Do all degree $n+1$ polynomials have rational roots?

Question

Do all degree $n+1$ polynomials have rational roots? It's hard to believe especially there are no clean algebraic solutions to polynomials with degrees $n\geq4$. How would you go about proving these polynomials have irrational roots?

Answer 1

Let $a_1,...,a_{n+1} \in \mathbb R \setminus \mathbb Q$ and consider

$p(x)=(x-a_1) \cdot ... \cdot (x-a_{n+1})$.

Then $p$ has degree $n+1$ but no rational root.