Data interpolation has many applications. Suppose we have been given the task of finding the degree 3 polynomial
P(x) = ax³ + bx² + cx + d
which satisfies P'(-1) = 3 and passes through the points (-1, 1), (0, 1), and (1,3).
1. Show that the coefficients of P, a, b, c, and d satisfy the following system of equations given the four data points above.
3a – 2b + c=3
-a + b - c+d =1
d=1
a+b+c+d= 3