Prove that if a polynomial f is constant for infinitely many arguments then it is constant.

Question

Let $f$ a polynomial with real coefficients. There exists a constant $ c $ such that $f(x)=c $ for infinitely many numbers $x$. Prove $f$ is constant.

I have no idea how to do this. I started by denoting $f=a_n x^n+...+a_0$ but I have no idea how to continue.

score 3 · Accepted Answer · answered Nov 24 '19 at 13:06

3

Let consider

$$p(x)=f(x)-c$$

and since $p(x)=0$ has at most $n$ solutions we have $p(x)=0 \implies f(x)=c$.

answered Nov 24 '19 at 13:06

user

154,566

This is so cool. I never thought it is that easy! Thank you so much. I will remember this method for sure. I will accept your answer after the question reaches the 15 minute limit – furfur Nov 24 '19 at 13:07
@furfur You are welcome! Bye – user Nov 24 '19 at 13:09
This is so cool! – Fomalhaut Nov 24 '19 at 13:10
furfur.Have a look at Benjamin's proof in Martin R's comment. – Peter Szilas Nov 24 '19 at 13:16

Prove that if a polynomial f is constant for infinitely many arguments then it is constant.

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