How to go from polynomial of third degree to multiplication of two smaller polynomials

Question

I have a basic calculus question which I should be able to do easily but I just can't remember how to tackle it. I'm working on my linear algebra exam and trying to find the eigenvalues for a certain matrix A. I got the following as the determinant:

$-\lambda^{3}+3\lambda^{2}+9\lambda-27$ and I want to get this in the form of $(\lambda+x_{1})^{2}(\lambda+x_{2})$. How to get there?

PS: the solution for this particular one is $x_{1}=-3$ and $x_{2}=3$, it case that might help you.

Answer 1

Using the rational root theorem, you can find those roots that you mentioned. They are $\pm3$, as you know. So, divide your original polynomial by $\lambda-3$ and then divide what you got by $\lambda+3$.

Answer 2

The polynomial can be factored by grouping: $$ -\lambda^{3}+3\lambda^{2}+9\lambda-27 = -\lambda^{2}(\lambda-3)+9(\lambda-3) = -(\lambda^{2}-9)(\lambda-3) = -(\lambda+3)(\lambda-3)^{2} $$