# Algebraic Identities

## What are Algebraic Identities for Class 8?

• Algebraic identities are algebraic equations which are always true for every value of variables in them.
• Algebraic identities have their application in the factorization of polynomials.
• They contain variables and constants on both sides of the equation.
• In an algebraic identity, the left-side of the equation is equal to the right-side of the equation.
• For example, (a+b)2 = a2+2ab+b2 , which is true for all the values of and b.

### Using Substitution Method.

• Substitution generally means putting numbers or values in the place of variables or letters.
• In the substitution method, an arithmetic operation is performed by substituting the values for the variables.
• For example, when we have x-2=4

When we substitute x= 6,

On the Right-hand side,

4

On the left hand-side,

x-2 = 6 – 2 = 4

Here, Right hand side = Left hand side which means (x-2) is an identity.

Suppose, (a+3) (a-3) = (a2-9)

Substituting a= 1

On the Right- hand side,

(a2-9) = (1-9) = -8

On the Left- hand side,

(a+3) (a-3) = (1+3) (1-3) = (4) (-2) = -8

Here, Right hand side = Left hand side which means that (a+3) (a-3) is an identity.

### Using Activity Method.

• In this method, the algebraic identity is verified geometrically by taking different values of a x and y.
• In the activity method, the identities are verified by cutting and pasting paper.
• To verify an identity using this method, you need to have a basic knowledge of Geometry.

### Identities Class 8 –

The standard identities class 8 are derived from the Binomial Theorem. The table below consists of some Standard identities in maths class 8.

 Identity I (a+b)2 = a2+2ab+b2 Identity II (a-b)2 = a2– 2ab+b2 Identity III a2-b2= (a+b) (a-b) Identity IV (x+a) (x+b) = x2+(a+b) x+ab Identity V (a+b+c)2= a2+b2+c2+ 2ab+2bc+2ca Identity VI (a+b)3= a3+b3+3ab(a+b) Identity VII (a-b)3= a3 -b3-3ab(a-b) Identity VIII a3 +b3+c3-3abc

### Now, you Might Think What a Binomial Theorem is!

• In algebra, the Binomial Theorem is defined as a way of expanding a binomial expression raised to a large power which might be troublesome.
• A polynomial equation with just two terms generally having a plus or a minus sign in between is known as a Binomial expression.

### A small Explanation for the Above Algebraic Identities for Class 8.

For example, let us take one of the basic identities,

(a+b)2 = a2+2ab+b2, which holds for all the values of a and b.

• An identity holds true for all the values of a and b.
• We can possibly substitute one instance of one side of the equality with its other side.
• In simple words, (a+b)2 can be replaced by a2+2ab+b2 and vice versa.
• These can be used as shortcuts which make manipulating algebra easier.

### Factoring Identities

The identities listed below in the table are factoring formulas for identities of algebraic expressions class 8.

 x2-y2 = (x+y) (x-y) x3-y3 = (x-y) (x2+xy+ y2) x3 +y3 = (x+y) (x2 -xy+ y2) x4-y4 = (x2-y2) (x2 + y2)

### Three – Variable Identities –

By manipulation of the various discussed identities

entities of algebraic expressions class 8 we get these three- variable identities.

 (x+y) (x+z) (y+z) = (x+y+z) (xy+yz+xz)-xyz x2 +y2+z2          = (x+y+z)2– 2(xy+yz+xz) x3 +y3+z3             = (x+y+z)(x2 + y2 +z2 -xy-xz-yz)

## The Four Basic Identities in Maths Class 8 have Been Listed Below.

 Identity I (a+b)2 = a2+2ab+b2 Identity II (a-b)2 = a2– 2ab+b2 Identity III a2-b2= (a+b) (a-b) Identity IV (x+a) (x+b) = x2+(a+b) x+ab

### Questions to be Solved on Identities Class 8

Question 1) Find the product of (x-1) (x-1)

Solution) We need to find the product (x-1) (x-1),

(x-1) (x-1) can also be written as (x-1)2.

We know the formula for (x-1)2, expand it

(a-b)2 = a2– 2ab+b2 where a= x, b=1

(x-1)2 = x2– 2x+1

Therefore, the product of (x-1) (x-1) is x2– 2x+1

Question 2) Find the product of (x+1) (x+1) as well as the value of it using x = 2.

Solution) We need to find the product (x+1) (x+1),

(x+1) (x+1) can also be written as (x+1)2.

We know the formula for (x+1)2, expand it

(a+b)2 = a2+ 2ab+b2 where a= x, b=1

(x+1)2 = x2+ 2x+1

Putting the value of x = 2 in equation 1,

(2)2+ 2(2) +1 = 9

Therefore, the product of (x+1) (x+1) is x2+ 2x+1 and the value of the expression is 9.

Question 3) Separate the constants and the variables from the given question.

-4, 4+x, 3x+4y, -5, 4.5y, 3y2+z

Solution) Variables are the ones which include any letter such as x, y, z etc along with the numbers.

In the given question,

Constants = -4, -5

Variables = 3x+4y, 4+x, 4.5y, 3y2+z

Question 4) Find the value of $\frac{{{x^2} – 1}}{5}$,at x = -1.

Solution) At x = -1,  $x = – 1,\frac{{{x^2} – 1}}{5}$

= $\frac{{{(-1)^2} – 1}}{5}$

= 0

Question 5) Find the value of x2+y2 – 10 at x=0 and y=0?

Solution) At x= 0 and y = 0,

x2+y2 – 10 = (0)2+(0)2 – 10

= -10

Question 6) Solve the following (x+2)2 using the concept of identities.

Solution) According to the identities and algebraic expression class 8,

We know the formula,

(a+b)2 = a2+2ab+b2

Where, a= x, b= 2

Let’s expand the given (x+2)2,

Therefore, (x+2)2 = x2+4x+4 is the solution.

Question 1) How Many Identities are there in Algebraic Expressions?

Answer) The algebraic identities for class 8 consists of three important identities. They are listed below-

 (a+b)2 = a2+2ab+b2 (a-b)2 = a2– 2ab+b2 a2-b2= (a+b) (a-b)

Question 2) Give the Difference Between Algebraic Identity and Expression?

Answer) An algebraic identity is equality which is true for all the values whereas an expression which consists of variables and constants is known as an algebraic expression. The value of the expression changes every time the values are changed.

Question 3) What is an Algebra Formula?

Answer) In mathematics algebra is a combination of both numbers as well as letters. In the algebra formula the numbers remain fixed as their value is known and the letters or alphabets are used to represent unknown quantities which need to be found out.