Find all real polynomials such that $p(x)=p(alpha x)$ for some $alpha>0$

for $p(x)$ we write
$$
p(x)=a_{0} x^{0}+a_{1} x^{1}+a_{2} x^{2}+ldots+a_{n} x^{n}=sum_{j=0}^{n} a_{j} x^{j}
$$

and for $p(alpha x)$
$$
p(alpha x)=a_{0}(alpha x)^{0}+a_{1}(alpha x)^{1}+a_{2}(alpha x)^{2}+ldots+a_{n}(alpha x)^{n}=sum_{k=0}^{n} a_{k}(alpha x)^{k}: alpha>0
$$

If $alpha=1$, then $p(x)=p(alpha x)$. I’m not sure how to prove when $alpha neq 1$ or how to apply induction in this.