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Factorize the expression $x^4-(2y-3z)^2$.
Given:
The given algebraic expression is $x^4-(2y-3z)^2$.
To do:
We have to factorize the expression $x^4-(2y-3z)^2$.
Solution:
Factorizing algebraic expressions:
Factorizing an algebraic expression implies 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.
$x^4-(2y-3z)^2$ can be written as,
$x^4-(2y-3z)^2=(x^2)^2-(2y-3z)^2$ [Since $x^4=(x^2)^2$]
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,
$x^4-(2y-3z)^2=(x^2)^2-(2y-3z)^2$
$x^4-(2y-3z)^2=[x^2+(2y-3z)][x^2-(2y-3z)]$
$x^4-(2y-3z)^2=(x^2+2y-3z)(x^2-2y+3z)$
Hence, the given expression can be factorized as $(x^2+2y-3z)(x^2-2y+3z)$.
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