Question

Factor the expression:

-\lambda ^{3}+4\lambda ^{2}+4\lambda -16

Answer 1

`-(\lambda -4)(\lambda -2)(\lambda +2)`

Explanation:

What we want to fator is:

`-\lambda^3 + 4 \lambda^2 + 4\lambda - 16`

There is no common factor, but let's factor it by grouping. The first two addends can be factor as follows:

`-\lambda^3 + 4 \lambda^2 = \lambda^2(-\lambda + 4) = -\lambda^2(\lambda - 4)`

the second addends can be factor as well:

`4\lambda - 16 = 4(\lambda- 4)`.

Then our original expression can be rewritten like

`-\lambda^3 + 4 \lambda^2 + 4\lambda -16=\lambda^2(\lambda - 4) + 4(\lambda - 4)`

And here the
`(\lambda-4)` is the common factor!

`-\lambda^2(\lambda - 4) + 4(\lambda - 4) = (\lambda - 4)(-\lambda^2 + 4)`

Finally, we can factor the quadratic expression as a difference of squares
`-\lambda^2 + 4 = 4 - \lambda^2 = (2+\lambda)(2-\lambda)`

Ant we get

`(\lambda - 4)(-\lambda^2 + 4)= (\lambda - 4)(\lambda + 2)(2-\lambda)`

now, we can extract the negative sign from
`(2-\lambda)`, and we get

`-(\lambda -4)(\lambda -2)(\lambda +2)`.