# Factorize the following expression:$x^4-625$.

Given:

The given algebraic expression is $x^4-625$.

To do:

We have to factorize the expression $x^4-625$.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression implies 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.

$x^4-625$ can be written as,

$x^4-625=(x^2)^2-(25)^2$             [Since $625=(25)^2$]

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,

$x^4-625=(x^2)^2-(25)^2$

$x^4-625=(x^2+25)(x^2-25)$

Now,

$(x^2-25)$ can be written as,

$(x^2-25)=x^2-5^2$

Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $x^2-(5)^2$.

$x^2-(5)^2=(x+5)(x-5)$.............(I)

Therefore,

$x^4-625=(x^2+25)(x+5)(x-5)$            [Using (I)]

Hence, the given expression can be factorized as $(x^2+25)(x+5)(x-5)$.

Updated on: 08-Apr-2023

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