Factorize the expression $a^4-(2b+c)^4$.

Given:

The given expression is $a^4-(2b+c)^4$.

To do:

We have to factorize the expression $a^4-(2b+c)^4$.

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.

$a^4-(2b+c)^4$ can be written as,

$a^4-(2b+c)^4=(a^2)^2-[(2b+c)^2]^2$             [Since $a^4=(a^2)^2, (2b+c)^4=[(2b+c)^2]^2$]

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,

$a^4-(2b+c)^4=(a^2)^2-[(2b+c)^2]^2$

$a^4-(2b+c)^4=[a^2+(2b+c)^2][a^2-(2b+c)^2]$

Now,

Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $a^2-(2b+c)^2$.

$a^2-(2b+c)^2=(a+2b+c)(a-2b-c)$.............(I)

Therefore,

$a^4-(2b+c)^4=[a^2+(2b+c)^2](a+2b+c)(a-2b-c)$            [Using (I)]

Hence, the given expression can be factorized as $[a^2+(2b+c)^2](a+2b+c)(a-2b-c)$.

Akhileshwar Nani

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Updated on: 08-Apr-2023

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