We'll find the interpolating polynomial passing through the three points , , , using the Vandermonde matrix.
For our polynomial, we'll take , , and .
Since we have 3 points, we can expect degree 2 polynomial.
So define our interpolating polynomial as:
So, to find the coefficients of our polynomial, we solve the system , .
In order to solve the system, we will use an augmented matrix based on the Vandermonde matrix, and solve for the coefficients using Gaussian elimination. Substituting in our and values, our augmented matrix is:
Then, using Gaussian elimination,
Our coefficients are , , and . So, the interpolating polynomial is
Now we add a point, , to our data set and find a new interpolation polynomial with this method.
Since we have 4 points, we will have degree 3 polynomial.
Thus our polynomial is ,
and we get the coefficients by solving the system .
Constructing our augmented matrix as before and using Gaussian elimination, we get:
Therefore, our polynomial is: