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# Algebra II Po lynomials: Operations and Functions - PowerPoint PPT Presentation

Algebra II Po lynomials: Operations and Functions.

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Presentation Transcript

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

click on the topic to go

to that section

Properties of Exponents Review

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

Review

Table of

Contents

Goals and Objectives

Students will be able to simplify complex expressions containing 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.

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??

. . .

. . .

. . .

.

(-3a3b2)(2a4b3)

-6a7b5

(-4p2q4n)(3p3q3n)

-12p5q7n2

Simplify:

Teacher

(-3a3b2)(2a4b3)

(-4p2q4n)(3p3q3n)

Work out:

.

xy3 x5y4

(3x2y3)(2x3y)

6x5y4

Teacher

x6y7

.

(3x2y3)(2x3y)

xy3 x5y4

(m4np)(mn3p2)

1

Simplify:

B

Teacher

A

m4n3p2

B

m5n4p3

C

mnp9

D

Solution not shown

D

(-3x3y)(4xy4)

-12x4y5

2

Simplify:

(-3x3y)(4xy4)

Teacher

A

x4y5

B

7x3y5

C

-12x3y4

D

Solution not shown

.

3

Work out:

2p2q3 4p2q4

D

2p2q3 4p2q4

8p4q7

Teacher

.

A

6p2q4

B

6p4q7

C

8p4q12

D

Solution not shown

Exponents

.

4

Simplify:

A

Teacher

A

50m6q8

B

15m6q8

C

50m8q15

D

Solution not shown

5

Simplify:

(-6a4b5)(6ab6)

B

Teacher

A

a4b11

B

-36a5b11

C

-36a4b30

D

Solution not shown

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

Teacher

Simplify:

Try...

Teacher

A

6

Divide:

Teacher

C

A

D

Solutions not shown

B

7

Simplify:

D

Teacher

C

A

D

Solution not shown

B

A

8

Work out:

Teacher

C

A

D

Solution not shown

B

9

Divide:

C

Teacher

C

A

D

Solution not shown

B

D

10

Simplify:

Teacher

C

A

D

Solution not shown

B

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

.

Simplify:

Teacher

Try:

Teacher

D

11

Work out:

Teacher

C

A

B

D

Solution not shown

12

Work out:

B

Teacher

C

A

D

Solution not shown

B

C

13

Simplify:

Teacher

C

A

D

Solution not shown

B

D

14

Simplify:

Teacher

C

A

D

Solution not shown

B

15

Simplify:

C

Teacher

C

A

D

Solution not shown

B

Negative and zero exponents:

Teacher

Spend time showing students how negatives and zeros occur.

Why is this? Work out the following:

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:

Simplify and write the answer in both forms.

Teacher

Simplify and write the answer in both forms.

Teacher

Teacher

Simplify:

Write the answer with positive exponents.

Teacher

B

16

Simplify and leave the answer with positive exponents:

Teacher

A

C

B

D

Solution not shown

A

17

Simplify. The answer may be in either form.

Teacher

A

C

B

D

Solution not shown

18

Write with positive exponents:

D

Teacher

A

C

B

D

Solution not shown

19

Simplify and write with positive exponents:

D

Teacher

A

C

B

D

Solution not shown

20

Simplify. Write the answer with positive exponents.

C

Teacher

A

C

B

D

Solution not shown

21

Simplify. Write the answer without a fraction.

A

Teacher

A

C

B

D

Solution not shown

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.

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

Teacher

Try...

Teacher

22

Simplify and write with positive exponents:

D

Teacher

A

C

B

D

Solution not shown

23

Simplify. Answer can be in either form.

B

Teacher

A

C

B

D

Solution not shown

24

Simplify and write with positive exponents:

B

Teacher

A

C

B

D

Solution not shown

25

Simplify and write without a fraction:

A

Teacher

A

C

B

D

Solution not shown

26

Simplify. Answer may be in any form.

D

Teacher

A

C

B

D

Solution not shown

27

Simplify. Answer may be in any form.

C

Teacher

A

C

B

D

Solution not shown

Simplify the expression:

A

C

B

D

D

Simplify the expression:

A

C

A

B

D

Polynomials

Table of

Contents

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

1) 3+2=5th degree

2) 5+1=6th degree

Solution

[This object is a teacher notes pull tab]

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

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:

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

click

click

click

click

Simplify

A

B

C

D

D

Simplify

A

B

C

A

D

Simplify

A

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.

A

C

B

C

D

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

A

B

B

C

D

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

Subtract

A

D

B

C

D

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

A

C

B

C

D

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

A

B

C

C

D

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

Simplify

A

C

B

C

D

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

Simplify

A

D

B

C

D

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

Simplify

A

D

B

C

D

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

Simplify

A

C

B

C

D

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

Simplify

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

B

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

by a Monomial

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

B

A

C

D

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

A

B

B

C

D

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

A

A

B

C

D

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

A

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

D

C

D

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

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

B

D

C

D

50  polynomials.

A

B

B

C

D

51  polynomials.

A

B

B

C

D

52  polynomials.

A

C

B

C

D

53  polynomials.

A

B

B

C

D

54  polynomials.

Find the area of a square with a side of

A

B

C

C

D

55  polynomials.

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

A

B

C

D

B

Teacher

sq. units

56  polynomials.

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

A

B

C

D

A

57  polynomials.

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

A

B

C

D

A

S  polynomials.pecial Binomial Products

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

C

C

D

59  polynomials.

A

D

B

C

D

60  polynomials.

What is the area of a square with sides ?

A

B

D

C

D

61  polynomials.

A

B

B

C

D

Problem is from:  polynomials.

and solution.

A-APR

Trina's

Triangles

Dividing a Polynomial by a Monomial  polynomials.

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.

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

Simplify

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

A

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

Simplify

C

A

B

D

C

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

Simplify

A

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

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

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

B

C

D

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

Divide the polynomial.

A

B

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

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

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

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

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

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

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

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

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

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

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

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

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

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

F

A Odd- Degree

B Odd- Function

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

F

C Even- Degree

D Even- Function

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

A Odd- Degree

B Odd- Function

D

Even- Function

E

F

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

F

A Odd- Degree

B Odd- Function

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

F

C Even- Degree

D Even - Function

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?

5 zeros

87 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

4 zeros

88 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

3 zeros

89 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

2 zeros

90 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

None

91 odd,but all of the exponents are odd.

How many zeros does the polynomial appear to have?

2 zeros

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

of Polynomial Functions

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?

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

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?

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

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?

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.

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

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

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

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

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

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

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

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

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

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

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

E 4

F

5

107 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

No

108 zeros.

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

A

0

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

D 3

110 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

Yes

111 zeros.

How many imaginary zeros does the polynomial  graphed have?

A

0

B

1

C

2

D

3

E

4

F

5

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

D 3

113 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

No

114 zeros.

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

A

0

B

1

C

2

D

3

E

4

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

C 2

116 zeros.

Do any of the zeros have a multiplicity of 2?

Yes

No

Yes

117 zeros.

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

A

0

B

1

C

2

D

3

E

4

C 2

F

5

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.

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

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

E

4

A 0

F

5

120 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

1

121 multiplicities, of the following polynomial:

How many distinct real zeros does the polynomial have?

A

0

B

1

C

2

D

3

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

A 0

E

4

F

5

123 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

2

124 multiplicities, of the following polynomial:

How many distinct real zeros does the polynomial have?

A

0

B

1

C

2

D

3

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

C 2

E

4

F

5

126 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

1

127 multiplicities, of the following polynomial:

How many distinct real zeros does the polynomial have?

A

0

B

5

C

6

D

7

B 5

E

8

F

9

128 multiplicities, of the following polynomial:

What is the multiplicity of x=1?

3

129 multiplicities, of the following polynomial:

How many distinct imaginary zeros does the polynomial have?

A

0

B

1

C

2

D

3

C 2

E

4

F

5

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.

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

3

D

C

2

D

3

E

4

131 polynomial.

How many REAL zeros does the polynomial

equation have?

A

0

B

1

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

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

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

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

C

-2, multiplicity of 1

D

-2, multiplicity of 2

E

, multiplicity of 1

F

, multiplicity of 2

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

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

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

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

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

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

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.

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

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

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

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.

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.

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.

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

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

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

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

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

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