Given that $1 – j$ and $–2 + j$ are roots of $x^5 + 2x^4 – x^3 – 2x^2 + 10x$, how many roots does this polynomial have and what are they?

Question

If a polynomial has a degree of 5, I know that it must have 5 roots. 2 out of the 5 roots are given as complex number, so does that mean there're 3 real roots left to be found ?

Answer 1

Assumtion: $j=\sqrt {-1} $

Since complex roots occur in conjugate pairs, the 4 roots would be $1-j$, $1+j$, $-2+j$ and $-2-j$.

One obvious root is $0$. Hence five roots.

Answer 2

As others have noted: since the coefficients are real, all non-real roots come in conjugate pairs, so you already know $4$ roots. But $0$ is an obvious root, so now we have all $5$ roots of the quintic. In particular, the $5$ roots are all distinct: $1\pm j,\,-2\pm j,\,0$.