Let $a\neq b$ $|a,b \in\mathbb N$ and let $P(x)=x^3+ax^2+bx+1$
Show that $P$ is irreducible over $\mathbb Q$.
I tried writing $(x^2+dx+e)(x+f)=x^3+ax^2+bx+1$ to find a contradiction. Found that $f$ is rational and $f=1/e$.
2026-04-04 13:42:27.1775310147
Prove that the following cubic polynomial is irreducible over $\mathbb Q$
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Hint. As $P(x)$ has degree $3$, if it were reducible, it would have a linear factor, i.e. it would have a rational root. This root can only be $1$ or $-1$, because the constant term is $1$ as well as the highest degree term coefficient. Then see if it is compatible with what you know about $a$ and $b$.