Question

How do you factor `8u^3-24u^2v+18uv^2`?

Answer 1

The answer is `=2u(2u-3v)^2`

Explanation:

We need

`a^2-2ab-b^2=(a-b)^2`

Therefore,

`8u^2-24u^2v+18uv^2`

`=2u(4u^2-12uv+9v^2)`

`=2u(2u-3v)(2u-3v)`

`=2u(2u-3v)^2`

Answer 2

`8u^3-24u^2v+18uv^2 = 2u(2u-3v)^2`

Explanation:

Given:

`8u^3-24u^2v+18uv^2`

Note that all of the terms are divisible by `2u`, so we can separate that out as a factor first.

Then the remaining quadratic is a perfect square trinomial, being of the form:

`A^2-2AB+B^2 = (A-B)^2`

with `A=2u` and `B=3v` ...

`8u^3-24u^2v+18uv^2 = 2u(4u^2-12uv+9v^2)`

`color(white)(8u^3-24u^2v+18uv^2) = 2u((2u)^2-2(2u)(3v)+(3v)^2)`

`color(white)(8u^3-24u^2v+18uv^2) = 2u(2u-3v)^2`