Find all polynomials $f(x)$ satisfying $f(x^2 ) + f(x) \cdot f(x + 1) = 0$ ∀ $x ∈ R$
Obvious polynomials are $f(x)=0$ and $f(x) = -1$
There are no linear polynomials satisfying the equation.
I took $f(x) = a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+…a_nx^n$
For $f(x)$ to satisfy the condition
$a_0(1+a_0+a_1+a_2+…a_n) = 0$
$a_n + {a_n}^2=0$
Taking simple values of $x$ like $x=1,x=0$ did not help and I am unable to proceed further.
Can anybody help?
Best Answer
Hint: Basic Complex Analysis tells us that the given equation holds for complex $x$ also. [I am referring to the Identity Theorem]. If $f$ has a root $x$ then the hypothesis shows that $x^{2}, x^{4},..$ are all zeros. But a polynomial can have only finite number of zeros (unless it is the zero polynomial). So the only possible roots are roots of unity and $0$.