If $U =\{ f \in P_3| f(-1)=f(1)=0\}$ Then is $P_3 = U⊕P_2$?

Question

I know that I want to be able to show that $U\cap P_2= \{0\}$ I was able to work out that given the conditions, an element of U should be of the form $ax^3 + bx^2 -ax -b$. Is this correct? If so, how do I proceed from here?

Answer 1

This is not true.

If $a=0$ and say $b=1$ then you have a polynomial $p(x)=x^2-1\in U$ which is also in $P_2$, so $U\cap P_2\ne \{0\}$