Slide 1

Pre-Calculus 30

Slide 2

PC30.10 PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree 5 with integral coefficients).

Slide 3

Slide 4

Slide 5

PC30.10 PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree 5 with integral coefficients).

Slide 6

Slide 7

What is a Polynomial Function?

Slide 8

A polynomial function has the form Where n = is a whole number x = is a variable The coefficients a n to a 0 are real numbers The degree of the poly function is n, the exponent of the greatest power of x The leading coefficient is a n the coefficient of the greatest power of x The constant term is a 0

Slide 9

Slide 10

Types of Polynomial Functions:

Slide 11

Slide 12

Each graph has at least one less change of direction then the degree of its function.

Slide 13

Characteristics of Polynomial Functions:

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Ex. 3.1 (p.114) #1-4 odds in each, 5-10 # 1-4 odds in each, 5-13 odds

Slide 22

PC30.10 PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree 5 with integral coefficients).

Slide 23

When only given the equation of a function what are some strategies that we have use to find x-intercepts? Sub in a zero and solve Factor Quadratic Formula Decompostion

Slide 24

Which of these strategies will work if our polynomial function have a degree greater than 2?

Slide 25

In this section we will look at some methods to completely factor a polynomial function with a degree greater than 2 in order to find the zeros (x-intercepts).

Slide 26

Long Division Long division of a polynomial is done just like your do with numbers but now you have variables

Slide 27

You use long division to divide a polynomial by a binomial: The dividend, P(x), which is the polynomial that is being divided The divisor, x-a, which is the binomial being divided into the polynomial The quotient, Q(x), which is the expression that results from the division The remainder, R, which is what is left over when the division is done.

Slide 28

Slide 29

Slide 30

Slide 31

If we graphed the polynomial function from example 2 what would the x-intercepts be?

Slide 32

Long division gives is factors of the polynomial function which when set equal to zero and solved are x-intercepts, zeros, or roots.

Slide 33

Synthetic Division: A short form of division that uses only the coefficients of the terms It involves fewer calculations

Slide 34

Slide 35

Remainder Theorem: When a polynomial P(x) is divided by a binomial x-a, the remainder is P(a) If the remainder is 0 then the binomial x-a is a factor of P(x) If the remainder is not 0 then the binomial x-a is NOT a factor of P(x)

Slide 36

Slide 37

Slide 38

Ex. 3.2 (p.124) #1,2,3-7 odds in each, 8-13 #2, 3-7 odds in each, 8, 9-17 odds

Slide 39

PC30.10 PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree 5 with integral coefficients).

Slide 40

Last day we looked at how dividing polynomial functions by a binomial shows us if that binomial is a factor or not We also discussed how we want our equations in factored form because that gives is the zeros/roots/x-intercepts Today we are going to extend that idea

Slide 41

Factor Theorem: The factor theorem states that x-a is a factor of a polynomial P(x) if and only if P(a)=0 If and only if means that the result works both ways. That is, If x-a is a factor then P(a)=0 If P(a)=0, then x-a is a factor of a polynomial P(x)

Slide 42

Slide 43

Slide 44

Slide 45

When factoring a polynomial function sometimes the most difficult part is deciding which values of a we should use when using long division, synthetic division or factor theorem.

Slide 46

Slide 47

This is referred to as the Integral Zero Theorem The integral zero theorem describes the relationship between the factors and the constant term of a polynomial. The theorem states that if x-a is a factor of a polynomial P(x) with integral coefficients, then a is a factor of the constant term of P(x) and x=a is a integral zero of P(x).

Slide 48

Slide 49

Factor by Grouping: If a polynomial P(x) has an even number of terms, it may be possible to group tow terms at a time and remove a common factor If the binomial that results from common factoring is the same for each pair of terms, then P(x) may be factored by grouping Will not always work!!!!

Slide 50

Steps to factoring Polynomial Functions: 1. Use the integral zero theorem to list possible integer values for zeros 2. You can use the factor theorem to determine if the values that are zeros (this take a lot of time so I dont suggest it) 2. Use one type of division to determine all the factors 3. Write equation in factored form

Slide 51

Slide 52

Slide 53

Slide 54

Ex. 3.3 (p.133) #1-7 odds in each, 8-14 evens #3-7 odds in each, 8-16 evens

Slide 55

PC30.10 PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree 5 with integral coefficients).

Slide 56

We are now going to use all the information we have learned in this unit to this point along with a little new into to Graph Polynomial Functions without a graphing calculator

Slide 57

Slide 58

Slide 59

Slide 60

Slide 61

Slide 62

Slide 63

Slide 64

Slide 65

Slide 66

Slide 67

Slide 68

Slide 69

Ex. 3.4 (p.147) #1-10 odds in questions with multiple parts, 12-18 evens #3-10 odds in questions with multiple parts, 11-23 odds