Algebra
# Functional Equations

The function $f$ satisfies $(x-y)f(x+y)-(x+y)f(x-y)+4xy(x^2-y^2)=0,$

whenever $|x| \neq |y |$.

Given that $f(1)=2,$ what is $f(x)$?

What is the polynomial $p(x)$ that satisfies $p(x^2+4)=\left(p(x)\right)^2+4 \text{ and } p(0)=0?$

A function $f$ from the positive integers to the positive integers satisfies the conditions

1) $f(n) = 2 f(n-1) + 3$,

2) $f(1) = 9 .$

What is the value of $f(9)$?

If $f(x)$ is a polynomial satisfying $27 f(x^3) -4f(x^2) - x^6 f(3x) + 46 = 0$, what is $f(10)?$