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 $(x^4+1)(x^2+1)(x+1)(x-1)$.

Akhileshwar Nani

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Updated on: 07-Apr-2023

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