Factorize $8x^3$ $-$ $24x^2y$ $+$ $54xy^2$ $-$ $162y^3$

The given expression $8x^3-24x^2y+54xy^2-162y^3$ is a polynomial that contains four algebraic terms. In this problem, we have to factorize this algebraic expression.

Group the terms of the expression

In the expression $8x^3-24x^2y+54xy^2-162y^3$, the first two terms have a common factor and the remaining two terms have another common factor. So, group the terms of the given algebraic expression on the basis of common factors.

$= \,\,\,$ $(8x^3-24x^2y)+(54xy^2-162y^3)$

Take out the common factors

In the first group, $8x^2$ is a common factor. Similarly, $54y^2$ is another common factor in the remaining two terms. They can be taken common from them as per distributive property of multiplication over subtraction.

$= \,\,\,$ $8x^2(x-3y)+54y^2(x-3y)$

Factorize the algebraic expression

In the simplified algebraic expression, $x-3y$ is a common factor and it can be taken out common from the terms as per distributive property of multiplication over addition to finish factorisation (or) factorization of the polynomial.

$= \,\,\,$ $(x-3y)(8x^2+54y^2)$

Therefore, the given algebraic expression $8x^3-24x^2y+54xy^2-162y^3$ is factored as $(x-3y)(8x^2+54y^2)$ by using factorization by grouping.