# Interpolation and Polynomial Approximation

The following data are given for a polynomial P (x) of unknown degree.

x      0     1     2      3
_______________________
P(x)     4     9    15    18

Determine the coefficient of x^3 in P(x) if all fourth-order forward differences are 1.

### Comments for Interpolation and Polynomial Approximation

 Mar 25, 2013 Polynomial Approximation by: Staff AnswerPart II worked out the equation for you two ways: using the (1) Newton's Form, and the (2) Vander monde Approachthe coefficient of x³ in P(x) = -⅔The complete 3rd degree polynomial function passing through the four points is: A graph of the function is shown below: Deriving the EquationNewton’s formThe advantage of using this method is that you can compute the coefficients easily, by hand.With four data points, the formula is: -----------------------------------------------------------

 Mar 25, 2013 Polynomial Approximation by: Staff ----------------------------------------------------------- Part II n = 4. The curve must go through four points. when x = 0, P(x) = 4 when x = 1, P(x) = 9 when x = 2, P(x) = 15 when x = 3, P(x) = 18 -----------------------------------------------------------

 Mar 25, 2013 Polynomial Approximation by: Staff -----------------------------------------------------------Part IIIThe final equation is:Vander monde Approach The standard form of an Interpolating Polynomial for four data points is the following third degree polynomial. Determine the value of the coefficients a₁,a₂, a₃, and a₄ as follows:Evaluate the function for the data point (0, 4) -----------------------------------------------------------

 Mar 25, 2013 Polynomial Approximation by: Staff ----------------------------------------------------------- Part IV Evaluate the function for the data point (1, 9) Evaluate the function for the data point (2, 15) Evaluate the function for the data point (3, 18) -----------------------------------------------------------

 Mar 25, 2013 Polynomial Approximation by: Staff ----------------------------------------------------------- Part V Determine the value of the coefficients a₁,a₂, a₃, and a₄ by solving the following system of four linear equations: The final derivation of the equation is: Thanks for writing. Staff www.solving-math-problems.com

 Apr 08, 2020 Second condition NEW by: Giant Butt What about the second condition of "if all fourth-order forward differences are 1"?