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Algebra II

Polynomials:

Operations and Functions

IMPORTANT TIP: Throughout this unit, it is 
extremely important that you, as a teacher, 
emphasize correct vocabulary and make sure 
students truly know the difference between 
monomials and polynomials. Having a solid 
understanding of rules that accompany each will 
give them a strong foundation for future math 
classes.

Teacher

2013-09-25

www.njctl.org


Table of Contents

click on the topic to go

to that section

Properties of Exponents Review

Adding and Subtracting Polynomials

Multiplying a Polynomial by a Monomial

Multiplying Polynomials

Special Binomial Products

Dividing a Polynomial by a Monomial

Dividing a Polynomial by a Polynomial

Characteristics of Polynomial Functions

Analyzing Graphs and Tables of Polynomial Functions

Zeros and Roots of a Polynomial Function

Writing Polynomials from its Zeros


Properties of Exponents

Review

Return to

Table of

Contents


Exponents

Goals and Objectives

Students will be able to simplify complex expressions containing exponents.


Exponents

Why do we need this?

Exponents allow us to condense bigger expressions into smaller ones. Combining all properties of powers together, we can easily take a complicated expression and make it simpler.



Exponents

Multiplying powers of the same base:

(x4y3)(x3y)

Have students think about what the expression means and then use the rule as a short cut.

For example: (x4y3)(x3y) = x x x x y y y x x x y = x7y4

The rules are a quick way to get an answer.

Teacher

Can you write this expression in another way??

. . .

. . .

. . .

.


Exponents

(-3a3b2)(2a4b3)

-6a7b5

(-4p2q4n)(3p3q3n)

-12p5q7n2

Simplify:

Teacher

(-3a3b2)(2a4b3)

(-4p2q4n)(3p3q3n)


Exponents

Work out:

.

xy3 x5y4

(3x2y3)(2x3y)

6x5y4

Teacher

x6y7

.

(3x2y3)(2x3y)

xy3 x5y4


Exponents

(m4np)(mn3p2)

1

Simplify:

B

Teacher

A

m4n3p2

B

m5n4p3

C

mnp9

D

Solution not shown


Exponents

D

(-3x3y)(4xy4)

-12x4y5

2

Simplify:

(-3x3y)(4xy4)

Teacher

A

x4y5

B

7x3y5

C

-12x3y4

D

Solution not shown


Exponents

.

3

Work out:

2p2q3 4p2q4

D

2p2q3 4p2q4

8p4q7

Teacher

.

A

6p2q4

B

6p4q7

C

8p4q12

D

Solution not shown


5m2q3 10m4q5

Exponents

.

4

Simplify:

A

Teacher

A

50m6q8

B

15m6q8

C

50m8q15

D

Solution not shown


Exponents

5

Simplify:

(-6a4b5)(6ab6)

B

Teacher

A

a4b11

B

-36a5b11

C

-36a4b30

D

Solution not shown


Exponents

Dividing numbers with the same base:

Have students think about what the expression means and then use the rule as a short cut.

For example: x4y3 = x x x x y y y = xy2

The rules are a quick way to get an answer.

Teacher

. . .

. . .

. . .

x3y

x x x y

= 4m


Exponents

Teacher

Simplify:


Exponents

Try...

Teacher


Exponents

A

6

Divide:

Teacher

C

A

D

Solutions not shown

B


Exponents

7

Simplify:

D

Teacher

C

A

D

Solution not shown

B


Exponents

A

8

Work out:

Teacher

C

A

D

Solution not shown

B


Exponents

9

Divide:

C

Teacher

C

A

D

Solution not shown

B


Exponents

D

10

Simplify:

Teacher

C

A

D

Solution not shown

B


Exponents

Power to a power:

Have students think about what the expression means and then use the rule as a short cut.

For example:

The rules are a quick way to get an answer.

Teacher

.


Exponents

Simplify:

Teacher


Exponents

Try:

Teacher


Exponents

D

11

Work out:

Teacher

C

A

B

D

Solution not shown


Exponents

12

Work out:

B

Teacher

C

A

D

Solution not shown

B


Exponents

C

13

Simplify:

Teacher

C

A

D

Solution not shown

B


Exponents

D

14

Simplify:

Teacher

C

A

D

Solution not shown

B


Exponents

15

Simplify:

C

Teacher

C

A

D

Solution not shown

B


Exponents

Negative and zero exponents:

Teacher

Spend time showing students how negatives and zeros occur.

Why is this? Work out the following:


Exponents

Sometimes it is more appropriate to leave answers with positive exponents, and other times, it is better to leave answers without fractions. You need to be able to translate expressions into either form.

Teacher

Write with positive exponents:

Write without a fraction:


Exponents

Simplify and write the answer in both forms.

Teacher


Exponents

Simplify and write the answer in both forms.

Teacher


Exponents

Teacher

Simplify:


Exponents

Write the answer with positive exponents.

Teacher


Exponents

B

16

Simplify and leave the answer with positive exponents:

Teacher

A

C

B

D

Solution not shown


Exponents

A

17

Simplify. The answer may be in either form.

Teacher

A

C

B

D

Solution not shown


Exponents

18

Write with positive exponents:

D

Teacher

A

C

B

D

Solution not shown


Exponents

19

Simplify and write with positive exponents:

D

Teacher

A

C

B

D

Solution not shown


Exponents

20

Simplify. Write the answer with positive exponents.

C

Teacher

A

C

B

D

Solution not shown


Exponents

21

Simplify. Write the answer without a fraction.

A

Teacher

A

C

B

D

Solution not shown


Exponents

Combinations

Teacher

Usually, there are multiple rules needed to simplify problems with exponents. Try this one. Leave your answers with positive exponents.

**Much of the time it is a good idea to simplify inside the parentheses first.


Exponents

When fractions are to a negative power, a short cut is to flip the fraction and make the exponent positive.

Teacher

Try...


Exponents

Two more examples. Leave your answers with positive exponents.

Teacher


Exponents

22

Simplify and write with positive exponents:

D

Teacher

A

C

B

D

Solution not shown


Exponents

23

Simplify. Answer can be in either form.

B

Teacher

A

C

B

D

Solution not shown


Exponents

24

Simplify and write with positive exponents:

B

Teacher

A

C

B

D

Solution not shown


Exponents

25

Simplify and write without a fraction:

A

Teacher

A

C

B

D

Solution not shown


Exponents

26

Simplify. Answer may be in any form.

D

Teacher

A

C

B

D

Solution not shown


Exponents

27

Simplify. Answer may be in any form.

C

Teacher

A

C

B

D

Solution not shown


28

Simplify the expression:

A

C

B

D

Pull for Answer

D


29

Simplify the expression:

Pull for Answer

A

C

A

B

D


Adding and Subtracting

Polynomials

Return to

Table of

Contents


Vocabulary

A term is the product of a number and one or more variables to a non-negative exponent.

The degree of a polynomial is the highest exponent contained in the polynomial, when more than one variable the degree is found by adding the exponents of each variable

Term

degree

degree=3+1+2=6


Identify the degree of the polynomials:

1) 3+2=5th degree

2) 5+1=6th degree

Solution

[This object is a teacher notes pull tab]


What is the difference between a monomial and a polynomial?

A monomial is a product of a number and one or more variables raised to non-negative exponents. There is only one term in a monomial.

For example: 5x2 32m3n4 7 -3y 23a11b4

A polynomial is a sum or difference of two or more monomials where each monomial is called a term. More specifically, if two terms are added, this is called a BINOMIAL. And if three terms are added this is called a TRINOMIAL.

For example: 5x2 + 7m 32m + 4n3 - 3yz5 23a11 + b4


Standard Form

The standard form of an polynomial is to put the terms in order from 
highest degree (power) to the lowest degree.

Example: is in standard form.

Rearrange the following terms into standard form:


Review from Algebra I

Monomials with the same variables and the same

power are like terms.

 Like Terms Unlike Terms

 4x and -12x  -3b and 3a

 x3y and 4x3y  6a2b and -2ab2



30

Simplify

A

B

Pull for Answer

C

D

D


31

Simplify

A

B

C

Pull for Answer

A

D


32

Simplify

A

Pull for Answer

B

C

C

D


To add or subtract polynomials, simply distribute the + or - sign to each term in parentheses, and then combine the like terms from each polynomial.

Example:

(2a2 +3a -9) + (a2 -6a +3)


Example: sign to each term in parentheses, and then combine the like terms from each polynomial.

(6b4 -2b) - (6x4 +3b2 -10b)


33 sign to each term in parentheses, and then combine the like terms from each polynomial.

Add

Pull for Answer

A

C

B

C

D


34 sign to each term in parentheses, and then combine the like terms from each polynomial.

Add

A

Pull for Answer

B

B

C

D


35 sign to each term in parentheses, and then combine the like terms from each polynomial.

Subtract

A

Pull for Answer

D

B

C

D


36 sign to each term in parentheses, and then combine the like terms from each polynomial.

Add

Pull for Answer

A

C

B

C

D


37 sign to each term in parentheses, and then combine the like terms from each polynomial.

Add

A

Pull for Answer

B

C

C

D


38 sign to each term in parentheses, and then combine the like terms from each polynomial.

Simplify

A

Pull for Answer

C

B

C

D


39 sign to each term in parentheses, and then combine the like terms from each polynomial.

Simplify

Pull for Answer

A

D

B

C

D


40 sign to each term in parentheses, and then combine the like terms from each polynomial.

Simplify

Pull for Answer

A

D

B

C

D


41 sign to each term in parentheses, and then combine the like terms from each polynomial.

Simplify

Pull for Answer

A

C

B

C

D


42 sign to each term in parentheses, and then combine the like terms from each polynomial.

Simplify

Pull for Answer

A

B

B

C

D


43 sign to each term in parentheses, and then combine the like terms from each polynomial.

What is the perimeter of the following figure? (answers are in units)

A

B

C

D

Pull for Answer

B


M sign to each term in parentheses, and then combine the like terms from each polynomial.ultiplying a Polynomial

by a Monomial

Return to

Table of

Contents


Review from Algebra I sign to each term in parentheses, and then combine the like terms from each polynomial.

Find the total area of the rectangles.

3

5

4

8

square units

square units


Review from Algebra I sign to each term in parentheses, and then combine the like terms from each polynomial.

To multiply a polynomial by a monomial, you use the 
distributive property together with the laws of exponents for multiplication.

Example: Simplify.

 -2x(5x2 - 6x + 8)

 (-2x)(5x2) + (-2x)(-6x) + (-2x)(8)

 -10x3 + 12x2 + -16x

 -10x3 + 12x2 - 16x


YOU TRY THIS ONE! sign to each term in parentheses, and then combine the like terms from each polynomial.

Remember...To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication.

 Multiply: -3x2(-2x2 + 3x - 12)

click to reveal

 6x4 - 9x2 + 36x


More Practice! Multiply to simplify. sign to each term in parentheses, and then combine the like terms from each polynomial.

1.

2. 

3. 

click

click

click


44 sign to each term in parentheses, and then combine the like terms from each polynomial.

What is the area of the rectangle shown?

A

Pull for Answer

B

A

C

D


45 sign to each term in parentheses, and then combine the like terms from each polynomial.

A

Pull for Answer

B

B

C

D


46 sign to each term in parentheses, and then combine the like terms from each polynomial.

A

Pull for Answer

A

B

C

D


47 sign to each term in parentheses, and then combine the like terms from each polynomial.

A

Pull for Answer

C

B

C

D


48 sign to each term in parentheses, and then combine the like terms from each polynomial.

Find the area of a triangle (A=1/2bh) with a base of 5y and a height of 2y+2. All answers are in square units.

A

B

Pull for Answer

D

C

D


Multiplying Polynomials sign to each term in parentheses, and then combine the like terms from each polynomial.

Return to

Table of

Contents


Review from Algebra I sign to each term in parentheses, and then combine the like terms from each polynomial.

Find the total area of the rectangles.

8

5

2

6

Area of the big rectangle

Area of the horizontal rectangles

Area of each box

sq.units


Review from Algebra I sign to each term in parentheses, and then combine the like terms from each polynomial.

Find the total area of the rectangles.

4

2x

x

3


To multiply a polynomial by a polynomial, you multiply each term of the first polynomials by each term of the second. Then, add like terms.

Some find it helpful to draw arches connecting the terms, others find it easier to organize their work using an area model. Each method is shown below. Note: The size of your area model is determined by how many terms are in each polynomial.

Example:

3x

2y

6x2

4xy

2x

12xy

4y

8y2



Review from Algebra I 
polynomials.

The FOIL Method can be used to remember how multiply two binomials.

To multiply two binomials, find the sum of ....

First terms Outer terms  Inner Terms  Last Terms

Example:

FirstOuterInnerLast


Try it! Find each product. 
polynomials.

1) 

2) 

click

click


More Practice! Find each product. 
polynomials.

3) 

4) 

click

click


49 
polynomials.

What is the total area of the rectangles shown?

A

Pull for Answer

B

D

C

D


50 
polynomials.

A

Pull for Answer

B

B

C

D


51 
polynomials.

A

Pull for Answer

B

B

C

D


52 
polynomials.

A

Pull for Answer

C

B

C

D


53 
polynomials.

A

Pull for Answer

B

B

C

D


54 
polynomials.

Find the area of a square with a side of

A

B

Pull for Answer

C

C

D


55 
polynomials.

What is the area of the rectangle (in square units)?

A

B

C

D

Pull for Answer

B


How would you find the area of the shaded region? 
polynomials.

Shaded Area = Total area - Unshaded Area

Teacher

sq. units


56 
polynomials.

What is the area of the shaded region (in square units)?

A

B

C

D

Pull for Answer

A


57 
polynomials.

What is the area of the shaded region (in square units)?

A

B

C

D

Pull for Answer

A


S 
polynomials.pecial Binomial Products

Return to

Table of

Contents


Square of a Sum 
polynomials.

 (a + b)2

 (a + b)(a + b)

   a2 + 2ab + b2

The square of a + b is the square of a plus twice the product of a and b plus the square of b.

Example: 


Square of a Difference 
polynomials.

 (a - b)2

 (a - b)(a - b)

   a2 - 2ab + b2

The square of a - b is the square of a minus twice the product of a and b plus the square of b.

Example: 


Product of a Sum and a Difference 
polynomials.

 (a + b)(a - b)

 a2 + -ab + ab + -b2  Notice the -ab and ab    a2 - b2 equals 0.

The product of a + b and a - b is the square of a minus the

square of b. 

Example:  outer terms equals 0.


Try It!  Find each product. 
polynomials.

1.   

2.  

3.  

click

click

click


58 
polynomials.

A

B

Pull for Answer

C

C

D


59 
polynomials.

A

Pull for Answer

D

B

C

D


60 
polynomials.

What is the area of a square with sides ?

A

B

Pull for Answer

D

C

D


61 
polynomials.

A

Pull for Answer

B

B

C

D


Problem is from: 
polynomials.

Click for link for commentary

and solution.

A-APR

Trina's

Triangles


Dividing a Polynomial by a Monomial 
polynomials.

Return to

Table of

Contents


To divide a polynomial by a monomial, make each term 
of the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.


Examples the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Click to Reveal Answer


62 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Simplify

Pull for Answer

D

A

B

C

D


63 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Simplify

C

A

B

D

Pull for Answer

A


64 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Simplify

C

A

B

D

Pull for Answer

C


65 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Simplify

A

Pull for Answer

A

B

C

D


Dividing a Polynomial the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

by a Polynomial

Return to

Table of

Contents


Long Division of Polynomials the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

To divide a polynomial by 2 or more terms,

long division can be used.

Recall long division of numbers.

or

Multiply

Subtract

Bring down

Repeat

Write Remainder over divisor


Long Division of Polynomials the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

To divide a polynomial by 2 or more terms,

long division can be used.

Multiply

Subtract

Bring down

Repeat

Write Remainder over divisor

-2x2+-6x

-10x +3

-10x -30

33


Examples the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Teacher Notes

[This object is a teacher notes pull tab]


Example the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Solution

[This object is a teacher notes pull tab]


Example: In this example there are "missing terms". the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

  Fill in those terms with zero coefficients before dividing.

click


Examples the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Teacher Notes

Make sure you distribute as you subtract!

[This object is a teacher notes pull tab]


66 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Divide the polynomial.

A

Pull for Answer

B

B

C

D


67 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Divide the polynomial.

A

B

Pull for Answer

B

C

D


68 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Divide the polynomial.

A

B

Pull for Answer

A

C

D


69 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Divide the polynomial.

Pull

Pull


70 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Divide the polynomial.

Pull

Pull


71 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Divide the polynomial.

Pull

Pull


Characteristics of the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Polynomial Functions

Return to

Table of

Contents


Polynomial Functions: the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Connecting Equations and Graphs


Relate the equation of a polynomial function to its graph. the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

A polynomial that has an even number for its 
highest degree is even-degree polynomial.

A polynomial that has an odd number for its 
highest degree is odd-degree polynomial.


Odd-Degree Polynomials the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Even-Degree Polynomials

Observations about end behavior?


Even-Degree Polynomials the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Positive Lead Coefficient

Negative Lead Coefficient

Observations about end behavior?


Odd-Degree Polynomials the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Positive Lead Coefficient

Negative Lead Coefficient

Observations about end behavior?


End Behavior of a Polynomial the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.


72 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A

odd and positive

B

odd and negative

C

even and positive

D

even and negative

Pull for Answer

D even and negative


73 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A

odd and positive

B

odd and negative

C

even and positive

D

even and negative

Pull for Answer

A odd and positive


74 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A

odd and positive

B

odd and negative

C

even and positive

D

even and negative

Pull for Answer

C even and positive


75 the polynomial into the numerator of a separate 
fraction with the monomial as the denominator.

Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A

odd and positive

B

odd and negative

C

even and positive

D

even and negative

Pull for Answer

B odd and negative


Odd-functions not only have the highest exponent that is odd,but all of the exponents are odd.

An even-function has only even exponents.

Note: a constant has an even degree ( 7 = 7x0)

Examples:


76 odd,but all of the exponents are odd.

Is the following an odd-function, an even-function, or neither?

A

Odd

B

Even

Pull for Answer

C

Neither

C Neither


77 odd,but all of the exponents are odd.

Is the following an odd-function, an even-function, or neither?

A

Odd

B

Even

C

Neither

Pull for Answer

B Even


78 odd,but all of the exponents are odd.

Is the following an odd-function, an even-function, or neither?

A

Odd

B

Even

Pull for Answer

C

Neither

A Odd


79 odd,but all of the exponents are odd.

Is the following an odd-function, an even-function, or neither?

A

Odd

B

Even

C

Neither

Pull for Answer

A Odd


80 odd,but all of the exponents are odd.

Is the following an odd-function, an even-function, or neither?

A

Odd

B

Even

C

Neither

Pull for Answer

C Neither


An odd-function has rotational symmetry about the origin. odd,but all of the exponents are odd.

Definition of an Odd Function


An even-function is symmetric about the y-axis odd,but all of the exponents are odd.

Definition of an Even Function


81 odd,but all of the exponents are odd.

Pick all that apply to describe the graph.

A

Odd- Degree

B

Odd- Function

C

Even- Degree

D

Even- Function

E

Positive Lead Coefficient

F

Negative Lead Coefficient

Pull for Answer

A Odd- Degree

B Odd- Function

E Positive Lead Coefficient


82 odd,but all of the exponents are odd.

Pick all that apply to describe the graph.

A

Odd- Degree

B

Odd- Function

C

Even- Degree

D

Even- Function

E

Positive Lead Coefficient

F

Negative Lead Coefficient

Pull for Answer

C Even- Degree

D Even- Function

E Positive Lead Coefficient


83 odd,but all of the exponents are odd.

Pick all that apply to describe the graph.

A

Odd- Degree

B

Odd- Function

C

Even- Degree

Pull for Answer

A Odd- Degree

B Odd- Function

F Negative Lead Coefficient

D

Even- Function

E

Positive Lead Coefficient

F

Negative Lead Coefficient


84 odd,but all of the exponents are odd.

Pick all that apply to describe the graph.

A

Odd- Degree

B

Odd- Function

C

Even- Degree

D

Even- Function

E

Positive Lead Coefficient

F

Negative Lead Coefficient

Pull for Answer

A Odd- Degree

B Odd- Function

E Positive Lead Coefficient


85 odd,but all of the exponents are odd.

Pick all that apply to describe the graph.

A

Odd- Degree

B

Odd- Function

C

Even- Degree

D

Even- Function

E

Positive Lead Coefficient

F

Negative Lead Coefficient

C Even- Degree

D Even - Function

F Negative Lead Coefficient

Pull for Answer


Zeros of a Polynomial odd,but all of the exponents are odd.

Zeros are the points at which the polynomial 
intersects the x-axis.

An even-degree polynomial with degree n, 
can have 0 to n zeros.

An odd-degree polynomial with degree n,

will have 1 to n zeros


86 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

Pull for Answer

5 zeros


87 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

Pull for Answer

4 zeros


88 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

Pull for Answer

3 zeros


89 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

Pull for Answer

2 zeros


90 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

Pull for Answer

None


91 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

Pull for Answer

2 zeros


Analyzing Graphs and Tables odd,but all of the exponents are odd.

of Polynomial Functions

Return to

Table of

Contents


A polynomial function can graphed by creating a table, graphing the points, and then connecting the points with a smooth curve.


How many zeros does this function appear to have? graphing the points, and then connecting the points with a smooth curve.


There is a zero at x = -1, a second between x = 1 and x = 2, and a 
third between x = 3 and x = 4. Can we recognize zeros given only a table?


Intermediate Value Theorem and a 
third between x = 3 and x = 4. Can we recognize zeros given only a table?

Given a continuous function f(x), every value between f(a) and f(b) exists.

Let a = 2 and b = 4,

then f(a)= -2 and f(b)= 4. 


For every x value between 2 and 4, there exists a y-value between -2 and 4.


The Intermediate Value Theorem justifies saying that there is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.


92 is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

How many zeros of the continuous polynomial given can be found using the table?

Pull for Answer

4 zeros


93 is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

Where is the least value of x at which a zero occurs on this continuous function? Between which two values of x?

A

-3

Pull for Answer

B -2, C -1

B

-2

C

-1

D

0

E

1

F

2

G

3

H

4


94 is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

How many zeros of the continuous polynomial given can be found using the table?

Pull for Answer

4 zeros


95 is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

What is the least value of x at which a zero occurs on this continuous function?

A

-3

B

-2

Pull for Answer

B -2

C

-1

D

0

E

1

F

2

G

3

H

4


96 is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

How many zeros of the continuous polynomial given can be found using the table?

Pull for Answer

2 zeros


97 is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

What is the least value of x at which a zero occurs on this continuous function? Give the consecutive integers.

Pull for Answer

A

-3

B -2, C -1

B

-2

C

-1

D

0

E

1

F

2

G

3

H

4


Relative Maximums and Relative Minimums is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4.

Relative maximums occur at the top of a local "hill".

Relative minimums occur at the bottom of a local "valley".

There are 2 relative maximum points at x = -1 and the other at x = 1

The relative maximum value is -1 (the y-coordinate).

There is a relative minimum at x =0 and the value of -2


How do we recognize "hills" and "valleys" or the relative 
maximums and minimums from a table?

In the table x 
goes from -3 to 
1, y is 
decreasing. As 
x goes from 1 
to 3, y 
increases. And 
as x goes from 
3 to 4, y 
decreases.

Can you find a connection between

y changing "directions" and the max/min?


When y switches from increasing to decreasing there is a 
maximum. About what value of x is there a relative max?

Relative Max:

click to reveal


When y switches from decreasing to increasing there is a 
minimum. About what value of x is there a relative min?

Relative Min:

click to reveal


Since this is a closed interval, the end points are also a 
relative max/min. Are the points around the endpoint higher or 
lower?

Relative Min:

click to reveal

Relative Max:

click to reveal


98 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative minimum 
occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

Pull for Answer

C -1

E 1


99 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative maximum occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

Pull for Answer

B -2

F 2


100 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative minimum 
occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

Pull for Answer

E 1

H 4


101 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative maximum occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

Pull for Answer

A -3

G 3


102 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative minimum 
occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

B -2

E 1

H 4

Pull for Answer


103 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative maximum occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

A -3

C -1

G 3

Pull for Answer


104 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative minimum 
occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

B -2

F 2

H 4

Pull for Answer


105 
relative max/min. Are the points around the endpoint higher or 
lower?

At about what x-values does a relative maximum occur?

A

-3

E

1

B

-2

F

2

C

-1

G

3

D

0

H

4

D 0

G 3

Pull for Answer


Finding Zeros of a 
Polynomial Function 
relative max/min. Are the points around the endpoint higher or 
lower?

Return to

Table of

Contents


Vocabulary 
relative max/min. Are the points around the endpoint higher or 
lower?

A zero of a function occurs when f(x)=0

An imaginary zero occurs when the solution to f(x)=0, contains complex numbers.


The number of the zeros of a polynomial, both real and imaginary, is equal to the degree of the polynomial.

This is the graph of a polynomial 
with degree 4. It has four unique 
zeros: -2.25, -.75, .75, 2.25

Since there are 4 real zeros

there are no imaginary zeros

4 - 4= 0


When a vertex is on the x-axis, that zero 
counts as two zeros.

This is also a polynomial of 
degree 4. It has two 
unique real zeros: -1.75 
and 1.75. These two 
zeros are said to have a 
Multiplicity of two.

Real Zeros -1.75 1.75

There are 4 real zeros, therefore, no imaginary zeros for this function.


106 zeros.

How many real zeros does the polynomial graphed have?

A

0

B

1

C

2

D

3

E

4

Pull for Answer

E 4

F

5


107 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

Pull for Answer

No


108 zeros.

How many imaginary zeros does this 8th degree polynomial have?

A

0

Pull for Answer

B

1

A 0

C

2

D

3

E

4

F

5


109 zeros.

How many real zeros does the polynomial graphed 
have?

A

0

B

1

C

2

D

3

E

4

F

5

Pull for Answer

D 3


110 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

Pull for Answer

Yes


111 zeros.

How many imaginary zeros does the polynomial 
graphed have?

A

0

B

1

C

2

D

3

E

4

F

5

Pull for Answer

A 0


112 zeros.

How many real zeros does this 5th-degree polynomial have?

A

0

B

1

C

2

D

3

E

4

F

5

Pull for Answer

D 3


113 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

Pull for Answer

No


114 zeros.

How many imaginary zeros does this 5th-degree polynomial have?

A

0

B

1

C

2

D

3

E

4

Pull for Answer

F

5

C 2


115 zeros.

How many real zeros does the 6th degree polynomial have?

A

0

B

1

C

2

D

3

E

4

F

5

Pull for Answer

C 2


116 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

Pull for Answer

Yes


117 zeros.

How many imaginary zeros does the 6th degree polynomial have?

A

0

B

1

C

2

D

3

E

4

Pull for Answer

C 2

F

5


Finding the Zeros without a graph: zeros.

Recall the Zero Product Property.

If ab = 0, then a = 0 or b = 0.

Find the zeros, showing the multiplicities, of the following polynomial.

or

or

or

There are four real roots: -3, 2, 5, 6.5 all with multiplicity of 1.

There are no imaginary roots.


Find the zeros, showing the multiplicities, of the following polynomial.

or

or

or

or

This polynomial has five distinct real zeros: -6, -4, -2, 2, and 3.

-4 and 3 each have a multiplicity of 2 (their factors are being squared)

There are 2 imaginary zeros: -3i and 3i. Each with multiplicity of 1.

There are 9 zeros (count -4 and 3 twice) so this is a 9th degree polynomial.


118 polynomial.

How many distinct real zeros does the polynomial have?

A

0

B

1

C

2

D

3

E

4

Pull for Answer

E 4

F

5


Find the zeros, both real and imaginary, showing the multiplicities, of the following polynomial:

click to reveal

This polynomial has

1 real root: 2

and 2 imaginary roots:

-1i and 1i. They are simple roots with 
multiplicities of 1.


119 multiplicities, of the following polynomial:

How many distinct imaginary zeros does the polynomial have?

A

0

B

1

C

2

D

3

Pull for Answer

E

4

A 0

F

5


120 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

Pull for Answer

1


121 multiplicities, of the following polynomial:

How many distinct real zeros does the polynomial have?

A

0

B

1

C

2

D

3

Pull for Answer

D 3

E

4

F

5


122 multiplicities, of the following polynomial:

How many distinct imaginary zeros does the polynomial have?

A

0

B

1

C

2

D

3

Pull for Answer

A 0

E

4

F

5


123 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

Pull for Answer

2


124 multiplicities, of the following polynomial:

How many distinct real zeros does the polynomial have?

A

0

B

1

C

2

D

3

Pull for Answer

D 3

E

4

F

5


125 multiplicities, of the following polynomial:

How many distinct imaginary zeros does the polynomial have?

A

0

B

1

C

2

D

3

Pull for Answer

C 2

E

4

F

5


126 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

Pull for Answer

1


127 multiplicities, of the following polynomial:

How many distinct real zeros does the polynomial have?

A

0

B

5

C

6

D

7

Pull for Answer

B 5

E

8

F

9


128 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

Pull for Answer

3


129 multiplicities, of the following polynomial:

How many distinct imaginary zeros does the polynomial have?

A

0

B

1

C

2

D

3

Pull for Answer

C 2

E

4

F

5


Find the zeros, showing the multiplicities, of the following polynomial.

To find the zeros, you must first write the polynomial in factored form.

Review from Algebra I

or

or

or

or

This polynomial has two distinct real zeros: 0, and 1.

There are 3 zeros (count 1 twice) so this is a 3rd degree polynomial.

1 has a multiplicity of 2 (their factors are being squared).

0 has a multiplicity of 1.

There are 0 imaginary zeros.


Find the zeros, showing the multiplicities, of the following polynomial.

or

or

or

This polynomial has 4 zeros.

There are two distinct real zeros: , both with a multiplicity of 1.

There are two imaginary zeros: , both with a multiplicity of 1.


130 polynomial.

How many possible zeros does the polynomial

function have?

A

0

B

1

Pull for Answer

3

D

C

2

D

3

E

4


131 polynomial.

How many REAL zeros does the polynomial

equation have?

A

0

B

1

Pull for Answer

3

D

C

2

D

3

E

4


132 polynomial.

What are the zeros of the polynomial function

, with multiplicities?

A

x = -2, mulitplicity of 1

B

x = -2, multiplicity of 2

Pull for Answer

x = -2, multiplicity 1

x = 3, multiplicity 1

x = 0, multiplicity 2

A

C

F

C

x = 3, multiplicity of 1

D

x = 3, multiplicity of 2

E

x = 0 multiplicity of 1

F

x = 0 multiplicity of 2


133 polynomial.

Find the zeros of the following polynomial equation, including multiplicities.

A

x = 0, multiplicity of 1

Pull for Answer

B

x = 3, multiplicity of 1

x = 2, multiplicity 1

x = 3, multiplicity 2

A

D

C

x = 0, multiplicity of 2

D

x = 3, multiplicity of 2


134 polynomial.

Find the zeros of the polynomial equation, including multiplicities

A

x = 2, multiplicity 1

B

x = 2, multiplicity 2

Pull for Answer

x = 2, multiplicity 1

x = -i, multiplicity 1

x = i, multiplicity 1

A

C

D

C

x = -i, multiplicity 1

D

x = i, multiplicity 1

E

x = -i, multiplcity 2

F

x = i, multiplicity 2


135 polynomial.

Find the zeros of the polynomial equation, including multiplicities

A

2, multiplicity of 1

B

2, multiplicity of 2

x = 2, multiplicity 1

x = -2, multiplicity 1

x = , multiplicity 1

x = , multiplicity 1

A

C

E

F

Pull for Answer

C

-2, multiplicity of 1

D

-2, multiplicity of 2

E

, multiplicity of 1

F

, multiplicity of 2


Find the zeros, showing the multiplicities, of the following polynomial.

To find the zeros, you must first write the polynomial in factored form.

However, this polynomial cannot be factored using normal methods.

What do you do when you are STUCK??

RATIONAL ZEROS THEOREM


RATIONAL ZEROS THEOREM polynomial.

Make list of POTENTIAL rational zeros and test it out.

Potential List:

Test out the potential zeros by using the Remainder Theorem.

Remainder Theorem

For a polynomial p(x) and a possible zero a, (x-a) is a factor of p(x) if and only if p(a) = 0.


Using the Remainder Theorem. polynomial.

1 is a distinct zero, therefore (x -1) is a factor of the polynomial.

Use POLYNOMIAL DIVISION to factor out.

When you find a distinct zero, write the zero in factored form and then complete polynomial division.

Teacher Notes

or

or

or

or

This polynomial has three distinct real zeros: -2, -1/3, and 1, each with a multiplicity of 1.

There are 0 imaginary zeros.


Find the zeros using the Rational Zeros Theorem, showing the multiplicities, of the following polynomial.

Potential List:

±

±1

Remainder Theorem

-3 is a distinct zero, therefore (x+3) is a factor.

Use POLYNOMIAL DIVISION to factor out.


or multiplicities, of the following polynomial.

or

or

or

This polynomial has two distinct real zeros: -3, and -1.

-3 has a multiplicity of 2 (their factors are being squared).

-1 has a multiplicity of 1.

There are 0 imaginary zeros.


136 multiplicities, of the following polynomial.

Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem

A

x = 1, multiplicity 1

Pull for Answer

B D

B

x = 1, mulitplicity 2

x = -3, multiplicity 1

x = 1, multiplicity 2

C

x = 1, multiplicity 3

D

x = -3, multiplicity 1

E

x = -3, multiplicity 2

F

x = -3, multiplicity 3


137 multiplicities, of the following polynomial.

Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem

A

x = -2, multiplicity 1

Pull for Answer

B

x = -2, multiplicity 2

A E

x = -2, multiplicity 1

x = -1, multiplicity 2

C

x = -2, multiplicity 3

D

x = -1, multiplicity 1

E

x = -1, multiplicity 2

F

x = -1, multiplicity 3


138 multiplicities, of the following polynomial.

Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem.

A

, multiplicity 1

A, B E

Pull for Answer

B

, multiplicity 1

multiplicity 1

multiplicity 1

x = 1, multiplicity 1

C

, multiplicity 1

D

, multiplicity 1

E

x = 1, multiplicity 1

F

x = -1, multiplicity 1


139 multiplicities, of the following polynomial.

Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem

Pull for Answer

C, E, H

A

x = 1, multiplicity 1

E

x = , multiplicity 1

F

x = , multiplicity 1

B

x = -1, multiplicity 1

G

x = , multiplicity 1

C

x = 3, multiplicity 1

H

x = , multiplicity 1

D

x = -3, multiplicity 1


140 multiplicities, of the following polynomial.

Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem

A

x = -1, mulitplicity 1

Pull for Answer

B, C, and D

B

x = -1, mulitplicity 2

C

x = , multiplicity 1

D

x = , multiplicity 1

E

x = , multiplicity 2

F

x = , multiplicity 2


141 multiplicities, of the following polynomial.

Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem

Pull for Answer

A, C, E, and G

A

x = -1, multiplicity 1

E

x = , multiplicity 1

F

x = , multiplicity 2

B

x = -1, multiplicity 2

G

x = , multiplicity 1

C

x = 1, multiplicity 1

H

x = , multiplicity 2

D

x = 1, multiplicity 2


Writing a Polynomial Function from its Given Zeros multiplicities, of the following polynomial.

Return to

Table of

Contents


Write the polynomial function of lowest degree using the given zeros, including any multiplicities.

x = -1, multiplicity of 1

x = -2, multiplicity of 2

x = 4, multiplicity of 1

Work backwards from the zeros to the original polynomial.

or

or

or

Write the zeros in factored form by placing them back on the other side of the equal sign.

or

or

or


142 given zeros, including any multiplicities.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

x = -.5, multiplicity of 1

x = 3, multiplicity of 1

x = 2.5, multiplicity of 1

Pull for Answer

A

A

B

C

D


143 given zeros, including any multiplicities.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

x = 1/3, multiplicity of 1

x = -2, multiplicity of 1

x = 2, multiplicity of 1

Pull for Answer

B

A

B

C

D


144 given zeros, including any multiplicities.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

x = 0, multiplicity of 3

x = -2, multiplicity of 2

x = 2, multiplicity of 1

x = 1, multiplicity of 1

x = -1, multiplicity of 2

Pull for Answer

C

A

B

C

D

E


145 given zeros, including any multiplicities.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

Pull for Answer

A

D

B

C

D


146 given zeros, including any multiplicities.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

Pull for Answer

D

A

B

C

D


147 given zeros, including any multiplicities.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

Pull for Answer

D

A

B

C

D


Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

x = -2

x = -1

x = 1.5

x = 3

x = 1.5

x = -2

x = 3

x = -1

or

or

or


When the sum of the real zeros, including multiplicities, 
does not equal the degree, the other zeros are imaginary.

This is a polynomial of degree 6. 
It has 2 real zeros and

4 imaginary zeros.

Real Zeros -2 2


148 
does not equal the degree, the other zeros are imaginary.

Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A

even and positive

Pull for Answer

C

B

even and negative

odd and positive

C

odd and positive

D

odd and negative


149 
does not equal the degree, the other zeros are imaginary.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

A

B

C

D

Pull for Answer

E

C

F


150 
does not equal the degree, the other zeros are imaginary.

Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A

odd and positive

B

odd and negative

C

even and positive

D

even and negative

Pull for Answer

C

even and positive


151 
does not equal the degree, the other zeros are imaginary.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

A

B

Pull for Answer

A

C

D


152 
does not equal the degree, the other zeros are imaginary.

Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

A

odd and positive

B

odd and negative

Pull for Answer

C

even and positive

A

odd and positive

D

even and negative


153 
does not equal the degree, the other zeros are imaginary.

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities.

A

B

C

Pull for Answer

C

D


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