Algebra II Polynomials: 2013-09-25 www.njctl.org Teacher Operations and

Algebra II Polynomials: 2013-09-25 www.njctl.org Teacher Operations and

Algebra II Polynomials: 2013-09-25 www.njctl.org Teacher Operations and Functions Table of Contents 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 click on the topic to go to that section 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. Properties of Exponents Exponents Multiplying powers of the same base: Can you write this expression in another way??

Teacher (x4y3) (x3y) Teacher Exponents Simplify: (-3a3b2) (2a4b3) (-4p2q4n)(3p3q3n) Exponents xy3. x5y4

(3x2y3) (2x3y) Teacher Work out: 1 Simplify: (m4np)(mn3p2) A m4n3p2 B

m5n4p3 C mnp9 D Solution not shown Teacher Exponents 2 Simplify:

(-3x3y)(4xy4) A x4y5 B 7x3y5 C -12x3y4 D Solution not shown Teacher

Exponents Work out: 2p2q3 . 4p2q4 3 A 6p2q4 B 6p4q7 C 8p4q12

D Solution not shown Teacher Exponents 4 Simplify: 5m2q3 10m4q5 . A 50m6q8

B 15m6q8 C 50m8q15 D Solution not shown Teacher Exponents 5 Simplify: (-6a4b5)(6ab6)

A a4b11 B -36a5b11 C -36a4b30 D Solution not shown Teacher Exponents

Dividing numbers with the same base: Teacher Exponents Simplify: Teacher Exponents Try... Teacher Exponents

6 Teacher Exponents Divide: A C B D Solutions not shown Exponents

Simplify: Teacher 7 A C B D Solution not shown Exponents

Work out: Teacher 8 A C B D Solution not shown Exponents Divide:

A C B D Teacher 9 Solution not shown Exponents Simplify: Teacher

10 A C B D Solution not shown Power to a power: Teacher Exponents

Simplify: Teacher Exponents Try: Teacher Exponents Exponents Work out: Teacher 11

A C B D Solution not shown Exponents Work out: Teacher 12

A C B D Solution not shown Exponents Simplify: Teacher 13 A

C B D Solution not shown Exponents Simplify: Teacher 14 A

C B D Solution not shown Exponents Simplify: Teacher 15 A C

B D Solution not shown Negative and zero exponents: Why is this? Work out the following: Teacher 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.

Write with positive exponents: Write without a fraction: Teacher Exponents Simplify and write the answer in both forms. Teacher Exponents Simplify and write the answer in both forms. Teacher Exponents

Simplify: Teacher Exponents Write the answer with positive exponents. Teacher Exponents 16 Simplify and leave the answer with positive exponents: A

C B D Solution not shown Teacher Exponents 17 Simplify. The answer may be in either form. A C

B D Solution not shown Teacher Exponents Exponents Write with positive exponents: A C

B D Teacher 18 Solution not shown 19 Simplify and write with positive exponents: A C B

D Solution not shown Teacher Exponents 20 Simplify. Write the answer with positive exponents. A C B

D Solution not shown Teacher Exponents 21 Simplify. Write the answer without a fraction. A B C D

Solution not shown Teacher Exponents Combinations Usually, there are multiple rules needed to simplify problems with exponents. Try this one. Leave your answers with positive exponents. Teacher Exponents When fractions are to a negative power, a short cut is to flip the fraction and make the exponent positive. Try...

Teacher Exponents Two more examples. Leave your answers with positive exponents. Teacher Exponents 22 Simplify and write with positive exponents: A C

B D Solution not shown Teacher Exponents 23 Simplify. Answer can be in either form. A C B

D Solution not shown Teacher Exponents 24 Simplify and write with positive exponents: A C B D

Solution not shown Teacher Exponents 25 Simplify and write without a fraction: A C B D Solution not shown

Teacher Exponents 26 Simplify. Answer may be in any form. A C B D Solution not shown

Teacher Exponents 27 Simplify. Answer may be in any form. A C B D Solution not shown Teacher

Exponents Simplify the expression: A B C D Pull for Answer 28 Simplify the expression:

A B C D Pull for Answer 29 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 nonnegative 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 Solution Identify the degree of the polynomials: 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 Combine these like terms using the indicated operation. clic k clic k clic k clic k

30 Simplify B C D Pull for Answer A 31 Simplify A

C D Pull for Answer B A B C D Simplify Pull for Answer

32 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: (6b4 -2b) - (6x4 +3b2 -10b) A B C D Pull for

Answer 33 Add A B C D Pull for Answer 34 Add A

B C D Subtract Pull for Answer 35 A B C D Pull for Answer

36 Add B C D Pull for Answer 37 Add A A B

C D Simplify Pull for Answer 38 A B C D Simplify Pull for Answer 39

A B C D Simplify Pull for Answer 40 A B C D Simplify Pull for

Answer 41 A B C D Simplify Pull for Answer 42 43 What is the perimeter of the following figure? (answers are in units)

A B C Pull for Answer D M ultiplying a Polynomial by a Monomial Return to Table of Contents

Review from Algebra I Find the total area of the rectangles. 3 5 8 4 square units square units Review from Algebra I 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! 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.

1. clic k 2. clic k 3. clic k A B C D

What is the area of the rectangle shown? Pull for Answer 44 B C D Pull for 45 A A

B C D Pull for Answer 46 A B C D

Pull for Answer 47 48 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. B C D Pull for Answer A

Multiplying Polynomials Return to Table of Contents Review from Algebra I Find the total area of the rectangles. 5 8 2 6 Area of the big rectangle

Area of the horizontal rectangles Area of each box sq.units Review from Algebra I Find the total area of the rectangles. 2x x 3 4 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 2x 6x2 4xy 4y 12xy

8y2 Example 2: Use either method to multiply the following polynomials. Review from Algebra I The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of .... First terms Outer terms Example: First Outer Inner Last

Inner Terms Last Terms Try it! Find each product. 1) clic k 2) clic k More Practice! Find each product. 3) clic

k 4) clic k 49 What is the total area of the rectangles shown? B C D Pull for Answer A

A B C D Pull for Answer 50 A B C

D Pull for Answer 51 A B C D Pull for Answer

52 A B C D Pull for Answer 53 54 Find the area of a square with a side of

B C D Pull for Answer A 55 What is the area of the rectangle (in square units)? A B C Pull for Answer

D Teacher How would you find the area of the shaded region? What is the area of the shaded region (in square units)? A B C D Pull for Answer 56

What is the area of the shaded region (in square units)? A B C D Pull for Answer 57 S pecial Binomial Products Return to Table of Contents

Square of a Sum (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 (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

(a + b)(a - b) a2 + -ab + ab + -b2 0. Notice the -ab and ab a2 - b2 equals 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. 1. clic k 2. clic

k 3. clic k B C D Pull for Answer 58 A

A B C D Pull for Answer 59 60 What is the area of a square with sides ?

B C D Pull for Answer A B C D Pull for Answer

61 A A-APR Trina's Triangles Problem is from: Click for link for commentary and solution. Dividing a Polynomial by a Monomial 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 Click to Reveal Answer Simplify A B C D

Pull for Answer 62 Simplify A C B D Pull for Answer 63

Simplify A C B D Pull for Answer 64 A B

C D Simplify Pull for Answer 65 Dividing a Polynomial by a Polynomial Return to Table of Contents

Long Division of Polynomials 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 To divide a polynomial by 2 or more terms, long division can be used. -2x2+-6x -10x +3

-10x -30 33 Multiply Subtract Bring down Repeat Write Remainder over divisor Teacher Notes Examples Solution Example Example: In this example there are "missing terms". Fill in those terms with zero coefficients before dividing.

click Teacher Notes Examples A B C D Divide the polynomial. Pull for Answer 66

67 Divide the polynomial. B C D Pull for Answer A 68 Divide the polynomial. B C

D Pull for Answer A Divide the polynomial. Pull Pull 69 Divide the polynomial. Pull 70

Divide the polynomial. Pull 71 Characteristics of Polynomial Functions Return to Table of Contents Polynomial Functions: Connecting Equations and Graphs Relate the equation of a polynomial function to its graph.

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. Even-Degree Polynomials Odd-Degree Polynomials Observations about end behavior? Even-Degree Polynomials Positive Lead Coefficient Negative Lead Coefficient Observations about end behavior? Odd-Degree Polynomials

Positive Lead Coefficient Negative Lead Coefficient Observations about end behavior? End Behavior of a Polynomial Lead coefficient is positive Left End End Even- Degree Polynomial Odd- Degree Polynomial Lead coefficient is negative

Right Left End End Right 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 72 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 73 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 74 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 75 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: Odd-function

Even-function Neither f(x)=3x5 -4x3 h(x)=6x4 +2x 2x2 +3 g(x)= 3x2 +4x -4 y=5x y=x2 y=6x -2 g(x)=7x7 +2x3

f(x)=3x10 7x2 r(x)= 3x5 +4x3 -2 Is the following an odd-function, an even-function, or neither? A Odd B Even C Neither

Pull for Answer 76 Is the following an odd-function, an even-function, or neither? A Odd B Even C

Neither Pull for Answer 77 Is the following an odd-function, an even-function, or neither? A Odd B Even C

Neither Pull for Answer 78 Is the following an odd-function, an even-function, or neither? A Odd B Even

C Neither Pull for Answer 79 Is the following an odd-function, an even-function, or neither? A Odd B Even

C Neither Pull for Answer 80 An odd-function has rotational symmetry about the origin. Definition of an Odd Function An even-function is symmetric about the y-axis Definition of an Even Function 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 81 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 82 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 83 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 84 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

85 Zeros of a Polynomial 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 How many zeros does the polynomial appear to have? Pull for Answer 86 How many zeros does the polynomial appear to have?

Pull for Answer 87 How many zeros does the polynomial appear to have? Pull for Answer 88 How many zeros does the polynomial appear to have? Pull for Answer 89

How many zeros does the polynomial appear to have? Pull for Answer 90 How many zeros does the polynomial appear to have? Pull for Answer 91 Analyzing Graphs and Tables 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. X Y -3 58 -2 19 -1

0 0 -5 1 -2 2 3 3 4 4

-5 How many zeros does this function appear to have? X Y -3 58 -2 19 -1

0 0 -5 1 -2 2 3 3 4 4

-5 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? X Y -3 58 -2 19 -1

0 0 -5 1 -2 2 3 3 4

4 -5 Intermediate Value Theorem 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. X

Y -3 58 -2 19 -1 0 0 -5

1 -2 2 3 3 4 4 -5 How many zeros of the continuous polynomial given can be found using the table? X

Y -3 -12 -2 -4 -1 1 0 3

1 0 2 -2 3 4 4 -5 Pull for Answer

92 Where is the least value of x at which a zero occurs on this continuous function? Between which two values of x? X Y -3 -12 A -3 B -2

-2 -4 C -1 -1 1 D 0 0

3 1 0 2 -2 E 1 F 2 3

4 G 3 4 -5 H 4 Pull for Answer 93

How many zeros of the continuous polynomial given can be found using the table? X Y -3 2 -2 0 -1 5

0 2 1 -3 2 4 3 4 4 -5

Pull for Answer 94 A What is the least value of x at which a zero occurs on this continuous function? -3 X Y -3

2 -2 0 B -2 C -1 -1 5 D

0 0 2 1 -3 2 4 3 4

4 -5 E 1 F 2 G 3 H 4

Pull for Answer 95 How many zeros of the continuous polynomial given can be found using the table? X Y -3 5 -2 1

-1 -1 0 -4 1 -5 2 -2 3

2 4 0 Pull for Answer 96 A What is the least value of x at which a zero occurs on this continuous function? Give the consecutive integers. -3 X

Y -3 5 -2 1 B -2 C -1 -1

-1 D 0 0 -4 1 -5 2 -2

3 2 4 0 E 1 F 2 G 3

H 4 Pull for Answer 97 Relative Maximums and Relative Minimums 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. X Y -3 5 -2

1 -1 -1 0 -4 1 -5 2 -2 3

2 4 0 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 X

Y -3 5 -2 1 -1 -1 0 -4 1

-5 2 -2 3 2 4 0 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 X Y -3 5 -2 1 -1 -1

0 -4 1 -5 2 -2 3 2 4 0

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 X Y -3 5

-2 1 -1 -1 0 -4 1 -5 2

-2 3 2 4 0 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 98 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 99 X Y A -3

E 1 -3 5 B -2 F 2 -2

1 C -1 G 3 -1 -1 0 -4 1

-5 2 -2 3 2 4 0 D 0

H 4 Pull for Answer 100 At about what x-values does a relative minimum occur? X Y A -3 E

1 -3 5 B -2 F 2 -2 1 C

-1 G 3 -1 -1 0 -4 1 -5

2 -2 3 2 4 0 D 0 H 4

Pull for Answer 101 At about what x-values does a relative maximum occur? A X Y -3 2 -2 0

-3 E B -2 F 2 C -1 G

3 -1 5 D 0 H 4 0 2 1

-3 2 4 3 4 4 -5 1 Pull for Answer

102 At about what x-values does a relative minimum occur? A X Y -3 2 -2 0 -3

E B -2 F 2 C -1 G 3 -1

5 D 0 H 4 0 2 1 -3

2 4 3 5 4 -5 1 Pull for Answer 103 At about what x-values does a relative maximum occur?

X Y -3 -12 A -3 E 1 B -2

F 2 -2 -4 C -1 G 3 -1

1 D 0 H 4 0 3 1 0 2

-2 3 4 4 -5 Pull for Answer 104 At about what x-values does a relative minimum occur? X Y

-3 -12 A -3 E 1 B -2 F

2 -2 -4 C -1 G 3 -1 1 D

0 H 4 0 3 1 0 2 -2

3 4 4 -5 Pull for Answer 105 At about what x-values does a relative maximum occur? Finding Zeros of a Polynomial Function Return to Table of Contents

Vocabulary 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 zero for this function. A 0 B

1 C 2 D 3 E 4 F 5 Pull for

Answer 106 How many real zeros does the polynomial graphed have? 107 Do any of the zeros have a multiplicity of 2? Yes Pull for Answer No A 0 B

1 C 2 D 3 E 4 F 5 Pull for

Answer 108 How many imaginary zeros does this 8th degree polynomial have? A 0 B 1 C 2 D

3 E 4 F 5 Pull for Answer 109 How many real zeros does the polynomial graphed have? 110 Do any of the zeros have a multiplicity of 2? Yes Pull for

Answer No A 0 B 1 C 2 D 3

E 4 F 5 Pull for Answer 111 How many imaginary zeros does the polynomial graphed have? A 0

B 1 C 2 D 3 E 4 F 5

Pull for Answer 112 How many real zeros does this 5th-degree polynomial have? 113 Do any of the zeros have a multiplicity of 2? Yes Pull for Answer No A 0

B 1 C 2 D 3 E 4 F

5 Pull for Answer 114 How many imaginary zeros does this 5th-degree polynomial have? A 0 B 1 C 2

D 3 E 4 F 5 Pull for Answer 115 How many real zeros does the 6th degree polynomial have?

116 Do any of the zeros have a multiplicity of 2? Yes Pull for Answer No A 0 B 1 C 2

D 3 E 4 F 5 Pull for Answer 117 How many imaginary zeros does the 6th degree polynomial have?

Finding the Zeros without a graph: 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. A 0 B 1 C

2 D 3 E 4 F 5 Pull for Answer 118 How many distinct real zeros does the polynomial have?

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. A 0 B 1

C 2 D 3 E 4 F 5 Pull for Answer

119 How many distinct imaginary zeros does the polynomial have? Pull for Answer 120 What is the multiplicity of x=1? A 0 B 1 C

2 D 3 E 4 F 5 Pull for Answer 121 How many distinct real zeros does the polynomial have?

A 0 B 1 C 2 D 3 E 4

F 5 Pull for Answer 122 How many distinct imaginary zeros does the polynomial have? Pull for Answer 123 What is the multiplicity of x=1? A 0

B 1 C 2 D 3 E 4 F

5 Pull for Answer 124 How many distinct real zeros does the polynomial have? A 0 B 1 C 2

D 3 E 4 F 5 Pull for Answer 125 How many distinct imaginary zeros does the polynomial have? Pull for

Answer 126 What is the multiplicity of x=1? A 0 B 5 C 6 D 7

E 8 F 9 Pull for Answer 127 How many distinct real zeros does the polynomial have? Pull for Answer 128 What is the multiplicity of x=1?

A 0 B 1 C 2 D 3 E 4

F 5 Pull for Answer 129 How many distinct imaginary zeros does the polynomial have? 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:

There are two imaginary zeros: , both with a multiplicity of 1. , both with a multiplicity of 1. A 0 B 1 C 2 D

3 E 4 Pull for Answer 130 How many possible zeros does the polynomial function have? A 0 B

1 C 2 D 3 E 4 Pull for 131 How many REAL zeros does the polynomial equation have?

A x = -2, mulitplicity of 1 B x = -2, multiplicity of 2 C x = 3, multiplicity of 1 D x = 3, multiplicity of 2 E

x = 0 multiplicity of 1 F x = 0 multiplicity of 2 Pull for Answer 132 What are the zeros of the polynomial function , with multiplicities? A x = 0, multiplicity of 1 B x = 3, multiplicity of 1

C x = 0, multiplicity of 2 D x = 3, multiplicity of 2 Pull for Answer 133 Find the zeros of the following polynomial equation, including multiplicities. A x = 2, multiplicity 1

B x = 2, multiplicity 2 C x = -i, multiplicity 1 D x = i, multiplicity 1 E x = -i, multiplcity 2 F x = i, multiplicity 2

Pull for Answer 134 Find the zeros of the polynomial equation, including multiplicities A 2, multiplicity of 1 B 2, multiplicity of 2 C -2, multiplicity of 1

D -2, multiplicity of 2 E F , multiplicity of 1 , multiplicity of 2 Pull for Answer 135 Find the zeros of the polynomial equation, including multiplicities 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 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.

Teacher Notes 1 is a distinct zero, therefore (x -1) is a factor of the polynomial. Use POLYNOMIAL DIVISION to factor out. 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, o

following polynomial. Potential List: Remainder Theorem -3 is a distinct zero, therefore (x+3) is a factor. Use POLYNOMIAL DIVISION to factor out. 1 or 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. A x = 1, multiplicity 1 B x = 1, mulitplicity 2 C x = 1, multiplicity 3

D x = -3, multiplicity 1 E x = -3, multiplicity 2 F x = -3, multiplicity 3 Pull for Answer 136 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem

A x = -2, multiplicity 1 B x = -2, multiplicity 2 C x = -2, multiplicity 3 D x = -1, multiplicity 1 E x = -1, multiplicity 2

F x = -1, multiplicity 3 Pull for Answer 137 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem A , multiplicity 1 B , multiplicity 1

C , multiplicity 1 D , multiplicity 1 E x = 1, multiplicity 1 F x = -1, multiplicity 1 Pull for Answer

138 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem. Pull for Answer 139 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem A x = 1, multiplicity 1 E x= B

x = -1, multiplicity 1 F x= C x = 3, multiplicity 1 D x = -3, multiplicity 1 G x= H

x= , multiplicity 1 , multiplicity 1 , multiplicity 1 , multiplicity 1 A x = -1, mulitplicity 1 B x = -1, mulitplicity 2

C x= , multiplicity 1 D x= , multiplicity 1 E x= , multiplicity 2 F

x= , multiplicity 2 Pull for Answer 140 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem Pull for Answer 141 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem A

x = -1, multiplicity 1 E x= , multiplicity 1 B x = -1, multiplicity 2 F x= , multiplicity 2 C

x = 1, multiplicity 1 G x= , multiplicity 1 D x = 1, multiplicity 2 H x= , multiplicity 2

Writing a Polynomial Function from its Given Zeros 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 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Pull for Answer

x = -.5, multiplicity of 1 x = 3, multiplicity of 1 x = 2.5, multiplicity of 1 A B C D 143 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Pull for Answer x = 1/3, multiplicity of 1

x = -2, multiplicity of 1 x = 2, multiplicity of 1 A B C D Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. x x x x x A B C

D E = = = = = 0, multiplicity of 3 -2, multiplicity of 2 2, multiplicity of 1 1, multiplicity of 1 -1, multiplicity of 2 Pull for Answer 144

Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C D Pull for Answer 145 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Pull for Answer

146 A B C D Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Pull for Answer 147 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 = -2 x = 1.5 x= 3 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

A even and positive B even and negative C odd and positive D odd and negative Pull for Answer

148 Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. 149 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C E F Pull for Answer

D A odd and positive B odd and negative C even and positive D even and negative Pull for

Answer 150 Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. 151 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. B C D Pull for Answer A

A odd and positive B odd and negative C even and positive D even and negative Pull for Answer

152 Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. 153 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. B C D Pull for Answer A

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