Good! Notice that p = p*x^0 is an even term, so that would technically be covered in the statement about coefficients of even terms.
What must be true about the coefficients a, b, ...., p to have f an even function (f(-x)=f(x))? odd function (f(-x)=-f(x))?
So for f(x) to be even, the coefficients with odd powers must=0 and those with even powers can be anything.
For f(x) to be odd, the coefficients with even powers must be 0 and those with odd powers can be anything. Also, p=0
Is this correct?