Algebra II Po lynomials: Operations and Functions www.njctl.org 2013-09-25 IMPORTANT TIP: Throughout this unit, it is extremely important that you, as.

Algebra II Po lynomials: Operations and Functions www.njctl.org 2013-09-25 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

Table of Contents Adding and Subtracting Polynomials Dividing a Polynomial by a Monomial Characteristics of Polynomial Functions Analyzing Graphs and Tables of Polynomial Functions Zeros and Roots of a Polynomial Function click on the topic to go to that section Multiplying a Polynomial by a Monomial Multiplying Polynomials Special Binomial Products Dividing a Polynomial by a Polynomial Properties of Exponents Review 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.

Properties of Exponents

Exponents Multiplying powers of the same base: (x4y3)(x3y) Teacher 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..... Can you write this expression in another way??

Exponents (-3a3b2)(2a4b3) Simplify: (-4p2q4n)(3p3q3n) Teacher (-3a3b2)(2a4b3) -6a7b5 (-4p2q4n)(3p3q3n) -12p5q7n2

Work out: Exponents xy3 x5y4. (3x2y3)(2x3y) Teacher xy3 x5y4. (3x2y3)(2x3y) 6x5y4 x6y 7

1Simplify: Am4n3p2 Bm5n4p3 Cmnp9 DSolution not shown (m4np)(mn3p2) Exponents Teacher B

2Simplify: Ax4y5 B7x3y5 C-12x3y4 DSolution not shown Exponents Teacher D (- 3x3y)(4xy 4) -12x4y5 (-3x3y)(4xy4)

3Work out: A6p2q4 B6p4q7 C8p4q12 DSolution not shown Exponents Teacher D 2p2q3 4p2q4 8p4q7. 2p2q3 4p2q4.

4Simplify: A50m6q8 B15m6q8 C50m8q15 DSolution not shown Exponents Teacher A. 5m2q3 10m4q5

5Simplify: Aa4b11 B-36a5b11 C-36a4b30 DSolution not shown (-6a4b5)(6ab6) Exponents Teacher B

Exponents Dividing numbers with the same base: Teacher 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.... x x x y x3 y = 4m

Exponents Simplify: Teacher

Exponents Try... Teacher

6Divide: A B C DSolutions not shown Exponents Teacher A

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

8Work out: A B C DSolution not shown Exponents Teacher A

9Divide: A B C DSolution not shown Exponents Teacher C

Exponents Teacher 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.. Power to a power:

Try: Exponents Teacher

11Work out: A B C DSolution not shown Exponents Teacher D

12Work out: A B C DSolution not shown Exponents Teacher B

13Simplify: A B C DSolution not shown Exponents Teacher C

15Simplify: A B C DSolution not shown Exponents Teacher C

Negative and zero exponents: Exponents Why is this? Work out the following: Teacher Spend time showing students how negatives and zeros occur.

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 Teacher Write the answer with positive exponents.

16Simplify and leave the answer with positive exponents: A B C DSolution not shown Exponents Teacher B

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

18Write with positive exponents: A B C DSolution not shown Exponents Teacher D

19Simplify and write with positive exponents: A B C DSolution not shown Exponents Teacher D

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

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

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

22Simplify and write with positive exponents: A B C DSolution not shown Exponents Teacher D

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

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

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

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

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

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

29Simplify the expression: A B C D Pull for Answer A

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: Solution [This object is a teacher notes pull tab] 1) 3+2=5th degree 2) 5+1=6th degree

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. 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 32m3n4 7 -3y 23a11b4 For example: 5x2 + 7m 32m + 4n3 - 3yz5 23a11 + b4

Standard Form Th e 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 TermsUnlike Terms 4x and -12x -3b and 3a x3y and 4x3y 6a2b and -2ab2 Review from Algebra I

Combine these like terms using the indicated operation. click

30Simplify A B C D Pull for Answer D

31Simplify A B C D Pull for Answer A

32Simplify A B C D Pull for Answer C

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)

33Add A B C D Pull for Answer C

34Add A B C D Pull for Answer B

35Subtract A B C D Pull for Answer D

42Simplify A B C D Pull for Answer B

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

M ultiplying a Polynomial by a Monomial Return to Table of Contents

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

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) 6x4 - 9x2 + 36x click to reveal

More Practice!Multiply to simplify. 1. 2. 3. click

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

45 A B C D Pull for Answer B

46 A B C D A Pull for Answer

47 A B C D C Pull for Answer

48Find 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 C D D Pull for Answer

Multiplying Polynomials Return to Table of Contents

Find the total area of the rectangles. 5 8 2 6 sq.units Area of the big rectangle Area of the horizontal rectangles Area of each box Review from Algebra I

Find the total area of the rectangles. 2x 4 x 3 Review from Algebra I

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. 2x 4y 3x2y 6x2 4xy 12xy 8y2 E xample:

Example 2: Use either method to multiply the following polynomials.

The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of.... First termsOuter terms Inner Terms Last Terms Example: First Outer Inner Last Review from Algebra I

Try it!Find each product. 1) 2) click

3 ) 4) More Practice!Find each product. click

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

50 A B C D B Pull for Answer

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

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

How would you find the area of the shaded region? Shaded Area = Total area - Unshaded Area sq. units Teacher

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

57What is the area of the shaded region (in square units)? A B C D A Pull for Answer

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 Notice the -ab and ab a2 - b2equals 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. 1. 2. 3. click

59 A B C D D Pull for Answer

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

Problem is from: Click for link for commentary and solution. A-APR Trina's Triangles

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

62Simplify A B C D D Pull for Answer

63Simplify A B C D A Pull for Answer

64Simplify A B C D C Pull for Answer

65Simplify A B C D A Pull for Answer

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. Multiply Subtract Bring down Repeat Write Remainder over divisor -2x2+-6x -10x +3 -10x -30 33

Examples Teacher Notes [This object is a teacher notes pull tab]

Example Solution [This object is a teacher notes pull tab]

Example: In this example there are "missing terms". Fill in those terms with zero coefficients before dividing. click

Examples Teacher Notes [This object is a teacher notes pull tab] Make sure you distribute as you subtract!

66Divide the polynomial. A B C D B Pull for Answer

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

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

69Divide the polynomial. Pull

Characteristics of Polynomial Functions Return to Table of Contents

Polynomial Functions: C onnecting 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 CoefficientNegative Lead Coefficient Observations about end behavior?

Odd-Degree Polynomials Observations about end behavior? Positive Lead CoefficientNegative Lead Coefficient

End Behavior of a Polynomial Lead coefficient is positive Left End Right End Lead coefficient is negative Left End Right End Even- Degree Polynomial Odd- Degree Polynomial

72Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Aodd and positive B odd and negative Ceven and positive Deven and negative D even and negative Pull for Answer

73Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Aodd and positive B odd and negative Ceven and positive Deven and negative A odd and positive Pull for Answer

74Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Aodd and positive B odd and negative Ceven and positive Deven and negative C even and positive Pull for Answer

75Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Aodd and positive B odd and negative Ceven and positive Deven and negative Bodd and negative Pull for Answer

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-functionEven-functionNeither f(x)=3x5 -4x3 +2x h(x)=6x4 -2x2 +3 g(x)= 3x2 +4x - 4 y=5xy=x2y=6x -2 g(x)=7x7 +2x3f(x)=3x10 -7x2r(x)= 3x5 +4x3 - 2

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

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

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

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

80Is the following an odd-function, an even-function, or neither? A Odd B Even C Neither CNeither Pull for Answer

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

81Pick 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 AOdd- Degree BOdd- Function EPositive Lead Coefficient Pull for Answer

82Pick 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 CEven- Degree DEven- Function EPositive Lead Coefficient Pull for Answer

83Pick 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 AOdd- Degree BOdd- Function FNegative Lead Coefficient Pull for Answer

84Pick 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 AOdd- Degree BOdd- Function EPositive Lead Coefficient Pull for Answer

85Pick 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 CEven- Degree DEven - Function FNegative Lead Coefficient Pull for Answer

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

86How many zeros does the polynomial appear to have? 5 zeros Pull for Answer

90How many zeros does the polynomial appear to have? None Pull for Answer

Analyzing Graphs and Tables of Polynomial Functions Return to Table of Contents

XY -358 -219 0 0-5 1-2 23 34 4-5 A polynomial function can graphed by creating a table, graphing the points, and then connecting the points with a smooth curve.

XY -358 -219 0 0-5 1-2 23 34 4-5 How many zeros does this function appear to have?

XY -358 -219 0 0-5 1-2 23 34 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?

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.

XY -358 -219 0 0-5 1-2 23 34 4-5 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.

92How many zeros of the continuous polynomial given can be found using the table? XY -3-12 -2-4 1 03 10 2-2 34 4-5 4 zeros Pull for Answer

93Where 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 D 0 E 1 F 2 G 3 H 4 XY -3-12 -2-4 1 03 10 2-2 34 4-5 B -2, C -1 Pull for Answer

94How many zeros of the continuous polynomial given can be found using the table? XY -32 -20 5 02 1-3 24 34 4-5 4 zeros Pull for Answer

95What is the least value of x at which a zero occurs on this continuous function? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -32 -20 5 02 1-3 24 34 4-5 B -2 Pull for Answer

96How many zeros of the continuous polynomial given can be found using the table? XY -35 -21 0-4 1-5 2-2 32 40 2 zeros Pull for Answer

97What is the least value of x at which a zero occurs on this continuous function? Give the consecutive integers. A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -35 -21 0-4 1-5 2-2 32 40 B -2, C -1 Pull for Answer

Relative Maximums and Relative Minimums R elative 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? XY -35 -21 0-4 1-5 2-2 32 40 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? XY -35 -21 0-4 1-5 2-2 32 40 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? XY -35 -21 0-4 1-5 2-2 32 40 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? XY -35 -21 0-4 1-5 2-2 32 40 Relative Min: Relative Max: click to reveal

98 At about what x-values does a relative minimum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 C -1 E 1 Pull for Answer

99At about what x-values does a relative maximum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 B -2 F 2 Pull for Answer

100 At about what x-values does a relative minimum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -35 -21 0-4 1-5 2-2 32 40 E 1 H 4 Pull for Answer

101At about what x-values does a relative maximum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -35 -21 0-4 1-5 2-2 32 40 A -3 G 3 Pull for Answer

102 At about what x-values does a relative minimum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -32 -20 5 02 1-3 24 34 4-5 B -2 E 1 H 4 Pull for Answer

103At about what x-values does a relative maximum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -32 -20 5 02 1-3 24 35 4-5 A -3 C -1 G 3 Pull for Answer

104 At about what x-values does a relative minimum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -3-12 -2-4 1 03 10 2-2 34 4-5 B -2 F 2 H 4 Pull for Answer

105At about what x-values does a relative maximum occur? A -3 B -2 C D 0 E 1 F 2 G 3 H 4 XY -3-12 -2-4 1 03 10 2-2 34 4-5 D 0 G 3 Pull for Answer

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 zeros for this function.

106How many real zeros does the polynomial graphed have? A 0 B 1 C 2 D 3 E 4 F 5 E 4 Pull for Answer

107Do any of the zeros have a multiplicity of 2? Yes No Pull for Answer

108How many imaginary zeros does this 8th degree polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 A 0 Pull for Answer

109 How many real zeros does the polynomial graphed have? A 0 B 1 C 2 D 3 E 4 F 5 D 3 Pull for Answer

110Do any of the zeros have a multiplicity of 2? Yes No Yes 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 A 0 Pull for Answer

112How many real zeros does this 5th-degree polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 D 3 Pull for Answer

113Do any of the zeros have a multiplicity of 2? Yes No Pull for Answer

114How many imaginary zeros does this 5th-degree polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 C 2 Pull for Answer

115How many real zeros does the 6th degree polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 C 2 Pull for Answer

116Do any of the zeros have a multiplicity of 2? Yes No Yes Pull for Answer

117How many imaginary zeros does the 6th degree polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 C 2 Pull for Answer

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 There are four real roots: -3, 2, 5, 6.5 all with multiplicity of 1. There are no imaginary roots. Finding the Zeros without a graph:

Find the zeros, showing the multiplicities, of the following polynomial. 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 How many distinct real zeros does the polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 E 4 Pull for Answer

Find the zeros, both real and imaginary, showing the multiplicities, of the following polynomial: This polynomial has 1 real root: 2 and 2 imaginary roots: -1i and 1i. They are simple roots with multiplicities of 1. click to reveal

119How many distinct imaginary zeros does the polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 A 0 Pull for Answer

120What is the multiplicity of x=1? 1 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 D 3 Pull for Answer

122How many distinct imaginary zeros does the polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 A 0 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 D 3 Pull for Answer

125How many distinct imaginary zeros does the polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 C 2 Pull for Answer

127 How many distinct real zeros does the polynomial have? A 0 B 5 C 6 D 7 E 8 F 9 B 5 Pull for Answer

129How many distinct imaginary zeros does the polynomial have? A 0 B 1 C 2 D 3 E 4 F 5 C 2 Pull for Answer

Find the zeros, showing the multiplicities, of the following polynomial. 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. Review from Algebra I To find the zeros, you must first write the polynomial in factored form.

Find the zeros, showing the multiplicities, of the following polynomial. or There are two distinct real zeros:, both with a multiplicity of 1. There are two imaginary zeros:, both with a multiplicity of 1. This polynomial has 4 zeros.

130How many possible zeros does the polynomial function have? A0 B1 C2 D3 E4 D Pull for Answer 3

131How many REAL zeros does the polynomial equation have? A0 B1 C2 D3 E4 D Pull for Answer 3

132What are the zeros of the polynomial function, with multiplicities? Ax = -2, mulitplicity of 1 Bx = -2, multiplicity of 2 Cx = 3, multiplicity of 1 Dx = 3, multiplicity of 2 Ex = 0 multiplicity of 1 Fx = 0 multiplicity of 2 ACF ACF x = -2, multiplicity 1 x = 3, multiplicity 1 x = 0, multiplicity 2 Pull for Answer

133Find the zeros of the following polynomial equation, including multiplicities. Ax = 0, multiplicity of 1 Bx = 3, multiplicity of 1 Cx = 0, multiplicity of 2 Dx = 3, multiplicity of 2 AD AD Pull for Answer x = 2, multiplicity 1 x = 3, multiplicity 2

134Find the zeros of the polynomial equation, including multiplicities Ax = 2, multiplicity 1 Bx = 2, multiplicity 2 Cx = -i, multiplicity 1 Dx = i, multiplicity 1 Ex = -i, multiplcity 2 Fx = i, multiplicity 2 ACD ACD Pull for Answer x = 2, multiplicity 1 x = -i, multiplicity 1 x = i, multiplicity 1

135Find the zeros of the polynomial equation, including multiplicities A2, multiplicity of 1 B2, multiplicity of 2 C-2, multiplicity of 1 D-2, multiplicity of 2 E, multiplicity of 1 F, multiplicity of 2 ACEFACEF Pull for Answer x = 2, multiplicity 1 x = -2, multiplicity 1 x =, multiplicity 1

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

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.

1 is a distinct zero, therefore (x -1) is a factor of the polynomial. Use POLYNOMIAL DIVISION to factor out. Using the Remainder Theorem. or This polynomial has three distinct real zeros: -2, -1/3, and 1, each with a multiplicity of 1. There are 0 imaginary zeros. When you find a distinct zero, write the zero in factored form and then complete polynomial division. Teacher Notes

Find the zeros using the Rational Zeros Theorem, showing the multiplicities, of the following polynomial. Potential List: ± ±1 -3 is a distinct zero, therefore (x+3) is a factor. Use POLYNOMIAL DIVISION to factor out. Remainder Theorem

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.

136Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem Ax = 1, multiplicity 1 Bx = 1, mulitplicity 2 Cx = 1, multiplicity 3 Dx = -3, multiplicity 1 Ex = -3, multiplicity 2 Fx = -3, multiplicity 3 B D Pull for Answer x = -3, multiplicity 1 x = 1, multiplicity 2

137Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem Ax = -2, multiplicity 1 Bx = -2, multiplicity 2 Cx = -2, multiplicity 3 Dx = -1, multiplicity 1 Ex = -1, multiplicity 2 Fx = -1, multiplicity 3 A E Pull for Answer x = -2, multiplicity 1 x = -1, multiplicity 2

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

139Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem Ax = 1, multiplicity 1 Bx = -1, multiplicity 1 Cx = 3, multiplicity 1 Dx = -3, multiplicity 1 Ex =, multiplicity 1 F G H C, E, H Pull for Answer

140Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem Ax = -1, mulitplicity 1 Bx = -1, mulitplicity 2 Cx =, multiplicity 1 D Ex =, multiplicity 2 F B, C, and D Pull for Answer

141Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem Ax = -1, multiplicity 1 Bx = -1, multiplicity 2 Cx = 1, multiplicity 1 Dx = 1, multiplicity 2 Ex =, multiplicity 1 Fx =, multiplicity 2 Gx =, multiplicity 1 Hx =, multiplicity 2 Pull for Answer A, C, E, and G

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 or Work backwards from the zeros to the original polynomial. Write the zeros in factored form by placing them back on the other side of the equal sign.

142Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C D x = -.5, multiplicity of 1 x = 3, multiplicity of 1 x = 2.5, multiplicity of 1 A Pull for Answer

143Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C D x = 1/3, multiplicity of 1 x = -2, multiplicity of 1 x = 2, multiplicity of 1 B Pull for Answer

144Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C D E 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 Pull for Answer

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

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 = -2 x = -1 x = 1.5 x = 3 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

148Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Aeven and positive Beven and negative Codd and positive Dodd and negative C Pull for Answer odd and positive

149Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C D E F C Pull for Answer

150Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Aodd and positive Bodd and negative Ceven and positive Deven and negative C Pull for Answer even and positive

151Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C D A Pull for Answer

152Determine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Aodd and positive Bodd and negative Ceven and positive Deven and negative A Pull for Answer odd and positive

153Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. A B C D C Pull for Answer

Algebra II Po lynomials: Operations and Functions www.njctl.org 2013-09-25 IMPORTANT TIP: Throughout this unit, it is extremely important that you, as.

Similar presentations

Presentation on theme: "Algebra II Po lynomials: Operations and Functions www.njctl.org 2013-09-25 IMPORTANT TIP: Throughout this unit, it is extremely important that you, as."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Algebra II Po lynomials: Operations and Functions www.njctl.org 2013-09-25 IMPORTANT TIP: Throughout this unit, it is extremely important that you, as.

Similar presentations

Presentation on theme: "Algebra II Po lynomials: Operations and Functions www.njctl.org 2013-09-25 IMPORTANT TIP: Throughout this unit, it is extremely important that you, as."— Presentation transcript:

Similar presentations

About project

Feedback