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Factorize the expression $x^3-2x^2y+3xy^2-6y^3$.
Given:
The given algebraic expression is $x^3-2x^2y+3xy^2-6y^3$.
To do:
We have to factorize the expression $x^3-2x^2y+3xy^2-6y^3$.
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 $x^3-2x^2y+3xy^2-6y^3$ by grouping similar terms and taking out the common factors.
The terms in the given expression are $x^3, -2x^2y, 3xy^2$ and $-6y^3$.
We can group the given terms as $x^3, 3xy^2$ and $-2x^2y, -6y^3$.
Therefore, by taking $x$ as common in $x^3, 3xy^2$ and $-2y$ as common in $-2x^2y, -6y^3$, we get,
$x^3-2x^2y+3xy^2-6y^3=x(x^2+3y^2)-2y(x^2+3y^2)$
Now, taking $(x^2+3y^2)$ common, we get,
$x^3-2x^2y+3xy^2-6y^3=(x-2y)(x^2+3y^2)$
Hence, the given expression can be factorized as $(x-2y)(x^2+3y^2)$.
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