Find all polynomials $f:R→R$ such that $f(x+2)=f(x)+2$
Find all polynomials $g:R→R$ such that $g(2x)=2g(x)$
Since the functions must be polynomials, I tried using an argument by degrees, but this did not lead me anywhere. Can someone help with some ideas?
I took the degrees of both sides to be d, but this doesn't help since both sides are of the same degree. In my previous question which can be seen on my profile (mentioned by @Scene), multiplying by a polynomial on both sides worked, but I am not able to use this here. –
Best Answer
For the first one, put $x =0$ to get (Notice that $f(0)$ is the constant term. Call it $c$): $$f(2) = f(0) + 2 = c+2$$ Put $x = 2$ to get $$f(4) = f(2) + 2 = c+2 + 2 = c+4$$ Show by induction on a natural number $n$ that $f(2n) = c+2n$. Define $$h(x):= f(x) - f(0) - x$$ Notice that $h(x)$ has a degree no greater than $f(x)$ (As pointed out by prets, this is true only if $\deg(f)\ge 1$, which is implied by $f(x+2) = f(x)+2$ ). Also, $h(x)$ has infinite roots (all even numbers), so $h(x) = 0$ from here.
For the second question, put $x = 0$ to get: $g(0) = 2g(0) \iff g(0) = 0$. So, we can write: $$g(x) = xp(x)$$ Thus, $$2xp(2x) = 2xp(x)$$ Suppose $x \neq 0$ then we have: $$p(2x) = p(x)$$
It can be shown that $p(x) = p(1)$ is some plain old constant.