Find the values of a cubical function graph

Question

I'm having some problem with this question, I could only find the value of [1-5] and [1-7], which is quite easy.

But in the others I'm having some difficulty.

Thanks in advance.

Answer 1

You can use Lagrange interpolation :

The Idea : If you know n+1 values at respectively n+1 disctinct points for a polynom of degree n , so you know the polynom completely. (Because of number of coefficient of polynom (argument of dimension)).

So for your points $ P(a_i) $ for i from 0 to 3 ,

Lagrange said :

$$ P(X)=\sum_{i=0}^3P(a_i) \prod_{j=0,j \neq i}^3\dfrac{(X-a_j)}{a_i-a_j} $$

Develop and win.

2) You can also solve your 4x4 system given by plugin values in the equation.