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Factorize the expression $2x^3y^2 - 4x^2y^3 + 8xy^4$.
Given:
The given expression is $2x^3y^2 - 4x^2y^3 + 8xy^4$.
To do:
We have to factorize the expression $2x^3y^2 - 4x^2y^3 + 8xy^4$.
Solution:
GCF:
A common factor of two or more numbers is a factor that is shared by the numbers. The greatest common factor (GCF) of those numbers is found by finding all common factors of the numbers and selecting the largest one.
The terms in the given expression are $2x^3y^2, -4x^2y^3$ and $8xy^4$.
The numerical coefficient of $2x^3y^2$ is $2$
The numerical coefficient of $-4x^2y^3$ is $4$
The numerical coefficient of $8xy^4$ is $8$
This implies,
$2=2\times1$
$4=2\times2$
$8=2\times2\times2$
GCF of $2, 4$ and $8$ is $2$
The common variables in the given terms are $x$ and $y$.
The power of $x$ in $2x^3y^2$ is $3$
The power of $x$ in $-4x^2y^3$ is $2$
The power of $x$ in $8xy^4$ is $1$
The power of $y$ in $2x^3y^2$ is $2$
The power of $y$ in $-4x^2y^3$ is $3$
The power of $y$ in $8xy^4$ is $4$
The monomial of common literals with the smallest power is $xy^2$
Therefore,
$2x^3y^2=2\times xy^2 \times (x^2)$
$-4x^2y^3=2\times xy^2 \times (-2xy)$
$8xy^4=2\times xy^2 \times (4y^2)$
This implies,
$2x^3y^2 - 4x^2y^3 + 8xy^4=2xy^2(x^2-2xy+4y^2)$
Hence, the given expression can be factorized as $2xy^2(x^2-2xy+4y^2)$.
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