Question 3.7.1: Find a cubic polynomial of the form y = A + Bx + Cx² + Dx³ t...
Find a cubic polynomial of the form
y = A + Bx + Cx² + Dx³
that interpolates the data points (-1, 4), (1, 2), (2, 1), and (3,16).
In a particular problem, it generally is simpler to use distinct capital letters rather than subscripted symbols to denote the coefficients. Here we want to find the values of A, B, C, and D so that y(- 1) = 4, y(1) = 2, y(2) = 1, and y(3) = 16. These conditions yield the four linear equations
A – B + C – D = 4
A + B + C + D = 2
A + 2B + 4C + 8D = 1
A + 3B + 9C + 27D = 16.
We readily reduce this system to the echelon form
A – B + C – D = 4
B + D = – 1
C + 2D = 0
D = 2,
and then back substitution yields A = 7, B = – 3, C = – 4, and D = 2. Thus the desired cubic polynomial is
y = 7 – 3 x – 4 x² + 2 x³
The graph of this cubic is shown in Fig. 3.7.2, along with the four original data points.
