1. Copyright © 2007 Pearson Education, Inc. Slide 3-1

2. Copyright © 2007 Pearson Education, Inc. Slide 3-2 Chapter 3: Polynomial Functions 3.1 Complex Numbers 3.2 Quadratic Functions and Graphs 3.3 Quadratic Equations and Inequalities 3.4 Further Applications of Quadratic Functions and Models 3.5 Higher Degree Polynomial Functions and Graphs 3.6 Topics in the Theory of Polynomial Functions (I) 3.7 Topics in the Theory of Polynomial Functions (II) 3.8 Polynomial Equations and Inequalities; Further Applications and Models

3. Copyright © 2007 Pearson Education, Inc. Slide 3-3 3.8 Polynomial Equations and Inequalities Methods for solving quadratic equations known to ancient civilizations 16 th century mathematicians derived formulas to solve third and fourth degree equations In 1824, Norwegian mathematician Niels Henrik Abel proved it impossible to find a formula to solve fifth degree equations Also true for equations of degree greater than five

4. Copyright © 2007 Pearson Education, Inc. Slide 3-4 3.8 Solving Polynomial Equations: Zero- Product Property ExampleSolve Solution Factor by grouping. Factor out x + 3. Factor the difference of squares. Zero-product property

5. Copyright © 2007 Pearson Education, Inc. Slide 3-5 3.8 Solving an Equation Quadratic in Form ExampleSolve analytically. Find all complex solutions. Solution Let t = x 2. Replace t with x 2. Square root property

6. Copyright © 2007 Pearson Education, Inc. Slide 3-6 3.8 Solving a Polynomial Equation ExampleShow that 2 is a solution of and then find all solutions of this equation. SolutionUse synthetic division. By the factor theorem, x – 2 is a factor of P(x).

7. Copyright © 2007 Pearson Education, Inc. Slide 3-7 3.8 Solving a Polynomial Equation To find the other zeros of P, solve Using the quadratic formula, with a = 1, b = 5, and c = –1,

8. Copyright © 2007 Pearson Education, Inc. Slide 3-8 3.8 Using Graphical Methods to Solve a Polynomial Equation ExampleLet P(x) = 2.45x 3 – 3.14x 2 – 6.99x + 2.58. Use the graph of P to solve P(x) = 0, P(x) > 0, and P(x) < 0. Solution

9. Copyright © 2007 Pearson Education, Inc. Slide 3-9 3.8 Complex nth Roots If n is a positive integer, k a nonzero complex number, then a solution of x n = k is called an nth root of k. e.g. ­ –2i and 2i are square roots of –4 since (  2i) 2 = –4 - –2 and 2 are sixth roots of 64 since (  2) 6 = 64 Complex nth Roots Theorem If n is a positive integer and k is a nonzero complex number, then the equation x n = k has exactly n complex roots.

10. Copyright © 2007 Pearson Education, Inc. Slide 3-10 3.8 Finding nth Roots of a Number ExampleFind all six complex sixth roots of 64. SolutionSolve for x.

11. Copyright © 2007 Pearson Education, Inc. Slide 3-11 3.8 Applications and Polynomial Models ExampleA box with an open top is to be constructed from a rectangular 12-inch by 20-inch piece of cardboard by cutting equal size squares from each corner and folding up the sides. (a)If x represents the length of the side of each square, determine a function V that describes the volume of the box in terms of x. (b)Determine the value of x for which the volume of the box is maximized. What is this volume? x x x x x x x x

12. Copyright © 2007 Pearson Education, Inc. Slide 3-12 3.8 Applications and Polynomial Models Solution (a)Volume = length  width  height (b)Use the graph of V to find the local maximum point. x  2.43 in, and the maximum volume  262.68 in 3.