# What are the 4 identities of algebraic expressions?

What are the 4 identities of algebraic expressions? Explain with examples

**Algebraic identities** are equations in algebra where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation.

They are satisfied with any values of the variables, they form the foundation of algebra and are helpful to perform computations in simple and easy steps.

The four basic algebra identities are as follows.

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

(a – b)2 = a2 – 2ab + b2

(a + b)(a – b) = a2 – b2

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

**Two Variable Identities**

The following are the identities in algebra with two variables. These identities can be easily verified by expanding the square/cube and doing polynomial multiplication.

For example,

to verify the first identity below,

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

In the same way, we can verify the other identities as well.

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

(a – b)2 = a2 – 2ab + b2

(a + b)(a – b) = a2 – b2

(a + b)3 = a3 +3a2b + 3ab2 + b3

(a – b)3 = a3 – 3a2b + 3ab2 – b3

Example:

Expand (2x + y)2.

Solution:

To expand the given expression, substitute a = 2x and b = y in (a + b)2 = a2 + 2ab + b2,

(2x + y)2 = (2x)2 + 2(2x)(y) + y2

= 4×2 + 4xy + y2

**Three Variable Identities**

The algebra identities for three variables also have been derived just the way the two variable identities were. Further, these identities are helpful to easily work across the algebraic expressions with the least number of steps.

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ac

a2 + b2 + c2 = (a + b + c)2 – 2(ab + bc + ac)

a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – ca – bc)

(a + b)(b + c)(c + a) = (a + b + c)(ab + ac + bc) – 2abc

Example: When a + b + c = 0, what is the value of a3 + b3 + c3?

Solution:

By one of the above identities,

a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – ca – bc)

Substituting (a + b + c) = 0, we get

a3 + b3 + c3 – 3abc = 0 (a2 + b2 + c2 – ab – ca – bc)

a3 + b3 + c3 – 3abc = 0

a3 + b3 + c3 = 3abc

They are also used for the factorization of polynomials. The below list presents a set of algebraic identities helpful for the factorization of polynomials.

a2 – b2 = (a – b)(a + b)

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

a3 – b3 = (a – b)(a2 + ab + b2)

a3 + b3 = (a + b)(a2 – ab + b2)

Example: a4 – b4 = (a2)2 – (b2)2

= (a2 – b2) (a2 + b2)

= (a – b)(a + b)(a2 + b2)