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Understanding Polynomial Functions: Theory, Graphs, and Zeros

Learn about polynomial functions of higher degree with modeling, graphing transformations, local extrema, zeros, multiplicities, and the Intermediate Value Theorem. Explore examples and practical applications.

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Understanding Polynomial Functions: Theory, Graphs, and Zeros

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  1. 2.3 Polynomial Functions of Higher Degree with Modeling

  2. The Vocabulary of Polynomials

  3. Example Graphing Transformations of Monomial Functions

  4. Example Graphing Transformations of Monomial Functions

  5. Cubic Functions

  6. Quartic Function

  7. Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n – 1 local extrema and at most n zeros.

  8. Example Applying Polynomial Theory

  9. Example Finding the Zeros of a Polynomial Function

  10. Example Finding the Zeros of a Polynomial Function

  11. Multiplicity of a Zero ofa Polynomial Function

  12. Zeros of Odd and Even Multiplicity If a polynomial function f has a real zero c of odd multiplicity, then the graph of f crosses the x-axis at (c, 0) and the value of f changes sign at x = c. If a polynomial function f has a real zero c of even multiplicity, then the graph of f does not cross the x-axis at (c, 0) and the value of f does not change sign at x = c.

  13. Example Sketching the Graph of a Factored Polynomial

  14. Intermediate Value Theorem If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other words, if y0 is between f(a) and f(b), then y0=f(c) for some number c in [a,b]. In particular, if f(a) and f(b) have opposite signs (i.e., one is negative and the other is positive, then f(c) = 0 for some number c in [a, b].

  15. Intermediate Value Theorem

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