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Chapter 3: Polynomial Functions

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

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Chapter 3: Polynomial Functions

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  1. 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

  2. 3.8 Polynomial Equations and Inequalities • Methods for solving quadratic equations known to ancient civilizations • 16th 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

  3. 3.8 Solving Polynomial Equations: Zero- Product Property Example Solve Solution Factor by grouping. Factor out x + 3. Factor the difference of squares. Zero-product property

  4. 3.8 Solving an Equation Quadratic in Form Example Solve analytically. Find all complex solutions. Solution Let t = x2. Replace t withx2. Square root property

  5. 3.8 Solving a Polynomial Equation Example Show that 2 is a solution of and then find all solutions of this equation. Solution Use synthetic division. By the factor theorem, x – 2 is a factor of P(x).

  6. 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,

  7. 3.8 Using Graphical Methods to Solve a Polynomial Equation Example Let P(x) = 2.45x3 – 3.14x2 – 6.99x + 2.58. Use the graph of P to solve P(x) = 0, P(x) > 0, and P(x) < 0. Solution

  8. 3.8 Complex nth Roots • If n is a positive integer, k a nonzero complex number, then a solution of xn= 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 xn= k has exactly n complex roots.

  9. 3.8 Finding nth Roots of a Number Example Find all six complex sixth roots of 64. Solution Solve for x.

  10. 3.8 Applications and Polynomial Models Example A 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. • 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. • 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

  11. 3.8 Applications and Polynomial Models Solution • Volume = length  width  height • Use the graph of V to find the local maximum point. x  2.43 in, and the maximum volume  262.68 in3.

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