# 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 $(x-y)(1-x-y)$.

Updated on: 09-Apr-2023

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