Standard Algebraic Identities

Algebraic Identities are algebraic equations that are always true for every value of the variable in them. The algebraic equations that are valid for all values of variables in them are called algebraic identities.

It is used for the factorization of polynomials. In this way, algebraic identities are used in the computation of algebraic expressions and in solving different polynomials. They contain variable and constant on both the side of polynomial i.e. LHS and RHS. In algebraic identity, LHS must be equal to RHS. 

This article provides you with the standard algebraic identities, including their examples, and methods to solve algebraic identities.

What are Identities?

Consider the equality (x + 1) (x +2) = x² + 3x + 2. One can evaluate both sides of this equality for some value of a, say x = 5. For x = 5,

• LHS = (x + 1) (x + 2) = (5 + 1) (5 + 2) = 6 × 7 = 42
• RHS = x² + 3x + 2 = 5² + 3 × 5 + 2 = 25 + 15 + 2 = 42

Thus, the values of the two sides of the equality are equal for a = 5. One can find that for any value of x, LHS = RHS. Such equality, true for every value of the variable in it, is called an identity. Thus, (x + 1) (x + 2) = x² + 3x + 2 is an identity.

Read More: 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

Example 4: Expand (3x – 4y)³ using standard algebraic identities?

Solution:

(3x – 4y)³ is of the standard form VII where a = 3x and b = 4y. 

We have,

(3x – 4y)³ = (3x)³ – (4y)³ – 3(3x)(4y)(3x – 4y) = 27x ³ –  64y ³ – 108x²y + 144xy ²

Standard Algebraic Identities Practice Problems

1. Solve (4x + 5)² using algebraic identities.

2. Factorize x² - 25 using algebraic identities.

3. Expand (2x - 3)³ using algebraic identities.

4. Find the product of (x + 2) (x - 2) using algebraic identities.

5. Expand (a + b + c)² using algebraic identities.

Summary - Standard Algebraic Identities

Algebraic identities are equations that hold true for all values of the variables involved, making them crucial tools in mathematics. They facilitate the factorization of polynomials and simplify the computation of algebraic expressions. Derived primarily from the Binomial Theorem, standard algebraic identities include expressions like (a + b)² = a² + b² + 2ab and (a - b)² = a² + b² - 2ab. These identities are instrumental in solving various polynomial equations and verifying mathematical expressions by ensuring the left-hand side (LHS) equals the right-hand side (RHS). Understanding and applying these identities is essential for efficiently tackling algebraic problems.

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