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.

Re: finding a polynomial from another

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 $\displaystyle g(x)= ax^2+ bx+ c$. The condition that $\displaystyle f(0)= a(0)^2+ b(0)+ c= 0$ tells you that c= 0. So $\displaystyle g(x)- g(x-1)= ax^2+ bx- a(x-1)^2- b(x-1)= ax^2+ bx- ax^2+ 2ax- a- bx+ b= 2ax+ b- a= 2x- 1$ 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.

Re: finding a polynomial from another

[Deleted]. HallsOfIvy's answer is better (I would recommend trying it with g(x) having the form $\displaystyle ax^3 + bx^2 +cx + d$ as practice for this particular question - obviously you'll see some redundancy, but you will get to the same answer).