Algebra for Newton's divided differences (Background: I'm learning about divided differences from this YouTube video).
What I'm looking for is to understand the steps to rearrange a particular equation from one form to another. The original equation is: $$f(x_2) = f(x_0) + \frac{f(x_1)-f(x_0)}{(x_1-x_0)}(x_2-x_0) + b_2(x_2-x_0)(x_2-x_1)$$
The equation when solved for $b_2$ is: $$b_2 = \frac{\frac{f(x_2)-f(x_1)}{x_2-x_1}-\frac{f(x_1)-f(x_0)}{x_1-x_0}}{x_2-x_0}$$
I've tried to solve for $b_2$ but keep messing up! Perhaps someone would be kind enough to show me the steps?
 A: Let's replace $f(x_i)$ with $y_i$, to reduce clutter. We start with
$$ y_2 = y_0 + \frac{y_1-y_0}{x_1 - x_0}(x_2 - x_0) + b_2(x_2-x_0)(x_2-x_1). $$
We want the factor involving $b_2$ on one side, and the rest on the other side. So let's move the first two terms from the right-hand side to the left-hand side:
$$ y_2 - y_0 - \frac{y_1 - y_0}{x_1 - x_0}(x_2 - x_0) = b_2(x_2 - x_0)(x_2 - x_1). $$
In order to isolate $b_2$, we need to divide by $(x_2 - x_0)(x_2 - x_1)$. Let's do it in two steps. First divide by $x_2 - x_0$:
$$ \frac{y_2 - y_0}{x_2 - x_0} - \frac{y_1 - y_0}{x_1 - x_0} = b_2(x_2 - x_1). $$
Now divide by $x_2 - x_1$:
$$ \frac{\frac{y_2 - y_0}{x_2 - x_0} - \frac{y_1 - y_0}{x_1 - x_0}}{x_2 - x_1} = b_2. $$
We got a different formula.
In order to get the formula you stated, we need to somehow come up with the expression $y_2 - y_1$. So let's subtract $y_1$ from both sides of the original equation:
$$ y_2 - y_1 = y_0 - y_1 + \frac{y_1-y_0}{x_1 - x_0}(x_2 - x_0) + b_2(x_2-x_0)(x_2-x_1). $$
We have
$$ y_0 - y_1 + \frac{y_1-y_0}{x_1-x_0}(x_2-x_0) = \frac{y_1-y_0}{x_1-x_0}(x_2 - x_0 - (x_1 - x_0)) = \frac{y_1-y_0}{x_1-x_0}(x_2 - x_1). $$
For this calculation, we wrote
$$ y_0 - y_1 = \frac{(y_0-y_1)(x_1-x_0)}{x_1-x_0}. $$
Substituting this simplification into the equation, we get
$$y_2 - y_1 = \frac{y_1-y_0}{x_1-x_0}(x_2-x_1) + b_2(x_2-x_0)(x_2-x_1).$$
Let's move everything to the correct side:
$$y_2 - y_1 - \frac{y_1-y_0}{x_1-x_0}(x_2-x_1) = b_2(x_2-x_0)(x_2-x_1).$$
Now divide first by $x_2 - x_1$ (looking toward our goal):
$$\frac{y_2 - y_1}{x_2 - x_1} - \frac{y_1-y_0}{x_1-x_0} = b_2(x_2-x_0).$$
Next divide by $x_2 - x_0$:
$$\frac{\frac{y_2 - y_1}{x_2 - x_1} - \frac{y_1-y_0}{x_1-x_0}}{x_2-x_0} = b_2.$$
And we're done.
