Chapter 6 Polynomial Models and Linear Systems Section 6.1 Solving Polynomial Equations

Transcription

1 P a g e 53 Chapter 6 Polynomial Models and Linear Systems Section 6.1 Solving Polynomial Equations Definition: Polynomial Function A polynomial in x has an expression of the form where n is a nonnegative integer and The highest power of x in a polynomial is called the degree (or order) of the polynomial. The term with the highest power of x, that is is called the leading term of the polynomial. Example 1. Are the following functions polynomials? A first degree polynomial is a linear function. A second degree polynomial is a quadratic function. A polynomial of degree 3 is a cubic polynomial. Theorem A polynomial of degree n has at most (n-1) bends in the graph and has at most n zeros (or n x-intercepts or n roots). Example 2. Solve for x in the equation. Method 1: Intersect Tool Method 2: Zero Tool 53

2 P a g e 54 Method 3: solve function Method 4: SOLVER Example 3. Suppose you are a product designer for a consulting firm. Your client is a packaging manufacturer that has acquired cheaply a large surplus of rectangular cardboard sheets of various sizes. This cardboard will be used to make open-topped popcorn trays for use in movie theaters. Each tray will be constructed by cutting equal squares out of the corners of a cardboard sheet and then folding up the remaining flaps to form a box. Now, starting with a 20cm by 30cm sheet of cardboard, we construct a popcorn tray. (a) If its volume is precisely 500cubic cm, what are its dimensions? (b) What size squares must be cut out in order to have a tray with largest possible volume? 54

3 P a g e 55 Example 7 on page 250: A 12-foot ladder leans across a 5-foot fence and just touches a tall wall standing 3 feet behind the fence. How far is the foot of the ladder from the bottom of the fence? Recall Pythagorean Theorem and Similar Triangle Theorem: 55

4 P a g e 56 Section 6.2 Solving pairs of Linear Equations 1. Calculate the determinant of a 2 by 2 matrix matrix is a rectangular array of numbers with m rows and n columns and enclosed by a set of brackets [ ]. 2. Method of Elimination Example 1: Use method of elimination to solve the linear system 56

5 P a g e The Determinant Method 4. Matrices Method 57

6 P a g e 58 Applications: Example 9 on page 267: You are walking down the street minding your own business when you spot a thick envelope lying on the side walk. It turns out to contain fives and twenties, a total of 61 bills with a total value of $875. How many of each type of bills is there? Example 10 on page 267: Suppose large quantities of both a 7% alcohol solution and a 17% alcohol solution are available. How many gallons of each must be mixed in order to get 42 gallons of a 13% alcohol solution? 58

7 P a g e 59 Section 6.3 Linear Systems of Equations 1. Calculate the determinant of a 3 by 3 matrix by expansion Example 1. (a) Calculate the following determinant by expansion along the first row. (b) Calculate the following determinant by expansion along the second column. 59

8 P a g e Method of Elimination Example 2. Use the method of Elimination to solve the system: 60

9 P a g e The Determinant Method 4. The matrices Method 61

10 P a g e 62 Example 3. Solve the linear system: Example 6. You are walking down the street minding your own business when you spot a small but heavy leather bag lying on the sidewalk. It turns out to contain U.S. Mint American Eagle gold coins of the following types: One-half-ounce gold coins that sell for $285 each, One-quarter-ounce gold coins that sell for $150 each, and One-tenth-ounce gold coins that sell for $70 each. A bank receipt found in the bag certifies that it contains 258 such coins with a total weight of 67 ounces and a total value of exactly $ How many coins of each type are there? 62

11 P a g e 63 Example 7. A commercial customer orders 81 gallons of paint containing equal amounts of red paint, green paint, and blue paint---and hence could be prepared by mixing 27 gallons of each. However, the store wishes to prepare this order by mixing three types of paint that are already available in large quantity: A greenish paint that is 75% green, 12.5% red and 12.5% blue paint; A bluish paint that is 60% blue, 20% green and 20% red paint; A reddish paint that is 50% red, 25% blue and 25% green pain. How many gallons of each are needed to prepare the customer s order? 63