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Factorize the expression $p^2q^2-p^4q^4$.
Given:
The given expression is $p^2q^2-p^4q^4$.
To do:
We have to factorize the expression $p^2q^2-p^4q^4$.
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.
$p^2q^2-p^4q^4$ can be written as,
$p^2q^2-p^4q^4=p^2q^2[1-p^2q^2]$ (Taking $p^2q^2$ common)
$p^2q^2-p^4q^4=p^2q^2[1^2-(pq)^2]$ [Since $p^2q^2=(pq)^2$]
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,
$p^2q^2-p^4q^4=p^2q^2[1^2-(pq)^2]$
$p^2q^2-p^4q^4=p^2q^2(1+pq)(1-pq)$
Hence, the given expression can be factorized as $p^2q^2(1+pq)(1-pq)$.
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