# 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.