I am working on a linear algebra problem where I have to diagonalize a matrix. The characteristic equation is $-\lambda ^3 – 3 \lambda^2 + 4$. I need to factor this in order to solve part of the problem but I was never taught how to factor polynomial with missing terms. I have tried using synthetic division and got $(\lambda-1)(- \lambda^2-4)$. The example in the books says the factored form is $-(\lambda-1)(\lambda+2)^2$. Am I on the right track? I can't see what the next step would be.
[Math] Factoring cubic polynomials with missing terms.
algebra-precalculusfactoring
Best Answer
You have to check which divisors of the constant term, $4$, are the roots of $-\lambda^3-3 \lambda^2+4=0$. Let $a$ be this divisor(if there are more than one, take one of them), then apply the Euclidean division of $-\lambda^3-3 \lambda^2+4$ and $\lambda-a$.
Then you get $-\lambda^3-3 \lambda^2+4=(\lambda-a) \cdot q$, where $q$ is a second degree polynomial, which you can find its roots using the disciminant.