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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 $(x^2+25)(x+5)(x-5)$.
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