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Factorize the expression $x^3-y^2+x-x^2y^2$.
Given:
The given expression is $x^3-y^2+x-x^2y^2$.
To do:
We have to factorize the expression $x^3-y^2+x-x^2y^2$.
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.
Here, we can factorize the expression $x^3-y^2+x-x^2y^2$ by grouping similar terms and taking out the common factors.
The terms in the given expression are $x^3, -y^2, x$ and $-x^2y^2$.
We can group the given terms as $x^3, x$ and $-y^2, -x^2y^2$.
Therefore, by taking $x$ as common in $x^3, x$ and $-y^2$ as common in $-y^2, -x^2y^2$, we get,
$x^3-y^2+x-x^2y^2=x(x^2+1)-y^2(1+x^2)$
$x^3-y^2+x-x^2y^2=x(1+x^2)-y^2(1+x^2)$
Now, taking $(1+x^2)$ common, we get,
$x^3-y^2+x-x^2y^2=(1+x^2)(x-y^2)$
Hence, the given expression can be factorized as $(1+x^2)(x-y^2)$.
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