finding a polynomial from another

Hi, given a polynomial, how can one find another from it satisfying a condition. E.g if a given polynomial equation is f(x) ,we need to find another polynomial , g(x) ,with the relation that g(0)=0 and g(x)-g(x-1)=f(x).
E.g if f(x)=2x-1 then the required g(x)=x^2 .

Do we need to use integration here?
Thanks.

It is clear that g must be a polynomial of degree no more than 1 higher than the degree of f (because the highest power of g(x) will cancel in "g(x)- g(x- 1)) . If f(x)= 2x- 1 then you can write . The condition that tells you that c= 0. So for all x. In order that two polynomials be the same for all x, they must have corresponding coefficients the same: 2a= 2 and b- a= -1. 2a= 2 gives a= 1 and then b- a= b- 1= -1 gives b= 0.

[Deleted]. HallsOfIvy's answer is better (I would recommend trying it with g(x) having the form as practice for this particular question - obviously you'll see some redundancy, but you will get to the same answer).