ALGEBRAIC IDENTITIES

 

Standard Algebraic Identities List

All standard Algebraic Identities are derived from the Binomial Theorem. There are a number of algebraic identities but few are standard that are listed below.

• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 + b2 – 2ab
• (a + b)(a – b) = a2 – b2
• (a + b)3=a3 + b3 + 3ab(a + b)
• (a – b)3=a3 – b3 – 3ab(a – b)
• (a + b + c)2=a2 + b2 + c2 + 2ab + 2bc + 2ca

Methods to Solve Algebraic Identities

• We can verify algebraic identities by substitution method, in which we can put values in variable places and try to make both sides equal. i.e LHS = RHS.

Example:

(a – 2) (a + 2) = a2 – 22

Now we will start putting value in place of a.

starting with a = 1, (-1) x (3) = -3

then we will put a = 2, 0 x 4 = 0

Here we got a = 1 and a = 2 as the value which satisfy the given question.

• Another method is by manipulating identities which are commonly used:

i. (a + b)2 = a2 + b2 + 2ab       

ii. (a – b)2 = a2+ b2 – 2ab     

iii. (a + b)(a – b) =a2 – b2     

iv. (x + a)(x + b) = x2 + (a + b)x + ab

Proof:

i. (a + b)2 = (a + b) (a + b)

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

                = a2 + ab + ab + b2

                = a2 + 2ab + b2

Hence, LHS = RHS.

ii. (a – b)2 = (a – b) (a – b)

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

                = a2 – ab – ba + b2

                = a2 – 2ab + b2

Hence, LHS = RHS.

iii. (a + b) (a – b) = a (a – b) + b (a – b)

                            = a2 – ab + ab – b2

                            = a2 – b2

Hence, LHS = RHS.

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Standard Algebraic Identities Examples

Example 1: Solve (2x + 3) (2x – 3) using algebraic identities?

Solution: 

By the algebraic identity (a + b)(a – b) = a2 – b2 

We can  re-write the given expression as

(2x + 3) (2x – 3) = (2x)2 – (3)2 = 4x2– 9

Example 2: Solve (3x + 5)2 using algebraic identities?

Solution: 

By algebraic identity

(a + b)2 = a2 + b2 + 2ab   

We can re-write the given expression as;

(3x + 5)2 = (3x)2 + 2(3x)5 + 52

(3x + 5)2 = 9x 2 + 30x + 25

Example 3: Find the product of (x + 1)(x + 1) using standard algebraic identities?

Solution: 

(x + 1)(x + 1) can be written as (x + 1)2. Thus, it is of the standard form I where a = x and b = 1. 

We have,

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