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Factorize the expression $x^3-x$.
Given:
The given algebraic expression is $x^3-x$.
To do:
We have to factorize the expression $x^3-x$.
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.
$x^3-x$ can be written as,
$x^3-x=x(x^2-1)$ (Taking $x$ common)
$x^3-x=x(x^2-1^2)$ [Since $1=1^2$]
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,
$x^3-x=x(x^2-1^2)$
$x^3-x=x(x+1)(x-1)$
Hence, the given expression can be factorized as $x(x+1)(x-1)$.
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