Chapter 3: Polynomial Functions

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. 3.3 Quadratic Equations and Inequalities Quadratic Equation in One Variable An equation that can be written in the form ax2 + bx + c = 0 where a, b, and c are real numbers with a 0, is a quadratic equation in standard form. • Consider the • zeros of • x-intercepts of • solution set of • Each is solved by finding the numbers that make

4. 3.3 Zero-Product Property • If a and b are complex numbers and ab = 0, then a = 0 or b = 0 or both equal zero. Example Solve Solution The solution set of the equation is {– 4,2}. Note that the zeros of P(x) = 2x2 + 4x – 16 are –4 and 2, and the x-intercepts are (– 4,0) and (2,0).

5. 3.3 Example of a Quadratic with One Distinct Zero Example Solve Solution Graphing Calculator Solution There is one distinct zero, 3. It is sometimes called a double zero, or double solution (root) of the equation.

6. 3.3 Solving x2 = k Square Root Property The solution of is Figure 26 pg 3-38

7. 3.3 Examples Using the Square Root Property Solve each equation. • (b) Graphing Calculator Solution (a) (b) This solution is an approximation.

8. 3.3 The Quadratic Formula • Complete the square on and rewrite P in the form To find the zeros of P, use the square root property to solve for x.

9. 3.3 The Quadratic Formula and the Discriminant The Quadratic Formula The solutions of the equation , where , are • The expression under the radical, is called the discriminant. • The discriminant determines whether the quadratic equation has • two real solutions if • one real solution if • no real solutions if

10. 3.3 Solving Equations with the Quadratic Formula Solve Analytic Solution Rewrite in the form ax2 + bx + c = 0 Substitute into the quadratic formula a = 1, b = – 4, and c = 2, and simplify.

11. 3.3 Solving Equations with the Quadratic Formula Graphical Solution With the calculator, we get an approximation for

12. 3.3 Solving a Quadratic Equation with No Solution ExampleSolve Analytic Solution Graphing Calculator Solution Figure 30 pg 3-43

13. 3.3 Possible Orientations for Quadratic Function Graphs (a > 0) • Similar for a quadratic function P(x) with a < 0.

14. 3.3 Solving a Quadratic Inequality Analytically Solve The function is 0 when x = – 3 or x = 4. It will be greater than 0 or less than zero for any other value of x. So we use a sign graph to determine the sign of the product (x + 3)(x – 4). For the interval (–3,4), one factor is positive and the other is negative, giving a negative product. The product is positive elsewhere since (+)(+) and (–)(–) is positive.

15. 3.3 Solving a Quadratic Inequality Graphically Verify the analytic solution to Y1 < 0 in the interval (–3,4). • Solving a Quadratic Inequality • Solve the corresponding quadratic equation. • Identify the intervals determined by its solutions. • Use a sign graph to determine the solution interval(s). • Decide whether or not the endpoints are included.

16. 3.3 Literal Equations Involving Quadratics ExampleSolve (a) (b) Solution (a) (b)

17. 3.3 Modeling Length of Life The survival rate after age 65 is approximated by where x is measured in decades. This function gives the probability that an individual who reaches age 65 will live at least x decades (10x years) longer. Find the age for which the survival rate is .5 for people who reach the age of 65. Interpret the result. Solution Let Y1 = S(x), Y2 = .5, and find the positive x-value of an intersection point. Discard x = – 3. For x 2.2, half the people who reach age 65 will live about 2.2 decades or 22 years longer to age 87.