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.
Consider
f(x)=a*x^n+b*x^(n-1)+...+nx+p
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))?
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So f(-x)=a*(-x)^n+b*(-x)^(n-1)+...+n(-x)+p.
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?