Algebraic Identities

An algebraic identities are an equality relating one or more variables that is true for all values of the variables. Identities are powerful tools in algebra that are used to simplify algebraic expressions, prove equations, and reveal deeper relationships between quantities.

In this article, we will discuss some algebraic identities.

What Are Algebraic Identities?

An algebraic identities are special equations in math where the left side is always equal to the right side.

An identity in math is an equation that holds true for all the values, even if you change the variables involved. In simpler terms, these are math rules that work for any numbers you plug in.

For example,

• (a + b)^2 = a^2 + 2ab + b^2
• (a^2 – b^2) = (a + b)(a – b)

Standard Algebraic Identities List

Some Standard Algebraic Identities list are given below:

1. (a + b)^2 = a^2 + 2ab + b^2
2. (a – b)^2 = a^2 – 2ab + b^2
3. (a^2 – b^2) = (a + b)(a – b)
4. (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
5. (a – b)^3 = a^3 – 3a^2b + 3ab^2 – b^3
6. a^3 + b^3 = (a + b)(a^2 – ab + b^2)
7. a^3 – b^3 = (a – b)(a^2 + ab + b^2)

Solved Examples of Algebraic Identities

Example

Factorize x^2 – 25 using the difference of squares algebraic identity.

Solution

x^2 – 25 is of the form Difference of Squares. So, we can use the identity

x^2 – 25 = (x + 5)(x – 5)

Therefore,

x^2 – 25 = (x + 5)(x – 5)

Example

Expand (2a + 3b)^2 using the algebraic identity (a + b)^2.

Solution

(2a + 3b)^2 can be expanded using the identity

(a + b)^2 = a^2 + 2ab + b^2

So,

(2a + 3b)^2 = (2a)^2 + 2(2a)(3b) + (3b)^2

= 4a^2 + 12ab + 9b^2

Example

Factorize (x^3 – 64) using standard algebraic identities.

Solution

(x^3 – 64) is a difference of cubes and can be factored using Identity VII:

(x^3 – 64) = (x – 4)(x^2 + 4x + 16)

Example

Expand (5x + 2)^3 using standard algebraic identities.

Solution:

(5x + 2)^3 is of the form Identity IV where a = 5x and b = 2.

So,

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

= 125x^3 + 150x^2 + 60x + 8

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