1. The profit from selling local ballet tickets depends on the ticket price. Using past receipts, we find that the profit can be modeled

Transcription

1 Warm-up 1. The profit from selling local ballet tickets depends on the ticket price. Using past receipts, we find that the profit can be modeled by the function p x = 15x x + 60, where x is the price in dollars of each ticket. We want to find the ticket price that gives the maximum profit, and also find that maximum profit. 2. Taylor and Miranda are performing on a magic dimensionchanging stage that is 20 yards long by 15 yards wide. The length is decreasing linearly (with time) at a rate of 2 yards per hour, and the width is increasing linearly (with time) at a rate of 3 yards per hour. When will the stage have the maximum area, and when will the stage disappear (have an area of 0 square yards)?

2 Chapter 2 Sections 6 and 7 Main Ideas Key Terms Using Algebra to Solve a Polynomial Equation General Theorems Polynomial Equation that has a Quadratic Form about Polynomial Equations

3 2.6 Solving Polynomial Functions by Factoring Objective Grouping Terms To solve polynomial equations by various methods of factoring, including the use of the rational root theorem. 3 2 x x x x x 4x x x x x x x 2 x 2 x 5 0 x 2, 5

4 Try it: x 3 + 6x 2 4x 24 = 0 (x 3 +6x 2 ) (4x + 24) = 0 x 2 (x + 6) 4(x + 6) = 0 (x 2 4)(x + 6) = 0 (x 2)(x + 2)(x + 6) = 0 x = 6, ±2

5 4 2 Solve: 2x x 3 0 by rewriting the quartic in quadratic form. 2 x 2 2 x 2 3 = 0 2 m 2 m 3 = 0 2m 2 m 3 = 0 2m 3 m + 1 = 0 m = 3, m = 1 2 Let m = x 2 x 2 = 3 2, x2 = 1 x x = 3 2, x = 1 6, i 2

6 Try it: 2x 4 = 7x x 4 + 7x 2 15 = 0 2 x x 2 15 = 0 2 m m 15 = 0 Let m = x 2 x 2 = 3 2, x2 = 5 2m 2 + 7m 15 = 0 2m 3 m + 5 = 0 x = 3 2, x = 5 m = 3 2, m = 5 x = ± 6 2, ±i 5

7 1.Solve by grouping: 2x 3 +8x 2 9x 36 = 0 2.Rewrite each equation in quadratic form: a. 2x x = 0 b. x 12 5x = 0

8 1.Solve by grouping: 2x 3 +8x 2 9x 36 = 0 (2x 3 +8x 2 ) (9x + 36) = 0 2x 2 (x + 4) 9(x + 4) = 0 2x 2 9 = 0 2x 2 = 9 (2x 2 9)(x + 4) = 0 x 2 = 9 2 x = ± 9 2 = ± 3 2 = ± x = ± 3 2 2, 4

9 2. Rewrite each equation in quadratic form: a. 2x x = 0 Let m = 2x 1 m 2 5 m + 4 = 0 b. x 12 5x = 0 Let m = x 6 m 2 5 m + 4 = 0

10 Section 2.7 General Results for Polynomial Equations Objective: To Apply General Theorems About Polynomial Equations

11 2.7 General Results for Polynomial Equations THM 1. The Fundamental Theorem of Algebra In the complex number system consisting of all real and imaginary numbers... P( x) is a polynomial of degree n ( n 0) with complex coefficients P( x) has exactly n roots (provided a double root is counted as two roots,...)

12 2.7 General Results for Polynomial Equations Theorem Equation 2ix 5 x (3 2 i) x 7 0 # of Roots x 11x 19x 25x x 4 x 2 x + 5 x 7 3 x 1 = 0 10

13 2.7 General Results for Polynomial Equations THM 2. Complex Conjugates Theorem a bi is an imaginary root of P( x) a bi is also a root THM 3. Suppose Px ( ) is a polynomial with rational coefficients, and a and b are rational numbers such that b is irrational. If a b is a root of P( x), then a b is also a root.

14 2.7 General Results for Polynomial Equations Theorems 2 & 3 1. A cubic equation with real coefficients has roots -1 and 3 + 2i. What is the third root? 2. A quadratic equation with integral coefficients has a root What is the other root? 3 2i 3 5 3

15 2.7 General Results for Polynomial Equations THM 4. Px ( ) is a polynomial of odd degree with real coefficients Px ( ) has at least one real root.

16 2.7 GENERAL RESULTS FOR POLYNOMIAL EQUATIONS THEOREM 4 1.How many times will a seventh degree polynomial cross the x-axis? 1 to 7 2. How many real roots can a quartic equation have? 0 to 4

17 2.7 GENERAL RESULTS FOR POLYNOMIAL EQUATIONS x + 5 x + 3 (x 6)(x + 5) = x 2 + 8x + 15 = x 2 x 30 x = 5, x = 3 x = 6, x = 5 If the roots are known what is the short cut to find the quadratic? 2 x x sum of the roots product of the roots 0

18 2.7 General Results for Polynomial Equations 2 x x sum of the roots product of the roots 0 Find a quadratic equation with roots 2 3i sum = 2 3i 2 3i 4 product = 2 3i 2 3i 13 x 2 4x 13 0

19 Quick Check: 2 x x sum of the roots product of the roots 0 What is the quadratic with roots of 3 and 7? What is the quadratic with a root of 1 + 4i?

20 Quick Check: 2 x x sum of the roots product of the roots 0 What is the quadratic with roots of 3 and 7? Sum = = 4 Product = 3 7 = 21 x 2 4x 21

21 2 x x sum of the roots product of the roots 0 What is the quadratic with a root of 1 + 4i? Sum = 1 + 4i + 1 4i = 2 Product = 1 + 4i 1 4i = 1 + 4i 4i 4i 2 = 1 16( 1) The other root is: 1 4i = 17 x 2 + 2x + 17

22 Homework Page 83 #1, 3, 5, 7, 23, 25, 33 Page 89 #13, 15, 19