please show all work so I understand how to answer this type of question:

Assume that {x_i} from i=0 to n are arbitrary real numbers, and g : [0,1]-R is a continuous function. Prove that there exist real numbers {c_i} from i=0 to n such that n 0 p(x)g(x)dx = cip (?) (zi), Vp EPn(R) i=0