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Copyright © 2011 Pearson, Inc. 2.4 Real Zeros of Polynomial Functions

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 2 What you’ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower Bounds … and why These topics help identify and locate the real zeros of polynomial functions.

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 3 Division Algorithm for Polynomials

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 4 Example Using Polynomial Long Division

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 5 Example Using Polynomial Long Division

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 6 Remainder Theorem

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 7 Example Using the Remainder Theorem

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 8 Example Using the Remainder Theorem

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 9 Factor Theorem

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 10 Fundamental Connections for Polynomial Functions For a polynomial function f and a real number k the following statements are equivalent: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. k is an x-intercept of the graph of y = f(x). 4. x – k is a factor of f(x).

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 11 Example Using Synthetic Division

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 12 Example Using Synthetic Division

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 13 Rational Zeros Theorem

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 14 Upper and Lower Bound Tests for Real Zeros

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 15 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 16 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 17 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 18 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 19 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 20 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 21 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 22 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 23 Example Finding the Real Zeros of a Polynomial Function

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 24 Quick Review

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Copyright © 2011 Pearson, Inc. Slide 2.4 - 25 Quick Review Solutions

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