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Factorize the algebraic expression $a(x-y)+2b(y-x)+c(x-y)^2$.
Given:
The given algebraic expression is $a(x-y)+2b(y-x)+c(x-y)^2$.
To do:
We have to factorize the expression $a(x-y)+2b(y-x)+c(x-y)^2$.
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.
Here, we can factorize the expression $a(x-y)+2b(y-x)+c(x-y)^2$ by taking out the common factors. The greatest common factor of an algebraic expression is the greatest factor that can be divided into each of the terms with no remainder.
The terms in the given expression are $a(x-y), 2b(y-x)$ and $c(x-y)^2$.
We can write $2b(y-x)$ as,
$2b(y-x)=-2b(x-y)$
We can observe that $(x-y)$ is common to all the three terms.
Therefore, by taking $(x-y)$ as common, we get,
$a(x-y)+2b(y-x)+c(x-y)^2=(x-y)[a-2b+c(x-y)]$
$a(x-y)+2b(y-x)+c(x-y)^2=(x-y)[a-2b+c(x)-c(y)]$
$a(x-y)+2b(y-x)+c(x-y)^2=(x-y)(a-2b+cx-cy)$
Hence, the given expression can be factorized as $(x-y)(a-2b+cx-cy)$.
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