Prove that $\gcd(P, P')$ is an irreducible polynomial over $\Bbb R.$

Question

Let $P(X)$ = $X^4 + 2X^3 + 3X^2 + 2X + 1$

Let $P'$ be the derivative of $P$.

1. Factor $P$ as a product of irreducible polynomials over $\Bbb R$.
2. Find all the real and complex roots of $P$. What are their multiplicities?

Answer 1

Hint:

Common roots of a polynomial $P$ and its derivative are multiple roots of $P$, i.e. roots of order $\ge 2$. Now $P$ has degree $2$ and the g.c.d. of $P$ and $P'$, which yields the (complex) multiple roots has degree $2$. Do you see what you can deduce?

Answer 2

It seems that $P$ = $(x^2+x+1)^2$ and roots are cube roots of $1$ with multiplicity $2$.