Question

Suppose the polynomial \( a + bx + cx^2 + dx^3 \) interpolates the data,
\[ (-1,1), (0,3), (1,2), (2,4). \] Then which one of the following statements is correct?

• A. \( a = -2c, d = -2b \)
• B. \( a = 2c, d = 2b \)
• C. \( b = 3c, a = 2d \)
• D. \( b = 2c, a = 3d \)

Hint:

When solving interpolation problems, set up a system of equations based on the given points and solve for the unknowns in the polynomial coefficients.

Answer 1

The Correct Option is A

To solve this problem, we need to find the coefficients of the polynomial \( a + bx + cx^2 + dx^3 \) that satisfies the given interpolation conditions. We can use the data points to set up a system of equations.
The given data points are:
- For \( (x = -1, y = 1) \), we get the equation: \( a - b + c - d = 1 \).
- For \( (x = 0, y = 3) \), we get the equation: \( a = 3 \).
- For \( (x = 1, y = 2) \), we get the equation: \( a + b + c + d = 2 \).
- For \( (x = 2, y = 4) \), we get the equation: \( a + 2b + 4c + 8d = 4 \).
By substituting \( a = 3 \) into the other equations, we obtain the following system of equations: \[ 3 - b + c - d = 1 \Rightarrow b - c + d = 2, \] \[ 3 + b + c + d = 2 \Rightarrow b + c + d = -1, \] \[ 3 + 2b + 4c + 8d = 4 \Rightarrow 2b + 4c + 8d = 1. \] Solving this system, we find: \[ a = -2c, d = -2b. \] Thus, the correct option is (A).