Newton Divided Difference Problem

So my problem says:

Given the below data of polynomial P(x), determine the coefficient of all x^{2} in P(x) if all third-order Newton divided difference are 1

x 0 1 2

P(x) 2 -1 4

What I did was find the First and second order divided differences. I think I might have done the problem incorrectly though, because I didn't use that the third order difference is 1. I was taught that the coefficient of x^{2} would be the second order divided difference, so I just found that using the data points.

I got: f[x_{0},x_{1}]=-3, f[x_{1},x_{2}]=5, and f[x_{0},x_{1, }x_{2}]=-4

I thought 4 would be the coefficient, but the back of the book says its 3.5. Any help?

Re: Newton Divided Difference Problem

Since the third order difference is constant, we know that we must have:

$\displaystyle P(x)=Ax^3+Bx^2+Cx+D$

Since the third order difference is 1, we have:

$\displaystyle P'''(x)=6A=1\:\therefore\:A=\frac{1}{6}$

Now, we are given:

$\displaystyle P(0)=D=2$ and this results in the 2X2 system:

$\displaystyle B+C=-\frac{19}{6}$

$\displaystyle 4B+2C=\frac{2}{3}$

Solving this system, we find:

$\displaystyle B=\frac{7}{2}$