Answer:
The interpolating polynomial is
.
Step-by-step explanation:
First, notice that we want to calculate the <em>interpolating polynomial</em> through the points (0,-5); (1,-4) ; (-1,-9); (2,-3). This means that we want to find a polynomial such that
, and .
We will have four equations, so our polynomial will be, at most, of degree 3. Let us write
The coordinates give us the following equations:
Notice that from the second equation we know that . Then, we obtained the linear system of equations
which is equivalent to
.
So, we have reduced our interpolation problem to solve a linear system of equations. Now, notice that if we add the first two equations of the system we obtain
that yields .
Then, our system becomes
which is equivalent to
.
Recall that now the first two equations are just the same, so we will use the first and third ones:
.
If we multiply the first one and add it to the second we get:
that yields .
Thus, substituting this value in the first equation:
which is equivalent to . Then, .
Summing up all our results we get that the interpolating polynomial is
.