suppose p(x) is in ker(T). what does this mean? it means T(p(x)) = 0 (the polynomial, not the number).

since T(p(x)) = x^{2}p(x), we have 2 possibilities:

a)x^{2}is the 0-polynomial. is this ever true?

b) p(x) is the 0-polynomial. what does this say about dim(ker(T))?

note that dim(P_{2}) = 3 (the "2" is misleading). what does the rank-nullity theorem tell you about dim(im(T))?