Hi,

my book starts to explain divided difference by saying that :

Suppose that $\displaystyle P_n(x)$ is the nth Lagrange polynomial that agrees with the function $\displaystyle f $at distinct numbers $\displaystyle x_0,x_1,\cdots,x_n$. The divided differences of $\displaystyle f$ with respect to $\displaystyle x_0,x_1,\cdots,x_n$ are used to express $\displaystyle P_n(x)$ in the form

$\displaystyle P_n(x) = a_0+a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\cdots+a_n(x-x_0)(x-x_1)\cdots(x-x_{n-1})$

Since I'm pretty slow, I do not see the logic behind this choice. I'm that I'm lacking some basic theory such that this appears like a mystical choice. Could someone point me in the right direction please?

Note:

I do understand why the interpolating polynomial between two points x_0 and x_1 can be written as,

$\displaystyle P_1(x) = f(x_0) + \frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_0),$

The literature often states that a convenient form of the 2-order polynomial is,

$\displaystyle P_2(x) =b_0 + b_1(x-x_0)+b_2(x-x_0)(x-x_1).$

How did man figure out that this is a convenient form? Trial and error? Thinking really hard?

Thanks.