Question

If the polynomial \[ p(x) = \alpha + \beta (x+2) + \gamma (x+2)(x+1) + \delta (x+2)(x+1)x \] interpolates the data \[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline -2 & 2 \\ -1 & -1 \\ 0 & 8 \\ 1 & 5 \\ 2 & -34 \\ \hline \end{array} \] then \( \alpha + \beta + \gamma + \delta = \underline{\hspace{1cm}}\) .

Hint:

For interpolation problems, substitute the data points into the polynomial equation and solve for the unknown coefficients.

Answer 1

Correct Answer: 1

Since the polynomial interpolates the data, we substitute the values of \( x \) and \( f(x) \) into the polynomial and solve for \( \alpha, \beta, \gamma, \delta \). After solving the system of equations, we get: \[ \alpha + \beta + \gamma + \delta = \boxed{1}. \]