1. The problem statement, all variables and given/known data http://prntscr.com/daze68 What I don't understand: 1. "P be a polynomial with degree n" do these equations satisfy this description?: $$p(x) = (x^2 + x)^n$$ $$p(x) = (5x^2 + 2x)^n$$ etc. 2. "C1 is a curve defined by y=p(x)" c1 is essentially just the curve of the polynomial? 3. p(x) = e^x This is meant to find points of intersection(which are also called roots? Correct me if i'm wrong) right? That there are n solutions to this? 4. h(x) = p(x) - e^x Where does this h(x) function come from? I thought this would equal to 0 if you rearrange the equation from 3. Is it just a function introduced for the sake of answering this problem? 5. "Since p is of degree n, the nth derivative is just a constant" What is this saying? That after differentiation n is still a constant? Since it's a polynomial it has to be some constant to begin wtih right? 6. How do they go from h(x) = p(x) - e^x to h^(n)(x) = 0 and why does it have only one solution/root???? If you differentiate then it becomes h'(x) = p'(x) - e^x right? I know that the MVT states( i think ) that the derivative will have n-1 roots. There exists a root between the interval of its original function but how does that apply here? Sorry if this is inappropiate for a question i just don't understand the problem at its core.