is

Assume

Recall the formula for sum of two cubes

Using the above formula, the expression becomes

Note that both *b* are positive. So, neither

Therefore we can cancel the term

So, the correct choice is (b).

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