# Find all polynomials $P(x)$ with real coefficients such that $P(x)=P(x-1)$.

Find all polynomials $P(x)$ with real coefficients such that $P(x)=P(x-1)$. Any elegant solutions to this somewhat (trivial?) question?

I think that since $P(x)=P(x-1)$ implies $P(x)=P(y)$ for all integers $x$ and $y$, $P(x)$ must be constant and also work for all reals. Is this valid? Any other solutions?

$P'$ has an infinite number of zeroes. Since $P'$ is a polynomial, and a non-zero polynomial has a finite number of roots, it is necessarily the zero polynomial. Thus $P$ is constant.
The polynomial $P(x)-P(0)$ has zeroes $x=0$, $x=1$, $x=2$, $\ldots$. A polynomial having infinitely many roots must be the zero polynomial. It follows that $P(x)=P(0)$ for all $x$, so $P$ is a constant.