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If $f$ is a polynomial with integer coefficients and $f(0)=0$, prove that $f$ is a constant

Question: If $f(x)=a_0x^0 + a_1x^1 + a_2x^2 + \cdots + a_nx^n$ with $a_n \in \mathbb{Z}$ is a polynomial with integer coefficients and $f(0)=0$, prove that $f(x)$ is a constant polynomial.
My attempt: Since $f(0)=0$, then $f(x)$ must be some power of $x$.
The problem is that I am not sure how to proceed from here.
Any help would be appreciated.
Thank you!

A:

It is clear that $f(x)$ must be of degree $0$.
So if $f(x)$ is not a constant we must have $f(x) = x^k$ for some $k>0$.
So let $k\ge1$. Then $0 = a_n f(1) = a_n \cdot 1^k$, so $a_n=0$ and therefore $f(x) = ax^k+b\in\Bbb Z[x]$.

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