Question

How do you simplify `(w+2t)(w^2-2wt+4t^2)`?

Answer 1

`(w+2t)(w^2-2wt+4t^2) = w^3+8t^3`

Explanation:

The given expression is in the form:

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

with `A=w` and `B=2t`.

This is recognisable as the factored form (using real coefficients) of the sum of cubes:

`A^3+B^3 = (A+B)(A^2-AB+B^2)`

If you wanted to multiply it out by hand, you could use distributivity as follows:

`(w+2t)(w^2-2wt+4t^2) = w(w^2-2wt+4t^2)+2t(w^2-2wt+4t^2)`

`color(white)((w+2t)(w^2-2wt+4t^2)) = w^3-color(red)(cancel(color(black)(2w^2t)))+color(green)(cancel(color(black)(4wt^2)))+color(red)(cancel(color(black)(2w^2t)))-color(green)(cancel(color(black)(4wt^2)))+8t^3`

`color(white)((w+2t)(w^2-2wt+4t^2)) = w^3+8t^3`