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Factorize the expression $x-y-x^2+y^2$.
Given:
The given expression is $x-y-x^2+y^2$.
To do:
We have to factorize the expression $x-y-x^2+y^2$.
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.
$x-y-x^2+y^2$ can be written as,
$x-y-x^2+y^2=x-y-(x^2-y^2)$
Here, we can observe that $x^2-y^2$ 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,
$x^2-y^2=(x+y)(x-y)$.............(I)
This implies,
$x-y-x^2+y^2=(x-y)-[(x+y)(x-y)]$ [Using (I)]
$x-y-x^2+y^2=(x-y)[1-(x+y)]$ (Taking $x-y$ common)
$x-y-x^2+y^2=(x-y)(1-x-y)$
Hence, the given expression can be factorized as $(x-y)(1-x-y)$.
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