Determine a polynomial function whose graph passes through the given points. 5: (0,-5); (1,-4) ; (-1,-9); (2,-3),

The interpolating polynomial is

.

Step-by-step explanation:

First, notice that we want to calculate the interpolating polynomial 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

.