Bell Work:• Factor the following expressions completely!

• 1)

• 2)

UNIT 4Polynomial and Rational Functions

LESSON 4.1Polynomial Functions with Degree Greater Than 2

Degree of a Polynomial• The degree of a polynomial is the highest order exponent

that the function has.

• The functions we will be looking at in this lesson have a degree that is greater than two, which means the functions will not be linear or quadratic.

Intermediate Value Theorem:• We can prove that there is a zero for a polynomial

function by using this theorem.

• If f(a) > 0 and f(b) < 0, then there must be a point between a and b that crosses the x-axis, which means there is a zero between a and b.

• Ex: If f(x) = 2x³ - 2, show that there is a zero for the function between -2 and 2.

Sketching the Graph• We can find all of the values for a function that are

positive and negative by first finding its zeros, then using that to sketch the graph.

• Ex: Find all values of x such that f(x) > 0 and f(x) < 0, then sketch the graph of f(x).

• 1)

• 2)

Bell Work:• Using the given functions below, find the zeros, determine

the intervals at which f(x) > 0 and f(x) < 0, then sketch the graph.

Bell Work:• Using the given functions below, find the zeros, determine

the intervals at which f(x) > 0 and f(x) < 0, then sketch the graph.

• 1) f(x) =

• 2) f(x) =

Class Work:• Pages 269-271 #’s 33, 35, 39, 41

Homework:• Pages 269-271 #’s 5-9 odds, 17-27 odds, 34, 36, 43

Bell Work:• Without using your calculator, please divide the following

by hand using long division:

• 1)

LESSON 4.2Dividing Polynomials

LESSON ESSENTIAL QUESTION• What are the different properties of division pertaining to

polynomials and how are the used?

Long Division of Polynomials• We can divide two polynomials using long division.

• Ex: Simplify:

• It is possible to get a remainder if the factor you are dividing does not go into the function evenly:

• Ex: Simplify:

Bell Work:• Divide each expression using long division:

• 1) divided by

• 2) divided by

• 3) Repeat #2, except divide by

More Examples:• Divide:

• If and , then use long division to find

Synthetic Division of Polynomials• To use synthetic division, you must be dividing by a

binomial in the form x – c.

• It is much easier to explain this method through examples:

• Ex: divided by

• Ex: divided by x + 3.

Remainder Theorem:• If a polynomial f(x) is divided by x – c, then the remainder

is f(c).

• Ex: If , then use the remainder theorem to find f(2).

• Ex: If , then use the remainder theorem to find f(4).

• What would it mean if the remainder were zero?

Factor Theorem:• A polynomial f(x) has a factor x – c if and only if f(c) = 0.

• Ex: Show that x – 4 is a factor of .

• Ex: Show that x + 6 is a factor of

Homework:• Pages 278-279 #’s 1, 2, 9, 11, 13, 15, 21, 23, 25, 29, 31,

35, 37

• Small Quiz Tomorrow:

• Finding Zeros, Sketching Graphs, Intervals• Long Division/Synthetic Division

Bell Work:• 1) Simplify using long division:

• 2) Repeat number 1 using synthetic division.

• 3) How can we prove that we divided correctly?

• 4) Prove that (x+2) is a factor of .

Class Work/Homework:• Pages 278-279 #’s 10, 12, 14, 16, 22, 24, 26, 28, 30, 36

Bell Work:

• 1) Divide by .

• 2) Prove that you found the right remainder.

• 3) Prove that x + 2 is a factor of:•

LESSON 4.3Zeros of Polynomials

LESSON ESSENTIAL QUESTION• How do we find the total number of zeros for a function,

and how do we determine the number of real/imaginary, positive/negative, rational/irrational zeros?

Multiplicity of Zeros• If a function f(x) has more than one zero at the same

point, that zero is said to have a multiplicity.

• Ex: If , then f(x) has zeros at -4 with a multiplicity of 3, and at 3 with a multiplicity of 2.

Finding the Polynomial• We can find a polynomial f(x) if we know its zeros and one

other point on the function.

• Ex: Find f(x) if f(-4) = -504 and it has zeros: -1 with multiplicity of 2, 3, 4

• Ex: Find f(x) if f(5) = 68 and it has zeros: ±3i, 0, 3

Finding Zeros• Find the zeros of f(x) and state the multiplicity for each.

• Ex:

• Ex:

• Ex:

Homework:• Pages 291-293 #’s 1 – 25 odds

Bell Work:• 1) Find f(x) if f(2) = -72 and it has zeros: -4, 0 with

multiplicity of 3, and 5

• 2) Find the zeros of f(x) and state the multiplicity for each using the function

Classwork/Homework:• Pages 291-293 #’s 2, 6, 10, 12, 14, 16, 18, 22, 24

• This assignment will be collected tomorrow at the beginning of class!!!

Bell Work:• Warning, the lesson you are about to learn is off the heezy

fo sheezy…viewer discretion may be advised.

LESSON 4.4Complex and Rational Zeros of Polynomials

Lesson Essential Question:• How do we determine the rational and complex solutions

to a polynomial function?

Find the function knowing the zeros…• Example: Find a polynomial function f(x) of degree 4 that

has zeros of (2 + i) and (-3i) that has a leading coefficient of 1.

• Example: Find a polynomial function f(x) of degree 5 that has zeros of (2), (5i) and (2 + 3i) that has a leading coefficient of 1.

Dear 4th Period Pre-Calculus Class,• I would like to apologize for a mistake that I made

yesterday. I incorrectly typed the notes into the PowerPoint slide which caused confusion at the end of class. I hope that some day you all can forgive me for this horrific and tragic error that I have made. I know it is hard to understand because you all see me as a superhuman math machine, but like all superheroes, I have flaws. Am I implying I am a superhero? I guess so. Anyway, back to the point. I sincerely apologize and beg for your forgiveness.

• Sincerely,

• Mr. Kelsey (aka Math Man aka The Fabulous Factorer)

Determining Possible Zeros…• When dealing with a large polynomial function, we cannot

always factor it easily. We can determine possible zeros by looking at the following:

Example:• Express the following function as a product of its factors:• Find all of the solutions for the equation.

Example:• Express the following function as a product of its factors

and then find all zeros:

• f(x) =

• Examples: Pages 301 – 302 #’s 16 and 22

Homework:• Pages 301 – 302 #’s 1, 5, 9, 15, 17, 19, 23

Bell Work:• Find the polynomial f(x) of degree 4 if you know it has a

leading coefficient of 1 and zeros of (-2i), (4) and (0).

• Find all of the solutions to the following equation:

Classwork/Homework:• Pages 301 – 302 #’s 2, 6, 10, 18, 20, 24

• This will be collected tomorrow!!!

Bell Work:• Express the following function as a product of its factors,

then find all of the zeros for the function:

Bell Work:• Warning: The lesson you are about to witness is off the

hook, which can be extremely dangerous and not appropriate for all audiences. Viewers discretion may be advised.

LESSON 4.5Behaviors of Rational Functions

Lesson Essential Question:• How do we determine the horizontal, vertical, and oblique

asymptotes for a rational function, and the behavior about those asymptotes?

Let’s try some examples…• Sketch the graphs of the following functions by hand! (No

Graphing Calculators!!!)

Bell Work:• Sketch the graph of the following function:

Holes!!!!!• If a factor from the numerator and denominator will cancel

out, then you do not have a vertical asymptote. Instead, you have a hole! This is a value that cannot exist because of the factor in the denominator, but would have a value based upon the simplified form of the function.

Vertical Asymptotes:• How do we find vertical asymptotes?

• Vertical asymptotes occur when we have undefined values for x in the denominator of a function.

• Example: , this would have vertical asymptotes at x = 2 and x = -2 because those values would cause the denominator to be zero.

Horizontal Asymptotes:• To find horizontal asymptotes, we must use the degree of the

polynomials for the numerator and denominator.

• Let’s assume that the highest degree exponent for the numerator is (n) and for the denominator is (k), so:

• If , then the x-axis (line y = 0) is the asymptote.

• If , then the line is the asymptote where a is the coefficient for the numerator and b is the coefficient for the denominator (highest term)

• If then there is no horizontal asymptote.

Bell Work:• Sketch the graph of the following function by hand (no

cheating with your graphing calculator):

Oblique Asymptotes:• If function in the numerator is one degree higher than the

denominator, then there will be an oblique asymptote.

• How do we find what the oblique asymptote is?

• Good question! It is the linear function you get when you divide the numerator by the denominator.

• Sketch the graph of:

Steps for Graphing Rational Functions:

• 1) Find the asymptotes.• 2) Determine if there are any holes.• 3) Sketch each region of the graph created by the

asymptotes.

• Example: Sketch the graph of

• Example: Sketch the graph of

Examples:• Sketch the graph of

Homework:• Pages 318 – 319 #’s 7, 9, 15, 17, 21, 25, 29, 33

Quiz on Rational Functions:• Sketch the graph of each function:

• 1)

• 2)

• 3)

Bell Work:• What are all of the asymptotes for the following function:

• Sketch the graph of the function.

• Rewrite the following function as a product of its factors, then find all of the zeros for the function.

Examples:• Rewrite the following function as a product of its factors:

• What are all of the zeros to the function?

• Find all of the asymptotes and sketch the graph of the following function:

Unit 4 Test Upcoming!• The Test will be on:

• Sketching Polynomial Functions Using its Zeros• Find Intervals in which f(x) is positive or negative• Division of Polynomials (Long and Synthetic)• Remainder and Factor Theorem• Finding Zeros of Polynomials (Real, Complex, Irrational)• Find the Polynomial based upon its zeros and an

additional point• Sketching graphs of Rational Functions

Class Work:• Pages 321-322 • #’s 3-6, 10-13, 15-20, 24 – 27, 29, 31, 33, 35