Chapter 3 Polynomials

Sections Covered:

3.1 Remainder and Factor Theorems3.2 Analyzing Polynomial Graphs

3.3 Zeros of Polynomials3.4 Fundamental theorem of Algebra

3.5 Graphs of Rational Functions

~ November 2013 ~Sun Mon Tue Wed Thu Fri Sat

14 B3.1 “The Remainder and Factor Theorems”Long Division

HW 3.1 Long Division

15 A3.1 “The Remainder and Factor Theorems”Long Division

HW 3.1 Long Division

16

17 18 B3.1 “The Remainder and Factor Theorems”Synthetic Division

HW 3.1 Synthetic Division

19 A3.1 “The Remainder and Factor Theorems”Synthetic Division

HW 3.1 Synthetic Division

20 BQuiz: Long Division3.2 “Analyzing Graphs of Polynomials”

HW 3.2

21 AQuiz: Long Division3.2 “Analyzing Graphs of Polynomials”

HW 3.2

22 B3.3 “Zeros of Polynomial Functions”Rational Zeroes

HW 3.3 Rational Zeros

23

24 25 A3.3 “Zeros of Polynomial Functions”Rational Zeroes

HW 3.3 Rational Zeros

26 B3.3 “Zeros of Polynomial Functions”Descarte’s Rule

HW 3.3 Descarte’s Rule

27 Thanksgiving Break

28 Thanksgiving Break

29 Thanksgiving Break

30

~ December 2013 ~1 2 A

3.3 “Zeros of Polynomial Functions”Descarte’s Rule

HW 3.3 Descarte’s Rule

3 BQuiz 3.1-3.3 Review

HW Review 3.1-3.3

4 AQuiz 3.1-3.3 Review

HW Review 3.1-3.3

5 B***Quiz: 3.1-3.3***

Pre-Lab Worksheet

6 A***Quiz: 3.1-3.3***

Pre-Lab Worksheet

7

8 9 B3.4 “The Fundamental Theorem of Algebra”

HW 3.4

10 A3.4 “The Fundamental Theorem of Algebra”

HW 3.4

11 BBungee Barbie LAB

12 ABungee Barbie LAB

13 B3.5 “Graphs of Rational Functions”

HW 3.5

14

15 16 A3.5 “Graphs of Rational Functions”

HW 3.5

17 BChapter 3 Test Review

HW Chapter 3 Test Review Packet

18 AChapter 3 Test Review

HW Chapter 3 Test Review Packet

19 B***Chapter 3 Test***

20 A***Chapter 3 Test***

21

Table of ContentsNotes 3.1: Long Division pg. 3-4

HW 3.1: Long Division pg. 5-6

Notes 3.1: Synthetic Division pg. 7-9

HW 3.1: Synthetic Division pg. 10-11

Notes 3.2: Analyzing Graphs pg. 12-15

HW 3.2: Analyzing Graphs pg. 16-17

Notes 3.3: Rational Roots pg. 18-19

HW 3.3: Rational Roots pg. 20-21

Notes 3.3: Descartes Rule pg. 22-24

HW 3.3: Descartes Rule pg. 25-26

Notes 3.4: Fund. Thm of Algebra pg. 27-28

HW 3.4: Fund. Thm of Algebra pg. 29-

30

Notes 3.5: Graphs of Rat. Fun. pg. 31-

33

HW 3.5: Graphs of Rat. Fun. pg. 34-36

HW Reflection Sheet pg. 37-38

3.1 Long Division

We’ve learned how to multiply polynomials. Now, we’ll learn how to DIVIDE polynomials.

There are 2 ways to divide: long division and synthetic division. First, let’s go back to grade school and review how to divide plain old regular numbers.

8 Check:

Now, let’s apply the same process to polynomials. Check your solution by multiplying the divisor by the quotient and adding the remainder.

EX 1]

EX 2]

You Try:

1. (m2 – 7m – 11) ÷ (m – 8)

2. (a2 – 28) ÷ (a – 5)

3. (2x2 – 17x – 38) ÷ (2x + 3)

4. (n3 + 7n2 + 14n + 3) ÷ (n + 2)

5. (-5k2 + k3 + 8k + 4) ÷ (-1 + k)

3.1 Synthetic Division and the Remainder and Factor Theorems

Synthetic division works differently.

Divide by x – 2.

1. First, the divisor must be in the form: (x – k). In our example, k = 2.

2. If a term is missing, you MUST use a zero as a place holder.

3. Write the leading coefficient. Then multiply diagonally and add vertically, multiply and add, etc. ***FILL IN BELOW****

4. The answer is interpreted as follows: Work backwards.

The last number is the remainder.The next number back is the constant.The next number back is the x coefficient.The next number back is the x coefficient.The next number back is the x coefficient. And so on.

To divide by x – 2.

2 1 2 -6 -9

x x c remainder

EX 3] Divide by using synthetic division.

The Remainder Theorem: If a polynomial is divided by , then the remainder equals .

Do not write two signs!

That means ________ is a root of the equation, a zero of the function, and an x-intercept on the graph!!

That also means that ________ is a factor of the

polynomial.

And it means that ________ is also a factor, also know as a reduced polynomial ora depressed polynomial.

Getting zero in synthetic substitution is a big deal!!

TRANSLATION: Do synthetic division, the remainder is your answer.

In the example at the top of the previous page, divide by x – 2,

we got x + 4x + 2 + .

The Remainder Theorem states that f (2) will be equal to –5!

Example:

The Factor Theorem:A polynomial function has a factor of if and only if . That is, is a factor if and only if c is a zero of P.

Solutions = Roots = Zeros = x-InterceptsWhen we find the roots of a polynomial equation, we are finding the places where the value of the function is ZERO. If f (x) = 0, then x is a root!!!!

Examples: If f(4) = 0 then x = 4 is a zero and (x – 4) is a factor.If f(-7) = 0 then x = -7 is a zero and (x + 7) is a factor.If f(3/2) = 0 then x = 3/2 is a zero and (2x – 3) is a factor.

Watch what happens when we find f (3) for the function f (x) = x2 + 2x − 15

3 1 2 −15 Is your remainder 0? _______________ _____________

EX 4] f (x) = x3 + x2 + 2x + 24. *******(NEED CALCULATOR)*******

Graph the function. (You will need to set your window.) There is one real root at .

Because the degree of the polynomial is 3, we know there are two other roots.They must be imaginary.

We will use synthetic substitution to divide out (x + 3). Then the quadratic formula will allow us to find the other roots.

−3 1 1 2 24 So, using the coefficients of the quotient, we write _______________ (x + 3)(x2 − 2x + 8) = 0

We already know . We use the quadratic formula on the second ( ) and get

= =

= = .

Therefore, , ,

Example Problems:

Use Synthetic Division:1. (m2 – 7m – 11) ÷ (m – 8)

2. (a2 – 28) ÷ (a – 5)

3. (2x2 – 17x – 38) ÷ (2x + 3)

4. (n3 + 7n2 + 14n + 3) ÷ (n + 2)

5. (-5k2 + k3 + 8k + 4) ÷ (-1 + k)

Use the Remainder Theorem to find P(c).

6. P(x) = 2x3 – x2 + 3x – 1 , c = 3 7. P(x) = 6x3 – x2 + 4x , c = -3 8. P(x) = -x3 + 3x2 + 5x + 30 , c = 8

Determine if

Do you remember “complex conjugates”?

3.2 Analyzing Graphs of Polynomials

3.3 Possible Rational ZerosNOTE: is a polynomial function.

ALL polynomial functions have graphs that are __________ _______________ __________ .

Zeros of function : x – values for which p(x) = 0

Roots of equation : x – values for which p(x) = 0

Multiple Zeros of a Polynomial Function:

If has as a factor exactly k times, then r is a zero of multiplicity k of .

muliplicity

Zero/Root

EX 1] Find the zeros of and state the multiplicity of each zero.

_____ occurs as a zero of multiplicity _____ .

_____ occurs as a zero of multiplicity _____ .

_____ occurs as a zero of multiplicity _____ .

_____ occurs as a zero of multiplicity _____ .

Even Root = Bounce Point

Odd Root = goes through

EX 2] Given .

a) Find the maximum number of roots. ________________

b) Which zero(s) create a bounce point? _________________

c) Which zero(s) does the graph go through? _________________

x

y )(xf

The Rational Zero Theorem:

WHY IS IT IMPORTANT: Narrows the search for rational zeros to a finite list.

If has integer coefficients

and is a rational zero (in lowest terms) of p, then

p is a factor of the constant term and q is a factor of the leading coefficient .

EX 3] Find the roots of .

HINT: Apply the Rational Root Theorem to find the possible rational roots!

What is p? ________

What is q? ________

HINT: Use synthetic division on to locate a root!(Use your calc to estimate a zero )

EX 4] Find the possible rational roots of .

HINT: Apply the Rational Root Theorem to find the possible rational roots!

What is p? ________

What is q? ________

3.3 Descartes’ Rule of Signs

Descartes’ Rule of Signs:

WHY IS IT IMPORTANT: Narrows down even further the possible positive and negative roots.

Let P(x) be a polynomial function with real coefficients and with terms arranged in order of decreasing powers of x.

The number of positive real zeros of P(x) is equal to the number of variations in the sign of P(x), or to that number decreased by an even integer.

The number of negative real zeros of P(x) is equal to the number of variations in sign of P(-x), or to that number decreased by an even integer.

Ex 5] Use Descartes’ Rule of signs to determine both the number of possible positive and the number of possible negative real zeros of each polynomial function.

a) b)

Number of variations: _________ Number of variations: _________

Number of possible positive real zeros: ________ Number of possible positive real zeros: ________

P(-x) = P(-x) =

Number of variations: _________ Number of variations: _________

Number of possible negative real zeros: ________ Number of possible negative real zeros: ________

Steps for Finding the Zeros of a Polynomial Function with Integer Coefficients:1) Gather General Information.

Determine the degree n of the polynomial function. The number of distinct zeros of the polynomial function is at MOST n.

Apply Descartes’ Rule of Signs to determine the number of possible negative real zeros of each polynomial.

2) Check rational zeros. Apply the Rational Zero Theorem to list rational numbers that are possible zeros. Use synthetic division to test the numbers in the list.

3) Work with the reduced/depressed polynomial. Each time a zero is found, obtain the reduced/depressed polynomial. Work to get a reduced polynomial of degree 2. Then, find its zeros by factoring or by applying the quadratic formula.

EX 6] Find the zeros of .

At most _________ zeros.

Rational Root Theorem – Possible rational zeros:

Descartes Rule:

Synthetic Division/Quad. Formula/Factoring

Quick Sketch

x

y

5

EX 7] Find the zeros of Quick Sketch

x

y

5

3.4 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra: If is a polynomial function of degree with complex coefficients,then has at least one complex zero.

The Linear Factor Theorem:If is a polynomial function of degree with leading coefficient ,then has exactly n linear factors.

where c1, c2, … , cn are complex numbers.(real and/or imaginary)

The Number of Zeros of a Polynomial Function Theorem:If is a polynomial function of degree , then has exactly n complex zeros, provided each zero is counted according to its multiplicity.

EX 1] Find all zeros of the polynomial function and write the polynomial as a product of linear factors.

The Conjugate Pair Theorem:If ( ) is a complex zero of a polynomial function with real coefficients,then the conjugate is also a complex zero of the polynomial function.

Example: If is a zero, then __________ is also a zero. If is a zero, then __________ is also a zero.

EX 2] Find a polynomial function that has the indicated zeros.

a) degree 3; 1, 2, and -3 as zeros

b) degree 3; real coefficients and zeros 2i and -3

c) degree 2; real coefficients and a zero is 3 – 7i

Practice: Find all the zeros for the equation .

Hint: The two complex zeros are:

3.5 Graphs of Rational Functions

Rational Function can be written in the form where and arepolynomials and is NOT the zero polynomial.

Note The domain of a rational function of x includes all real numbers except the x-values that would make the denominator equal to zero.

The Graph of a Rational Function:1. Removable Discontinuity (Hole in the graph)

occurs when and have a common factor

2. Non-removable Discontinuity (Vertical Asymptote) occurs when the denominator equals zero

3. Horizontal Asymptote the value that the function approaches as x increases without bound

a. If the degree of the numerator < the degree of the denominator; y = 0 (the x-axis) is the horizontal asymptote

b. If the degree of the numerator = the degree of the denominator; is the H.A.

c. If the degree of the numerator > the degree of the denominator; there is NO horizontal asymptote

4. x-intercept zero(s) of the numerator

5. y-intercept the value of

6. Slant Asymptote occurs when the degree of the numerator is EXACTLY one more than the degree of its denominatorex:

**Use long division to find the equation of the slant asymptote. The slant asymptote will always be linear! Do NOT include the remainder1. Graph Advanced Functions - 3.5 Notes

Hole in graph: __________

V.A. __________

H.A. __________

x

y

x-intercept __________

y-intercept __________

slant asymptote __________

domain __________ range ___________

2. Graph

Hole in graph: __________

V.A. __________

H.A. __________

x-intercept __________

y-intercept __________

slant asymptote __________

domain __________ range ___________

3. Graph

Hole in graph: __________

V.A. __________

H.A. __________

x-intercept __________

y-intercept __________

slant asymptote __________

domain __________ range ___________

x

y

x

y

4. Graph

Hole in graph: __________

V.A. __________

H.A. __________

x-intercept __________

y-intercept __________

slant asymptote __________

domain __________ range ___________

x

y

Chap. 3 Homework Completion Sheet Name: _________________________ Block: _______

Prerequisite Skills: 1. Long Division 2. Factoring 3. Using Complex Numbers 4. Solving Using

Quadratic Formula

Date Assignment Score(out of 20)

NotesWhat am I confident about?

Specifically, what was difficult or confusing?Am I weak on a prerequisite skill?