Interpolation of a function at 4 points

Question

Assume that the cubic polynomial a + bx + cx^2 + dx^3 interpolates a function f(x) at the four points (0,2), (1,-1), (2,1), (3,3). I'm trying to do a question that asks me to write down a system of four equations in four unknowns that has as unique solutions the coefficients a,b,c,d. How would I create four such equations for these points? I understand that since there are four points there must be four equations, but i'm not sure how to make them.

Answer 1

So we know that $f(2)=a+2b+4c+8d$, by definition. But we also know that $f(2)=1$ (that's what you're given).

Thus one of the four equations is $a+2b+4c+8d=1$.

See if you can work out the rest (click the spoiler below if you want answers):

$a=2,a+b+c+d=-1,a+3b+9c+27d=3$