Problem 1 Consider the polynomial interpolation for the following data points x -2 0 1 2 y -5 1 1 7

(a). Write down the linear system in matrix form for solving the coefficients ai (i = 0, · · · , n) of the polynomial pn(x).

(b). Use the Lagrange interpolation process to obtain a polynomial to approximate these data points (simplify your answer).

Problem 2 (a). Use the Lagrange interpolation process to obtain a polynomial of least degree that assumes these values: x 0 2 3 4 y 7 11 28 63

(b). For the points in the Table of (a), find the Newton’s form of the interpolating polynomial. Show that the two polynomials obtained are identical, although their forms may differ.

(c). The polynomial p(x) = x 4 − x 3 + x 2 − x + 1 has the values shown.

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