How to find all polynomials with rational coefficients$f(x)=a_nx^n+\cdots+a_1x+a_0$, $a_i\in \mathbb Q$, such that $$\forall r\in\mathbb R\setminus\mathbb Q,\quad f(r)\in\mathbb R\setminus\mathbb Q.$$ thanks in advance
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The only candidates are those polynomials $f(x)\in\mathbb Q[x]$ that are factored over $\mathbb Q$ as product of first degree polynomials (this is because if $\deg f>1$ and $f$ is irreducible then all of its roots are irrationals.)
The first degree polynomials have this property. Can you see that these are all?
(Hint: The polynomial $f(x)+q$, for suitable $q\in\mathbb Q$, is not a product of first degree polynomials)