Let $\ f(x) \in \mathbb{R}[x]$ be a 7-th degree polynomial, such that.
$$ f(0)=0 \land f(i)=-3i $$ $$ f'(0)=0 \land f'(i)=-21 $$
Find all the complex roots of $ f(x)-3x^7$. Find all possible factorizations of $ f(x)-3x^7 $ in $\mathbb{C}[x]$
The problem is I'm missing a root.
Let $g(x)=f(x)-3x^7$. It's easy to see that 0 is a double root of g. Also i and -i, because i is a double root too (no more than evaluating is required to reach this conclusion).
Then I have 6 roots: two times 0, two times i, two times -i. The next root must be real, because if not its conjugate should also be there and g(x) is a polynomial of the 7-th degree too, so that can't be.
But god knows I don't know how to find it. Even though the first part of the problem is done here, I still can't factorize the polynomial in $\mathbb{C}$ if I don't have all the roots.
Any help?