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