How to prove $x^5 + x^4 + x^3 + x^2 + 1 = 0$ has no rational roots?

Question

I know I can set $x = \frac{p}{q}$ and then multiply everything and then show there is a contradiction but is there an easier way to do this? like by some theorem?

Answer 1

Steps:

1. Use rational root theorem

2. See possible solutions would be $1$ and $−1$.

3. Evaluate them to conclude there is no rational roots.