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 =−𝟏𝟓 𝒙 𝟐 +𝟔𝟎𝟎𝒙+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 dimension-changing 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)? 

Chapter 2 Sections 6 and 7 Learning Targets Factor to Solve a Polynomial Equation Apply General Theorems about Polynomial Equations

In your group, can you find the answer to any of the following problems? 1. Factor: 𝑥 3 +5 𝑥 2 −4𝑥−20=0 2. Solve: 2 𝑥 4 − 𝑥 2 −3=0 3. Find a quadratic equation that has roots 2±3𝑖

2.6 Solving Polynomial Functions by Factoring

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

2 𝑥 2 2 − 𝑥 2 −3=0 Let 𝑚= 𝑥 2 2 𝑚 2 − 𝑚 −3=0 𝑥 2 = 3 2 , 𝑥 2 =−1 2 𝑚 2 −𝑚−3=0 𝑥= 3 2 ,𝑥= −1 2𝑚−3 𝑚+1 =0 𝑚= 3 2 , 𝑚=−1

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

Rewrite each equation in quadratic form: Solve by grouping: Rewrite each equation in quadratic form: 𝟐𝒙 𝟑 +𝟖 𝒙 𝟐 −𝟗𝒙−𝟑𝟔=𝟎 a. 𝟐𝒙−𝟏 𝟐 −𝟓 𝟐𝒙−𝟏 +𝟒=𝟎 b. 𝒙 𝟏𝟐 −𝟓 𝒙 𝟔 +𝟒=𝟎

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

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

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

2.7 General Results for Polynomial Equations

2.7 General Results for Polynomial Equations Theorem 1 𝑥 4 𝑥−2 𝑥+5 𝑥−7 3 𝑥−1 =0 10

2.7 General Results for Polynomial Equations

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

If the roots are known what is the short cut to find the quadratic? 2.7 General Results for Polynomial Equations 𝑥+5 𝑥+3 (𝑥−6)(𝑥+5) = x2 + 8x + 15 = x2 – x – 30 𝒙=−𝟓, 𝒙=−𝟑 𝒙=𝟔, 𝒙=−𝟓 If the roots are known what is the short cut to find the quadratic?

2.7 General Results for Polynomial Equations

Quick Check: What is the quadratic with roots of –3 and 7? What is the quadratic with a root of –1 + 4i?

𝒙 𝟐 −𝟒𝒙−𝟐𝟏 Sum = −3+7=4 Product = −3∗7=−21 Quick Check: What is the quadratic with roots of –3 and 7? Sum = −3+7=4 Product = −3∗7=−21 𝒙 𝟐 −𝟒𝒙−𝟐𝟏

𝒙 𝟐 +𝟐𝒙+𝟏𝟕 What is the quadratic with a root of –1 + 4i? Sum = −1+4𝑖+ The other root is: −𝟏−𝟒𝒊 Sum = −1+4𝑖+ −𝟏−𝟒𝒊=−𝟐 Product = −1+4𝑖 −1−4i =𝟏+𝟒𝒊−𝟒𝒊− 𝟒𝒊 𝟐 =𝟏−𝟏𝟔(−𝟏) =𝟏𝟕 𝒙 𝟐 +𝟐𝒙+𝟏𝟕

Homework HW A:Page 83 #1, 3, 5, 7, 23, 25, 33 Page 89 #13, 15, 19 HW B:Page 83 #5,7,23,25,33,35,39,41* Page 89 #13, 19, 23

Cubic Efficiency