Math Problem Statement

Consider the following points. (−2, 28), (−1, 0), (0, −6), (1, −8), (2, 0) (a) Determine the polynomial function of least degree whose graph passes through the given points. p(x) =

(b) Sketch the graph of the polynomial function, showing the given points.

Solution

Here is the graph of the polynomial function p(x)=x4x3+x23x6p(x) = x^4 - x^3 + x^2 - 3x - 6. The red points on the graph correspond to the given points (2,28)(-2, 28), (1,0)(-1, 0), (0,6)(0, -6), (1,8)(1, -8), and (2,0)(2, 0).

Would you like more details on how this graph was constructed or have any other questions?

Here are some related questions you might find useful:

  1. What is the general form of a polynomial of degree nn?
  2. How do you determine the degree of a polynomial based on given points?
  3. What are the roots of the polynomial p(x)=x4x3+x23x6p(x) = x^4 - x^3 + x^2 - 3x - 6?
  4. How can you use interpolation to find a polynomial function?
  5. What is the significance of the leading coefficient in a polynomial function?
  6. How does the degree of a polynomial affect the shape of its graph?
  7. What are the key features of the graph of a polynomial function?
  8. How do you determine if a polynomial function has symmetry?

Tip: When finding a polynomial function from points, always check if the number of points corresponds to the degree of the polynomial you're solving for.

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Math Problem Analysis

Mathematical Concepts

Polynomial functions
Interpolation

Formulas

Lagrange interpolation formula

Theorems

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Suitable Grade Level

High School