Polynomial and its derivative have a common factor?

Question

When is $gcd(p(x),p'(x))\ne 1$ where $p(x)$ is a polynomial?

That is when does the derivative of a polynomial and the polynomial has a common factor?

By when i mean some condition for the coefficients and degree of polynomial.

Obviously valid for $kx^n$.

(I came accross this when i had to determine in a problem that derivative of a rational function had to have a perfect square in the denominator)

Answer 1

Hint: working in $\mathbb C$, we have $$p(x) = \prod_{i=1}^n\left(x-\alpha_i\right)^{n_i}$$where the $\alpha_i$ are the distinct roots of $p(x)$ in $\mathbb C$.

• What is $p'(x)$?
• If $gcd(p(x), p'(x)) = 1$, what can you say about the roots of $p$?

Extra hint:

What happens if $n_i > 1$ for some $i$