Question

Finding the interpolating polynomial for the function

Answer 1

Answer:

  f(x) = x^3 -2x^2 -3x +7

Step-by-step explanation:

Cubic polynomial regression using your favorite tool (graphing calculator, spreadsheet, or web site) will tell you the interpolating polynomial is ...

  f(x) = x^3 -2x^2 -3x +7

_____

You can use Lagrange polynomial interpolation. It gives the function as the sum of four factored cubics:

  $f(x)=3\cdot\dfrac{(x-2)(x-3)(x-4)}{(1-2)(1-3)(1-4)}+1\cdot\dfrac{(x-1)(x-3)(x-4)}{(2-1)(2-3)(2-4)}+\\\\\text{ }\qquad7\cdot\dfrac{(x-1)(x-2)(x-4)}{(3-1)(3-2)(3-4)}+27\cdot\dfrac{(x-1)(x-2)(x-3)}{(4-1)(4-2)(4-3)}$

Or, you can write equations for the coefficients a, b, c, d of ...

  ax^3 +bx^2 +cx +d = f(x)

These would be ...

  a + b + c + d = 3

  8a +4b +2c +d = 1

  27a +9b +3c +d = 7

  64a +16b +4c +d = 27

Your friendly linear equation solver will tell you ...

  (a, b, c, d) = (1, -2, -3, 7) . . . matches the equation shown above