Find a basis for the subspace $ = \{{ ∈ 4[] | (1) = (−1) = 0}\}$, 4[]. Justify your answer.
I tried several things. I'm having a hard time understanding how to use what I've been given.
I get the general polynom of like this: $a + bx + cx^2 + dx^3 + ex^4$
The solution for p(1) = 0 is
$a + b + c + d + e$
The solution for p(-1) = 0 is
$a - b + c - d + e$
What do I do? When I compare the two I get that $b = -d$
Do I need to do more? How do I use what I've been given correctly. Thanks
Think about what kind of polynomials satisfy the given condition.. \begin{align*}p \in W \iff \exists q\in P_2[x] : p(x) = (x-1)(x+1)q(x). \end{align*} I'm sure you can list a few. Also notice that if you have a set of polynomials in which no two polynomials have the same degree, then it must be linearly independent! E.g. $\{1,x,x^2\}.$