# Algebraic Expressions and Identities (Weightage None%)   Share

## NCERT Article for Class 8 Maths Chapter 3 Algebraic Expressions and Identities

Have you remember Chapter 2 on Linear Equations in one variable? If you forgot, then it’s time to revise as we will build upon the Linear equations. Please go to this click to revise Linear Equations in One Variable”Link”. Let’s discuss what does it mean by the term expression. Expressions are formed from a combination of variables and constants. 3x+7, 2x2+9, 8xy+x, 2x2 + 5y2 are few examples of expressions.

Let’s discuss the components of an expression by taking an example of expression 9x+10

• Terms - In 9x+10 there are two terms namely 9x and 10. Hence terms are added to form expressions. Let’s take another example, In 6x - 26 we have 2 terms namely 6x and -26.

• Factors - Terms themselves can be formed as the product of factors, in expression 9x+10, the term 9x is the product of its two factors 9 and x. While term 5 has only one factor which is 5.

• Coefficients - The numerical factor of a term is called coefficient OR in other words, we can say the number which is multiplied to variables is called coefficient. In the expression 9x+10, 9 is the coefficient of x.

What are Monomial, Binomial, Trinomial and Polynomial expressions?

• Monomials - Expressions that have only one term. Example - 4x, 5xy , -9z, 7xy2, 54xyz etc.

• Binomials - Expressions that have two terms. Example - x+y, 5xy+z, 9xy2 + xz, 61xy+xyz, etc

• Trinomials - Expressions that have three terms. Example - x+y+z, 5xy+z+xy, 9xy2 + xz+99y, 61xy+xyz+x, etc

• Polynomials - Expressions that have one or more terms with non-zero coefficients are polynomials. Example - 4x, x+y+z, 5xy+z, 9xy2 + xz+99y+10x+7y2+6yz, 61xy+xyz+73x+xz+53yz, etc

Let’s discuss another concept of Like and Unlike Terms.

 Terms that have the same type of variables are known as Like Terms. Terms which have different type of variables are Unlike Terms Examples are given below. Like Terms Unlike Terms 7x and 88x 55xy and - 4yx 9x^2 and -x^2 7x and 55xy -4xy and x^2 -9z and -9x

Operations on Algebraic Expressions

Example- Add Expression 1 =>  3x2 +7x - 2 and Expression 2 => 4x - 13

Let’s carry out addition of these two expressions.

You can see that the only like terms have been added

Subtraction -> Just like addition, Only Like Terms are subtracted when we subtract one expression from another.

Example - Subtract 2x^2 + 7x from 3x^3 +9x^2 + 16x -22

Let’s carry out the subtraction operation.

As you can see only like terms have been subtracted

Multiplication -> Multiplication operation is different from addition and subtraction and it can be applied between Like as well as, Unlike terms.

Type 1 Multiplication - Multiplying Monomials by Monomials

Example 1 -  Multiply 9a by 2a

=> 9ax2a

=> 9xax2xa                                           (rewriting in product form)

=> 18xa2             (multiply number to number and variable to variable)

=> 18a2

So product of 9a and 2a is 18a2

Example 2 - Multiply 4x2 , 16xy and 7yz

Solution: We need to simplify

Or,   (rewriting in product form)

Or, (multiply number to number and variable to variable)

Or,

Or,

So the multiplication 4x2, 16xy and 7z is

Type 2 Multiplication - Multiplying Monomials by Polynomials

Example 1 -  Multiply 3x by 7x+2y+z

=>

=> Use distributive law for solving this question.

=>

=>

Example 2 -  Multiply 2x+4y by 5x+3y+6z

=> Use distributive law for solving this question

=>

=>

=>

=>

=>

The product of 2x+4y and 5x+3y+6z is

Algebraic Identities

What is Identity?

An identity is an equality that holds true regardless of the values chosen for its variables. They are used in simplifying or rearranging algebra expressions. Or in simple words, an equality where LHS = RHS always is called an Identity.

Standard Identities - Identities which we use more often are called standard identities, They are mostly binomial or trinomial in nature. Let's discuss some standard identities.

Identity 1 - (a+b)2 = a2 + b2 + 2ab

Let's derive this identity.

(a+b)2 = (a+b)x(a+b)

= a(a+b) + b(a+b)

= a2 +ab + ba +b2

=> (a+b)2 = a2 + b2 + 2ab

Hence LHS = RHS for every value of a and b and this is an identity

Identity 2(a-b)2 = a2 + b2 - 2ab

Let's derive this identity.

(a-b)2 = (a-b)x(a-b)

= a(a-b) - b(a-b)

= a2 -ab - ba +b2

=> (a+b)2 = a2 + b2 - 2ab

Hence LHS = RHS for every value of a and b and this is an identity

Identity 3 - (a-b)(a+b) = a2 - b2

Let's derive this identity.

(a-b)(a+b) = (a-b)x(a-b)

= a(a+b) - b(a+b)

= a2 +ab - ba -b2

=> (a-b)(a+b) = a2 - b2

Hence LHS = RHS for every value of a and b and this is an identity

Note - Identities 1,2 and 3 are known as standard identities

One more important identity which is very useful is (x+a)(x+b) = x2 + x(a+b) + ab

It's derivation is also similar. (x+a)(x+b) = x(x+b) + a(x+b)

= x2 + xb + ax + ab

= x2 + x(a+b) + ab

Hence LHS = RHS for every value of xa and b and this is an identity

Let's learn to apply these identities

Example 1 - Find (105) ?

Solution : 105 can be written as (100+5)

=> (105)2 = (100+5)2

Let's apply the identity 1, (a+b)2 = a2 + b2 + 2ab

Here a = 100 and b = 5

(100+5)2 = 1002 + 52 + 2x100x5

= 10000 + 25 + 1000

= 11025

Example 2 - Find (97) ?

Solution : 97 can be written as (100-3)

=> (97)2 = (100-3)2

Let's apply the identity 2, (a-b)2 = a2 + b2 - 2ab

Here a = 100 and b = 3

(100-3)2 = 1002 + 32 - 2x100x3

= 10000 + 9 - 600

= 9409

Example 3 - Find ( 3m + 2m)(3m - 2m)

Let's apply the identity 3,  (a-b)(a+b) = a2 - b2

( 3m + 2m)(3m - 2m) = (3m)2 - (2m)2

= 9m2 - 4m2

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