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Factorize the expression $x^8-1$.
Given:
The given algebraic expression is $x^8-1$.
To do:
We have to factorize the expression $x^8-1$.
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.
$x^8-1$ can be written as,
$x^8-1=(x^4)^2-(1)^2$
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,
$x^8-1=(x^4)^2-(1)^2$
$x^8-1=(x^4+1)(x^4-1)$
Now,
$(x^4-1)$ can be written as,
$(x^4-1)=(x^2)^2-(1)^2$ [Since $1=1^2$]
Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $(x^2)^2-(1)^2$.
$(x^2-1^2)^2=(x^2+1)(x^2-1)$.............(I)
$(x^2-1)$ can be written as,
$(x^2-1)=(x)^2-(1)^2$ [Since $1=1^2$]
Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $(x)^2-(1)^2$
$x^2-1^2=(x+1)(x-1)$..................(II)
Therefore, using (I) and (II), we get,
$x^8-1=(x^4+1)(x^2+1)(x+1)(x-1)$
Hence, the given expression can be factorized as $(x^4+1)(x^2+1)(x+1)(x-1)$.