# Factorize the expression $x^4-1$.

Given:

The given expression is $x^4-1$.

To do:

We have to factorize the expression $x^4-1$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression means writing the expression as a product of two or more factors. Factorization is the reverse of distribution.

An algebraic expression is factored completely when it is written as a product of prime factors.

$x^4-1$ can be written as,

$x^4-1=(x^2)^2-(1)^2$             [Since $1^2=1$]

Here, we can observe that the given expression is a difference of two squares. So, by using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize the given expression.

Therefore,

$x^4-1=(x^2)^2-(1)^2$

$x^4-1=(x^2+1)(x^2-1)$

Now,

$x^2-1$ can be written as,

$x^2-1=x^2-1^2$

Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $x^2-1^2$.

$x^2-1^2=(x+1)(x-1)$.............(I)

Therefore,

$x^4-1=(x^2+1)(x+1)(x-1)$            [Using (I)]

Hence, the given expression can be factorized as $(x^2+1)(x+1)(x-1)$.

Updated on: 08-Apr-2023

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