Start Up Day 25Start Up Day 25Factor each Polynomial:1. 2.

Chapter 5, Sections 1&2Chapter 5, Sections 1&2• Objective: SWBAT determine zeros and

sketch the graphs of polynomial functions, with and without the use of technology.

• Essential Question: How can the understanding of zeros and end behavior help create a function graph?

• Home Learning: p. 285 # 20, 22, 26, 30, 38, 47-49 and p. 293 #7, 10, 14, 15, 18, 19, 20, 23, 27, 31, 32 and 35 AND TAKE NOTES--View 5-4 Virtual Nerd Video: Polynomial Long Division via the online textbook

What is a Polynomial What is a Polynomial Function?Function?

How do we Classify a How do we Classify a Polynomial Function?Polynomial Function?

End Behavior? End Behavior? The degree of a polynomial function affects the

shape of its graph and determines themaximum number of turning points, or places where

the graph changes direction.

It affects the end behavior , or the directions of the graph to the far left and to the far right.

A function is increasing when the y -values increase as x -values increase. A function is

decreasing when the y -values decrease as x -values increase.

10 min Partner Activity:10 min Partner Activity:• Cut out your Polynomial Cards• Categorize them into FOUR separate

groups by looking their general shape.• Identify the DEGREE (highest power

of “x”) of each.• Identify the “a” or “The leading

coefficient” of each.• Any Conclusion?

Which way will it go?Which way will it go?Use the Leading Coefficient Test to

describe the End Behavior.DEGREE OF POLYNOMIAL (Highest

Exponent)”n ”+a -a

Even DEGREE Up and Up Down and Down

Odd DEGREE Up on the right & Down on the left

Up on the left and down on the right

Match them up:Match them up:

How Many How Many Turning Points? Zeros Turning Points? Zeros or x-or x-

interceptsintercepts??

Turning Points—Relative minimums and/or maximums OR “stationary points”At most, a graph of a polynomial function to the “n”

power, has “n-1” turning Points

ZEROS (or x-intercepts)•At most, a graph of a polynomial function

to the “n” power, has “n” real zeros.– An even power function may not have any

real zeros– An odd power function always has at least

one real zero

Problem 2: End BehaviorProblem 2: End Behavior GOT IT? Consider the leading term (degree and coefficient). What is the end behavior of the graph?

Problem 3: A Cubic FunctionProblem 3: A Cubic Function

Problem 4: Determine the Problem 4: Determine the degree given Data Pointsdegree given Data Points

3

5.2 Problem 1—Factoring!5.2 Problem 1—Factoring!

Problem 2—From Factors to Problem 2—From Factors to Zeros!Zeros!

Problem 3—Creating your own Problem 3—Creating your own polynomial, given the zerospolynomial, given the zeros

Problem 4—Multiplicity?Problem 4—Multiplicity?

Multiplicity? Partner Up--Try Multiplicity? Partner Up--Try these Few to see what they do!these Few to see what they do!

The Big PictureThe Big Picture

• End Behavior?• Zeros? How many

and identify all• Turning Points?

How many might there be?

• Sketch it• Check your graph

Start Up Day 26Start Up Day 261. Factor x3 + 4x2 − 5x

2. What are the zeros of y = (x +1)(x − 3)(x + 2)?

3. Create a cubic polynomial function in standard form with zeros 4, −1, and 2?

.

Chapter 5, Sections 3 & 4Chapter 5, Sections 3 & 4• Objective: SWBAT Solve Polynomial Equations by

Factoring and by Graphing; Divide Polynomials by using long and/or synthetic division; Apply the Factor and Remainder Theorems

• Essential Question: How is polynomial division helpful in determining the factors and solutions of a polynomial equation? How do we use the Factor and Remainder Theorems?

• Home Learning: Worksheet 5.3, Worksheet 5.4 + TAKE NOTES--View Problem #1 From Lesson 5-5 via the online textbook.

Lessons 5.3 & 5.4Lessons 5.3 & 5.4In Search of the ALL ZerosIn Search of the ALL Zeros

• Sometimes your polynomial will be factorable!

• Consider setting your polynomial = 0

• Factor your polynomial completely

• Use the ZERO PRODUCT PROPERTY and solve for all the possible “x ” values

• These will be your x-intercepts or your zeros

24 22 xxxf

120

22022

24

xx

xx

Sketch the Curve: Sketch the Curve: 1. Set “=0”, FACTOR,

determine & Plot your “zeros”

2. Use your knowledge of “end behavior”.

3. Calculate a few additional points.

4. Connect with a smooth curve.

• Zeros with multiplicity of 2?

• These will be a “bounce” point.

• Zeros with multiplicity of 3?

• These will be a “twist” point (inflection)

24 22 xxxf

Real or Imaginary?Real or Imaginary?

Polynomial Long DivisionPolynomial Long Division

Synthetic DivisionSynthetic DivisionA Tool for Sketching the Graphs of A Tool for Sketching the Graphs of

Polynomial Functions of Higher Polynomial Functions of Higher DegreeDegree

In Search of Real Zeros and other points on the curve

Setting your polynomial=0 and Factoring will allow you to find the real zeros BUT is not always possible!

The Rational Zero Test gives us a start!

If your polynomial has integer coefficients, every rational zero of “f” has the form:

p/q

Where “p” is a factor of the constant term

And “q” is a factor of the leading coefficient

The Rational Root The Rational Root Theorem--Using p/qTheorem--Using p/q

• The possible Real Zeros are:

• The factors of the constant term”12” divided by the factors of the leading coefficient “1”

• These values are the only possible x-intercepts!

• “2” seems like a good starting point!

1234)( 23 xxxxfGiven F(x)

1

12,6,4,3,2,1

The First step in synthetic division is to set up your problem by placing your “x ” value on the shelf in the upper left hand box.

Then, using the coefficients only

Place each term’s coefficient from the highest power term to the lowest, in order, from left to right.

Make sure to include a “0” for any missing powers of “x ”.

GETTING STARTED:

1234)( 23 xxxxf

Given f(x), find f(2) using synthetic division

2 1 -3 -4 12

1

Notice the ordering has changed to allow for the polynomial to be written from the highest power to the lowest power of “x”.

Drop the first coefficient straight down.

1234)( 23 xxxxf

2 1 -3 -4 12

2

1 -1

The next step is to multiply your “x” value by the bottom line element, place the resultant on the shelf and then add. Place your sum under the shelf and repeat this process until your done!

1234)( 23 xxxxf

2 1 -3 -4 12

2 -2 -12

1 -1 -6 0

F(2)=0

Which means that x=2 is a zero of the function, since the remainder is “0”!

This is called the “FACTOR THEOREM”

(X-2) IS A FACTOR OF THE POLYNOMIAL.

1234)( 23 xxxxf

Try to use synthetic division to evaluate f(1)

1 1 -3 -4 12

1 -2 -6

1 -2 -6 6

This means that f(1)=6

OR (1,6) is a point on your graph

This is known as the “REMAINDER THEOREM”

OVERVIEW—Create your own Graphic Organizer

•The End Behavior is determined by the leading coefficient and the degree of the polynomial.

•The number of possible x-intercepts is determined by the degree of the polynomial

•If the polynomial is factorable, factoring can help you find these x-intercepts or the “ZEROS” of the polynomial function.

•The number of possible turning points is determined by the degree minus one.

•Points on the curve can be found by plugging in “x” values or by using Synthetic Division-p/q gives you the possible real zero values