Final answer:
To find the cubic polynomial interpolating the points using Newton's interpolation, we build a divided differences table and use the results to construct the polynomial in the Newton form.
Step-by-step explanation:
To determine the cubic polynomial that interpolates the data points (−1,−7), (0,−3), (1,−1), and (2, 1-1) using Newton's interpolation and divided differences, we first compute the divided differences table based on the given points.
The divided differences table is constructed as follows (ignoring the typo in the last point which should be (2,y)):
- f[x0] = -7
- f[x0, x1] = (−3 - (−7))/((0 - (−1))) = 4
- f[x1] = -3
- f[x0, x1, x2] = (f[x1, x2] - f[x0, x1])/((1 - (−1)))
- f[x1, x2] = (−1 - (−3))/((1 - 0)) = 2
- f[x2] = -1
- Continuing this process, we obtain the necessary coefficients to construct the interpolating polynomial.
The polynomial is structured as follows:
P(x) = f[x0] + f[x0, x1](x - x0) + f[x0, x1, x2](x - x0)(x - x1) + ...
By populating the coefficients calculated above into the polynomial, we can obtain the required cubic polynomial that interpolates the given points.