How do I find the coefficient of $x^2$ in this polynomial, given its value at three points? We are given the following data about a polynomial $P(x)$ of unknown degree: $P(0) = 2$, $P(1) = −1$, $P(2) = 4$. 
Determine the coefficient of $x^2$ in $P(x)$ if all the third-order differences are $1$.

Since all third order differences are $1$ we know that this polynomial is of degree $3$ and leading coëfficiënt $\frac{1}{6}$.
$P(0) = 2$, so we know $P(x) = \frac{1}{6} x^3 + b x^2 + cx + 2$.
Because $P(1) = -1$ and $P(2) = 4$  we know that :
$ \frac{1}{6} + b + c + 2 = -1$
$ \frac{8}{6} + 4b + 2c + 2 = 4$.
The solution is $b = \frac {7}{2}$ and $c= \frac {-20} {3}$.
