How to prove that $x^3+2x^2+x-1$ has no rational root?

Question

I need to prove that $x^3+2x^2+x-1$ has no rational root.

Actually, the question was to prove that the polynomial is irreducible in $\mathbb Q[x]$. Now since $\mathbb Q$ is a field so any polynomial of degree $3$ is irreducible iff it has no root in $\mathbb Q$. So I only need to prove that the given polynomial has no root in $\mathbb Q$.

But I am not able to prove this. Can anyone help me out? It's really very urgent as I have an exam tomorrow on this topic.

Thanks in advance.

Answer 1

Suppose $x=p/q$ is a rational root, with $p$ and $q$ coprime. Then since $x(x+1)^2=1$, $\frac pq(\frac pq+1)^2=1$.

Thus, $p(p+q)^2=q^3$. Let $r$ be a prime factor of $q$. Then since $r|p(p+q)^2$, $r|p$ or $r|p+q$. Either way, $r|p$, a contradiction. Thus, we must have $q=1$, but there are no integer roots.