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4-5 Locating zeros of a polynomial function

Chapter 4: Polynomial and Rational Functions. 4-5 Locating zeros of a polynomial function. 4-5 Locating Zeros of a Polynomial Function. Example 1: The Location Principal.

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4-5 Locating zeros of a polynomial function

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  1. Chapter 4: Polynomial and Rational Functions 4-5 Locating zeros of a polynomial function

  2. 4-5 Locating Zeros of a Polynomial Function Example 1: The Location Principal Suppose f(x) represents a polynomial function with real coefficients. If a and b are two numbers with f(a) negative and f(b) positive, then the function has at least one real zero between a and b. (the graph must cross the x-axis between a and b) f(x) = -2x2 + 8x –1

  3. 4-5 Locating Zeros of a Polynomial Function Problem 1: Determine between which consecutive integers the real zeros of each function are located. f(x) = x4 – 4x3 + 8x2 + 36x – 9

  4. 4-5 Locating Zeros of a Polynomial Function Example 2: Approximate the real zeros of the function below to the nearest tenth. f(x) = x4 – 4x3 + 8x2 + 36x – 9

  5. 4-5 Locating Zeros of a Polynomial Function Problem 2: Approximate the real zeros of the function below to the nearest tenth. f(x) = 18x2 – 27x – 35

  6. 4-5 Locating Zeros of a Polynomial Function An upper bound limit is an integer greaten than or equal to the greatest real zero of a function. A lower bound is an integer less than or equal to the least real zero of a function. Upper Bound Theorem: Suppose c is a positive real number and P(x) is divided by x – c. If the resulting quotient and remainder have no change in sign, then P(x) has no real zero greater than c. Thus, x is an upper bound of the zeros P(x) Lower Bound Theorem: If c is an upper bound of the zeros of P(-x), then –c is a lower bound of the zeros of P(x).

  7. 4-5 Locating Zeros of a Polynomial Function Example 3: Use the Upper Bound Theorem to find an integral upper bound and the Lower Bound Theorem to find an integral lower bound of the zeros of f(x) = 2x4 – 3x – 6

  8. 4-5 Locating Zeros of a Polynomial Function Problem 3: Use the Upper Bound Theorem to find an integral upper bound and the Lower Bound Theorem to find an integral lower bound of the zeros of f(x) = x4 – 3x3 – 2x2 + 3x – 5

  9. 4-5 Locating Zeros of a Polynomial Function Practice 1: Use the Upper Bound Theorem to find an integral upper bound and the Lower Bound Theorem to find an integral lower bound of the zeros of f(x) = -2x3 + 4x2 + 1

  10. 4-5 Locating Zeros of a Polynomial Function Practice 2: Use the Upper Bound Theorem to find an integral upper bound and the Lower Bound Theorem to find an integral lower bound of the zeros of f(x) = 3x4 – x3 – 8x2 – 3x - 20

  11. 4-5 Locating Zeros of a Polynomial Function HW 4-5 pg. 240 #13-31 odd

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