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