Find all real polynomials $P(x)$ which satisfy the equation$ P(x)P(-x)=P(x^2-1)$

Question

Find all real polynomials $P(x)$ having only real zeros and which satisfy the equation

$$P(x)P(-x)=P(x^2-1)$$

Please explain me the process and refer some books to learn polynomials.

Thanks in advance !

Answer 1

Of course $P(x) =0$ (zero polynomial) or $P(x)=1$ are solutions to this.

Let us consider the non-trivial case.

Hint:

Suppose $x=a$ is a zero of this polynomial, i.e. $P(a)=0$, then $$P(a^2-1)=0.$$ But this will lead to infinitely many zeros of the polynomial unless $a=a^2-1$. This solves to $$a=\frac{1\pm\sqrt{5}}{2}.$$ Thus
$$P(x)=x^2-x-1.$$ is a solution. Now try to use this to find others.