An algebra question

Question

1. The problem statement, all variables and given/known data
This problem is a part of a set of problems, I've solved all of them and this problem remains

1. If f(x) is a non-zero polynomial of degree d, show that f(x) - f(x-1) is a polynomial of degree d - 1. (For the purposes of this result, it is convenient to define the degree of the zero polynomial to be -1.)

2.If f(x) = g(x) + h(x), show that f(x) - f(x-1) = g(x) - g(x-1) + h(x) - h(x-1)

3. Use the binomial theorem to calculate x^(k+1)/(k+1) - (x-1)^(k+1)/(k+1)

(Answer is: x^k - (kx^(k-1)/2) + (kC2)x^(k-2)/3 - ...... - kx^2/2 + x - 1/(k+1))

 4. Suppose that for all i <= k one knows a polynomial f_i such that f_i(x) - f_i(x-1) = x^i . The last problem gave us fi for i = 0, 1, 2, 3, and 4. Combine your work on this problem in order to show you how to calculate . fk+1

I have solved the first 3 problems but I have no idea about how to solve the fourth one whatsoever, it is so confusing

Answer 1

1)

 

I think there is a typo somewhere in this part.

 

 

2)

 

f(x) - f(x - 1) = [g(x) - g(x - 1)] + [h(x) - h(x - 1)]

 

 

I simply put this in distributive form using brackets.  As you can see, the form is consistent.  You are just taking the difference between the functions in the interval  [x-1 , x].

 

 

 

4)

 

I would plug in all the value of i into the function.