Find a basis for all of the even polynomials and all of the odd polynomials in $P_4(R)$.
So, firstly, I know that a function is called even if f(−x) = f(x) for all x, and a function is called odd if f(−x) = −f(x) for all x. Thus, an odd function will have some leading coefficient like a$x^n$, where n is odd (and vice-versa for even). However, finding a basis for this is tricky. Can anyone help me out please?
Hint: $P_4(\mathbb{R})$ has (standard) basis $$ \{ 1,x,x^2,x^3,x^4\} $$ Which of these do you need to describe even polynomials? What happens if you throw an odd degree term in?