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 $(x-2y)(x^2+3y^2)$.