Slide 2 / 257 Algebra II

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1 New Jersey enter for Teaching and Learning Slide / 57 Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. lick to go to website: Slide / 57 lgebra II Polynomials: Operations and Functions Table of ontents click on the topic to go to that section Properties of Review dding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying Polynomials Special inomial Products ividing a Polynomial by a Monomial ividing a Polynomial by a Polynomial haracteristics of Polynomial Functions nalyzing Graphs and Tables of Polynomial Functions Zeros and Roots of a Polynomial Function Writing Polynomials from its Zeros Slide / 57

2 Slide / 57 Properties of Review Return to Table of ontents Slide 5 / 57 Goals and Objectives Students will be able to simplify complex expressions containing exponents. Slide 6 / 57 Why do we need this? allow us to condense bigger expressions into smaller ones. ombining all properties of powers together, we can easily take a complicated expression and make it simpler.

3 Slide 7 / 57 Properties of Slide 8 / 57 Multiplying powers of the same base: an you write this expression in another way?? (xy)(xy) Simplify: (-ab)(ab) (-pqn)(pqn) Slide 9 / 57

4 Slide / 57 Work out: xy.x5y (xy)(xy) Slide / 57 Simplify: (mnp)(mnp) mnp m5np mnp9 Solution not shown Simplify: (-xy)(xy) xy5 7xy5 -xy Solution not shown Slide / 57

5 Slide / 57 Work out: pq. pq 6pq 6pq7 8pq Solution not shown Slide / 57 Simplify: 5mq. mq5 5m6q8 5m6q8 5m8q5 Solution not shown Slide 5 / 57 5 Simplify: (-6ab5)(6ab6) ab -6a5b -6ab Solution not shown

6 Slide 6 / 57 ividing numbers with the same base: Simplify: Slide 7 / 57 Slide 8 / 57 Try...

7 6 ivide: Solutions not shown 7 Simplify: Solution not shown 8 Work out: Solution not shown Slide 9 / 57 Slide / 57 Slide / 57

8 Slide / 57 9 ivide: Solution not shown Simplify: Solution not shown Power to a power: Slide / 57 Slide / 57

9 Simplify: Try: Slide 5 / 57 Slide 6 / 57 Slide 7 / 57 Work out: Solution not shown

10 Slide 8 / 57 Work out: Solution not shown Slide 9 / 57 Simplify: Solution not shown Slide / 57 Simplify: Solution not shown

11 5 Simplify: Solution not shown Negative and zero exponents: Slide / 57 Slide / 57 Why is this? Work out the following: Slide / 57 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:

12 Simplify and write the answer in both forms. Simplify and write the answer in both forms. Simplify: Slide / 57 Slide 5 / 57 Slide 6 / 57

13 Slide 7 / 57 Write the answer with positive exponents. 6 Simplify and leave the answer with positive exponents: Solution not shown 7 Simplify. The answer may be in either form. Solution not shown Slide 8 / 57 Slide 9 / 57

14 Slide / 57 8 Write with positive exponents: Solution not shown Slide / 57 9 Simplify and write with positive exponents: Solution not shown Solution not shown Slide / 57 Simplify. Write the answer with positive exponents.

15 Slide / 57 Solution not shown ombinations Usually, there are multiple rules needed to simplify problems with exponents. Try this one. Leave your answers with positive exponents. Simplify. Write the answer without a fraction. When fractions are to a negative power, a short cut is to flip the fraction and make the exponent positive. Try... Slide / 57 Slide 5 / 57

16 Two more examples. Leave your answers with positive exponents. Slide 7 / 57 Solution not shown Simplify and write with positive exponents: Slide 8 / 57 Solution not shown Simplify. can be in either form. Slide 6 / 57

17 Slide 9 / 57 Simplify and write with positive exponents: Solution not shown Slide 5 / 57 5 Simplify and write without a fraction: Solution not shown Slide 5 / 57 6 Simplify. may be in any form. Solution not shown

18 Slide 5 / 57 7 Simplify. may be in any form. Solution not shown Slide 5 / 57 8 Simplify the expression: Slide 5 / 57 9 Simplify the expression:

19 Slide 55 / 57 dding and Subtracting Polynomials Return to Table of ontents Slide 56 / 57 Vocabulary term is the product of a number and one or more variables to a non-negative exponent. Term 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 degree degree=++=6 Slide 57 / 57 Solution Identify the degree of the polynomials:

20 Slide 57 () / 57 Solution Identify the degree of the polynomials: ) +=5th degree ) 5+=6th degree [This object is a teacher notes pull tab] What is the difference between a monomial and a polynomial? Slide 58 / 57 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: 5x mn 7 -y ab polynomial is a sum or difference of two or more monomials where each monomial iscalled a term. More specifically, if two terms are added, this is called a INOMIL. nd if three terms are added this is called a TRINOMIL. For example: 5x + 7m m + n - yz5 a + b 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: Slide 59 / 57

21 Slide 6 / 57 Review from lgebra I Monomials with the same variables and the same power arelike terms. Like Terms x and -x xy and xy Unlike Terms -b and a 6ab and -ab Slide 6 / 57 ombine these like terms using the indicated operation. Slide 6 / 57 Simplify

22 Slide 6 / 57 Simplify Slide 6 / 57 Simplify 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: (a +a -9) + (a -6a +) Slide 65 / 57

23 Slide 66 / 57 Example: (6b -b) - (6x +b -b) Slide 67 / 57 dd Slide 68 / 57 dd

24 Slide 69 / 57 5 Subtract Slide 7 / 57 6 dd Slide 7 / 57 7 dd

25 Slide 7 / 57 8 Simplify Slide 7 / 57 9 Simplify Slide 7 / 57 Simplify

26 Slide 75 / 57 Simplify Slide 76 / 57 Simplify Slide 77 / 57 What is the perimeter of the following figure? (answers are in units)

27 Slide 78 / 57 Multiplying a Polynomial by a Monomial Return to Table of ontents Slide 79 / 57 Review from lgebra I Find the total area of the rectangles. 5 8 square units square units Review from lgebra I To multiply a polynomial by a monomial, you use the distributive property together with the laws of exponents for multiplication. Example: Simplify. -x(5x - 6x + 8) (-x)(5x) + (-x)(-6x) + (-x)(8) -x + x + -6x -x + x - 6x Slide 8 / 57

28 Slide 8 / 57 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: -x(-x + x - ) click to reveal 6x - 9x + 6x Slide 8 / 57 More Practice!. Multiply to simplify. click. click. click Slide 8 / 57 What is the area of the rectangle shown?

29 Slide 8 / 57 5 Slide 85 / 57 6 Slide 86 / 57 7

30 8 Find the area of a triangle (=/bh) with a base of 5y and a height of y+. ll answers are in square units. Slide 87 / 57 Slide 88 / 57 Multiplying Polynomials Return to Table of ontents Slide 89 / 57 Review from lgebra I Find the total area of the rectangles rea of the big rectangle rea of the horizontal rectangles rea of each box sq.units

31 Slide 9 / 57 Review from lgebra I Find the total area of the rectangles. x x 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. Slide 9 / 57 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: x y x y 6x x y x y 8y Example : Use either method to multiply the following polynomials. Slide 9 / 57

32 Slide 9 / 57 Review from lgebra I The FOIL Method can be used to remember how multiply two binomials. To multiply two binomials, find the sum of... First terms Example: Outer terms First Outer Inner Try it! Inner Terms Last Terms Last Slide 9 / 57 Find each product. ) click ) More Practice! click Find each product. ) click ) click Slide 95 / 57

33 Slide 96 / 57 9 What is the total area of the rectangles shown? Slide 97 / 57 5 Slide 98 / 57 5

34 Slide 99 / 57 5 Slide / 57 5 Slide / 57 5 Find the area of a square with a side of

35 Slide / What is the area of the rectangle (in square units)? Slide / 57 How would you find the area of the shaded region? Slide / What is the area of the shaded region (in square units)?

36 Slide 5 / What is the area of the shaded region (in square units)? Slide 6 / 57 Special inomial Products Return to Table of ontents Slide 7 / 57 Square of a Sum (a + b) (a + b)(a + b) a + ab + b The square of a + b is the square of a plus twice the product of a and b plus the square of b. Example:

37 Slide 8 / 57 Square of a ifference (a - b) (a - b)(a - b) a - ab + b The square of a - b is the square of a minus twice the product of a and b plus the square of b. Example: Slide 9 / 57 Product of a Sum and a ifference (a + b)(a - b) a + -ab + ab + -b a - b Notice the -ab and ab equals. The product of a + b and a - b is the square of a minus the square of b. Example: Try It!. outer terms equals. Find each product. click. click. click Slide / 57

38 Slide / Slide / What is the area of a square with sides Slide / 57?

39 Slide / PR Trina's Triangles Slide 5 / 57 Problem is from: lick for link for commentary and solution. Slide 6 / 57 ividing a Polynomial by a Monomial Return to Table of ontents

40 Slide 7 / 57 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 Slide 8 / 57 lick to Reveal Slide 9 / 57 6 Simplify

41 Slide / 57 6 Simplify Slide / 57 6 Simplify Slide / Simplify

42 Slide / 57 ividing a Polynomial by a Polynomial Return to Table of ontents Long ivision of Polynomials Slide / 57 To divide a polynomial by or more terms, long division can be used. Recall long division of numbers. or Multiply Subtract ring down Repeat Write Remainder over divisor Long ivision of Polynomials To divide a polynomial by or more terms, long division can be used. -x+-6x -x + -x - Multiply Subtract ring down Repeat Write Remainder over divisor Slide 5 / 57

43 Slide 6 / 57 Notes Examples Slide 6 () / 57 Notes Examples [This object is a teacher notes pull tab] Slide 7 / 57 Solution Example

44 Slide 7 () / 57 Solution Example [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. Slide 8 / 57 Slide 9 / 57 Notes Examples

45 Slide 9 () / 57 Notes Examples Make sure you distribute as you subtract! [This object is a teacher notes pull tab] Slide / ivide the polynomial. Slide / ivide the polynomial.

46 Slide / ivide the polynomial. Slide / ivide the polynomial. Pull Students type their answers here Slide / 57 7 ivide the polynomial. Pull Students type their answers here

47 Slide 5 / 57 7 ivide the polynomial. Pull Students type their answers here Slide 6 / 57 haracteristics of Polynomial Functions Return to Table of ontents Slide 7 / 57 Polynomial Functions: onnecting Equations and Graphs

48 Slide 8 / 57 Relate the equation of a polynomial function to its graph. polynomial that has an even number for its highest degree is even-degree polynomial. polynomial that has an odd number for its highest degree is odd-degree polynomial. Even-egree Polynomials Odd-egree Polynomials Slide 9 / 57 Observations about end behavior? Even-egree Polynomials Positive Lead oefficient Negative Lead oefficient Observations about end behavior? Slide / 57

49 Slide / 57 Odd-egree Polynomials Positive Lead oefficient Negative Lead oefficient Observations about end behavior? Slide / 57 End ehavior of a Polynomial Lead coefficient is positive Left End Lead coefficient is negative Right End Left End Right End Even- egree Polynomial Odd- egree Polynomial odd and positive odd and negative even and positive even and negative 7 etermine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Slide / 57

50 Slide / 57 7 etermine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. odd and positive odd and negative even and positive even and negative Slide 5 / 57 7 etermine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. odd and positive odd and negative even and positive even and negative Slide 6 / 57 odd and positive odd and negative even and positive even and negative 75 etermine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative.

51 Slide 7 / 57 Odd-functions not only have the highest exponent that is odd,but all of the exponents are odd. n even-function has only even exponents. Note: a constant has an even degree ( 7 = 7x) Examples: Odd-function Even-function f(x)=x5 -x +x h(x)=6x -x + Neither g(x)= x +x - y=5x y=x y=6x - g(x)=7x7 +x f(x)=x -7x r(x)= x5 +x - Odd Even Neither Odd Even Neither 77 Is the following an odd-function, an even-function, or neither? Slide 8 / Is the following an odd-function, an even-function, or neither? Slide 9 / 57

52 Odd Even Neither Odd Even Neither Odd Even Neither 8 Is the following an odd-function, an even-function, or neither? Slide 5 / Is the following an odd-function, an even-function, or neither? Slide 5 / Is the following an odd-function, an even-function, or neither? Slide 5 / 57

53 n odd-function has rotational symmetry about the origin. Slide 5 / 57 efinition of an Odd Function Slide 5 / 57 n even-function is symmetric about the y-axis efinition of an Even Function Slide 55 / 57 Odd- egree Odd- Function Even- egree Even- Function E Positive Lead oefficient F Negative Lead oefficient 8 Pick all that apply to describe the graph.

54 Slide 56 / 57 Odd- egree Odd- Function Even- egree Even- Function E Positive Lead oefficient F Negative Lead oefficient 8 Pick all that apply to describe the graph. Slide 57 / 57 Odd- egree Odd- Function Even- egree Even- Function E Positive Lead oefficient F Negative Lead oefficient 8 Pick all that apply to describe the graph. Slide 58 / 57 Odd- egree Odd- Function Even- egree Even- Function E Positive Lead oefficient F Negative Lead oefficient 8 Pick all that apply to describe the graph.

55 Slide 59 / 57 Odd- egree Odd- Function Even- egree Even- Function E Positive Lead oefficient F Negative Lead oefficient 85 Pick all that apply to describe the graph. Slide 6 / 57 Zeros of a Polynomial Zeros are the points at which the polynomial intersects the x-axis. n even-degree polynomial with degree n, can have to n zeros. n odd-degree polynomial with degree n, will have to n zeros Slide 6 / How many zeros does the polynomial appear to have?

56 Slide 6 / How many zeros does the polynomial appear to have? Slide 6 / How many zeros does the polynomial appear to have? Slide 6 / How many zeros does the polynomial appear to have?

57 Slide 65 / 57 9 How many zeros does the polynomial appear to have? Slide 66 / 57 9 How many zeros does the polynomial appear to have? Slide 67 / 57 nalyzing Graphs and Tables of Polynomial Functions Return to Table of ontents

58 polynomial function can graphed by creating a table, graphing the points, and then connecting the points with a smooth curve. X Y How many zeros does this function appear to have? X Y There is a zero at x = -, a second between x = and x =, and a third between x = and x =. an we recognize zeros given only a table? X Y Slide 68 / 57 Slide 69 / 57 Slide 7 / 57

59 Slide 7 / 57 Intermediate Value Theorem Given a continuous function f(x), every value between f(a) and f(b) exists. Let a = and b =, then f(a)= - and f(b)=. For every x value between and, there exists a y-value between - and. The Intermediate Value Theorem justifies saying that there is a zero between x = and x = and that there is another between x = and x =. X Y Slide 7 / 57 X Y How many zeros of the continuous polynomial given can be found using the table? - Slide 7 / 57

60 E F G H - -5 Slide 7 / 57 9 Where is the least value of x at which a zero occurs on this continuous function? etween which two values of x? X Y - Slide 75 / 57 X Y How many zeros of the continuous polynomial given can be found using the table? X Y E - F G H What is the least value of x at which a zero occurs on this continuous function? Slide 76 / 57

61 Slide 77 / 57 X Y How many zeros of the continuous polynomial given can be found using the table? E -5 F - G H 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 relative maximum points at x = - and the other at x = The relative maximum value is - (the y-coordinate). There is a relative minimum at x = and the value of - Slide 78 / What is the least value of x at which a zero occurs on this continuous function? Give the consecutive integers. X Y - Slide 79 / 57

62 How do we recognize "hills" and "valleys" or the relative maximums and minimums from a table? In the table x goes from - to, y is decreasing. s x goes from to, y increases. nd as x goes from to, y decreases. X Y Slide 8 / 57 an you find a connection between y changing "directions" and the max/ min? When y switches from increasing to decreasing there is a maximum. bout what value of x is there a relative max? Relative Max: click to reveal X Y When y switches from decreasing to increasing there is a minimum. bout what value of x is there a relative min? Relative Min: click to reveal X Y Slide 8 / 57 Slide 8 / 57

63 Since this is a closed interval, the end points are also a relative max/min. re the points around the endpoint higher or lower? Relative Min: click to reveal Relative Max: click to reveal X Y Slide 8 / 57 - E - F - G H 98 t about what x-values does a relative minimum occur? Slide 85 / 57 - E - F - G H 99 t about what x-values does a relative maximum occur? Slide 8 / 57

64 - E X Y F G H - E X Y F G H Slide 87 / 57 t about what x-values does a relative maximum occur? Slide 86 / 57 t about what x-values does a relative minimum occur? Slide 88 / 57 X Y E - F - G H -5 t about what x-values does a relative minimum occur?

65 Slide 89 / 57 X Y E - F - G H t about what x-values does a relative maximum occur? X Y E - F - G H - -5 X Y E - F - G H t about what x-values does a relative maximum occur? Slide 9 / 57 t about what x-values does a relative minimum occur? Slide 9 / 57

66 Slide 9 / 57 Finding Zeros of a Polynomial Function Return to Table of ontents Vocabulary Slide 9 / 57 zero of a function occurs when f(x)= n imaginary zero occurs when the solution to f(x)=, 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. It has four unique zeros: -.5, -.75,.75,.5 Since there are real zeros there are no imaginary zeros - = Slide 9 / 57

67 Slide 95 / 57 When a vertex is on the x-axis, that zero counts as two zeros. This is also a polynomial of degree. It has two unique real zeros: -.75 and.75. These two zeros are said to have a Multiplicity of two. Real Zeros There are real zeros, therefore, no imaginary zeros for this function. Slide 96 / 57 E F 5 6 How many real zeros does the polynomial graphed have? Slide 97 / 57 7 o any of the zeros have a multiplicity of? Yes No

68 E F 5 9 How many real zeros does the polynomial graphed have? E F 5 Slide 99 / 57 Slide 98 / 57 8 How many imaginary zeros does this 8th degree polynomial have? Slide / 57 o any of the zeros have a multiplicity of? Yes No

69 E F 5 Slide / 57 E F 5 How many real zeros does this 5th -degree polynomial have? Slide / 57 How many imaginary zeros does the polynomial graphed have? Slide / 57 o any of the zeros have a multiplicity of? Yes No

70 Slide / 57 E F 5 How many imaginary zeros does this 5th -degree polynomial have? Slide 5 / 57 E F 5 5 How many real zeros does the 6th degree polynomial have? Slide 6 / 57 6 o any of the zeros have a multiplicity of? Yes No

71 Slide 7 / 57 E F 5 7 How many imaginary zeros does the 6th degree polynomial have? Slide 8 / 57 Finding the Zeros without a graph: Recall the Zero Product Property. If ab =, then a = or b =. Find the zeros, showing the multiplicities, of the following polynomial. or or or There are four real roots: -,, 5, 6.5 all with multiplicity of. 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, -, -,, and. - and each have a multiplicity of (their factors are being squared) There are imaginary zeros: -i and i. Each with multiplicity of. There are 9 zeros (count - and twice) so this is a 9th degree polynomial. Slide 9 / 57

72 E F 5 Slide / 57 8 How many distinct real zeros does the polynomial have? Find the zeros, both real and imaginary, showing the multiplicities, of the following polynomial: Slide / 57 click to reveal This polynomial has real root: and imaginary roots: -i and i. They are simple roots with multiplicities of. Slide / 57 E F 5 9 How many distinct imaginary zeros does the polynomial have?

73 Slide / 57 What is the multiplicity of x=? E F 5 Slide 5 / 57 E F 5 How many distinct imaginary zeros does the polynomial have? Slide / 57 How many distinct real zeros does the polynomial have?

74 Slide 6 / 57 What is the multiplicity of x=? E F 5 Slide 8 / 57 E F 5 5 How many distinct imaginary zeros does the polynomial have? Slide 7 / 57 How many distinct real zeros does the polynomial have?

75 Slide 9 / 57 6 What is the multiplicity of x=? E 8 F 9 Slide / 57 7 How many distinct real zeros does the polynomial have? Slide / 57 8 What is the multiplicity of x=?

76 Slide / 57 E F 5 9 How many distinct imaginary zeros does the polynomial have? Find the zeros, showing the multiplicities, of the following polynomial. Slide / 57 To find the zeros, you must first write the polynomial in factored form. Review from lgebra I or or or or This polynomial has two distinct real zeros:, and. There are zeros (count twice) so this is a rd degree polynomial. has a multiplicity of (their factors are being squared). has a multiplicity of. There are imaginary zeros. Find the zeros, showing the multiplicities, of the following polynomial. or or or This polynomial has zeros. There are two distinct real zeros: There are two imaginary zeros:, both with a multiplicity of., both with a multiplicity of. Slide / 57

77 Slide 5 / 57 How many possible zeros does the polynomial function have? E Slide 6 / 57 How many REL zeros does the polynomial equation have? E Slide 7 / 57 What are the zeros of the polynomial function, with multiplicities? x = -, multiplicity of x =, multiplicity of x =, multiplicity of E x = multiplicity of F x = multiplicity of x = -, mulitplicity of

78 Find the zeros of the following polynomial equation, including multiplicities. Slide 8 / 57 x =, multiplicity of x =, multiplicity of x =, multiplicity of x =, multiplicity of Find the zeros of the polynomial equation, including multiplicities Slide 9 / 57 x =, multiplicity x =, multiplicity x = -i, multiplicity x = i, multiplicity E x = -i, multiplcity F x = i, multiplicity 5 Find the zeros of the polynomial equation, including multiplicities, multiplicity of -, multiplicity of -, multiplicity of E F, multiplicity of, multiplicity of, multiplicity of Slide / 57

79 Find the zeros, showing the multiplicities, of the following polynomial. Slide / 57 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 STUK?? RTIONL ZEROS THEOREM Slide / 57 RTIONL ZEROS THEOREM Make list of POTENTIL 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) =. Slide / 57 Using the Remainder Theorem. Notes is a distinct zero, therefore (x -) is a factor of the polynomial. Use POLYNOMIL IVISION to factor out. or or or or This polynomial has three distinct real zeros: -, -/, and, each with a multiplicity of. There are imaginary zeros.

80 Slide / 57 Find the zeros using the Rational Zeros Theorem, showing the multiplicities, of the following polynomial. Potential List: ± ± Remainder Theorem - is a distinct zero, therefore (x+) is a factor. Use POLYNOMIL IVISION to factor out. Slide 5 / 57 or or or or This polynomial has two distinct real zeros: -, and -. - has a multiplicity of (their factors are being squared). - has a multiplicity of. There are imaginary zeros. x =, multiplicity x =, mulitplicity x =, multiplicity x = -, multiplicity E x = -, multiplicity F x = -, multiplicity 6 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem Slide 6 / 57

81 7 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem x = -, multiplicity Slide 7 / 57 x = -, multiplicity x = -, multiplicity x = -, multiplicity E x = -, multiplicity F x = -, multiplicity, multiplicity, multiplicity Slide 8 / 57 8 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem., multiplicity, multiplicity E x =, multiplicity F x = -, multiplicity 9 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem x =, multiplicity E x= x = -, multiplicity F x= x =, multiplicity G x= x = -, multiplicity H x=, multiplicity, multiplicity, multiplicity, multiplicity Slide 9 / 57

82 Slide / 57 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem x = -, mulitplicity x = -, mulitplicity x=, multiplicity x=, multiplicity E x=, multiplicity F x=, multiplicity Slide / 57 Find the zeros of the polynomial equation, including multiplicities using the Rational Zeros Theorem x = -, multiplicity E x=, multiplicity x = -, multiplicity F x=, multiplicity x =, multiplicity G x=, multiplicity x =, multiplicity H x=, multiplicity Slide / 57 Writing a Polynomial Function from its Given Zeros Return to Table of ontents

83 Write the polynomial function of lowest degree using the given zeros, including any multiplicities. Slide / 57 x = -, multiplicity of x = -, multiplicity of x =, multiplicity of 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 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Slide / 57 x = -.5, multiplicity of x =, multiplicity of x =.5, multiplicity of x = /, multiplicity of x = -, multiplicity of x =, multiplicity of Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Slide 5 / 57

84 x =, multiplicity of x = -, multiplicity of x =, multiplicity of x =, multiplicity of x = -, multiplicity of Slide 6 / 57 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. E Slide 7 / 57 5 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. 6 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Slide 8 / 57

85 Slide 9 / 57 7 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Slide 5 / 57 x = - x = - x = - x =.5 x= x =.5 x= x = - 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 real zeros and imaginary zeros. Real Zeros - Slide 5 / 57

86 even and positive Slide 5 / 57 8 etermine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. even and negative odd and positive odd and negative 9 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Slide 5 / 57 E F 5 etermine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. odd and positive odd and negative even and negative even and positive Slide 5 / 57

87 5 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Slide 55 / 57 5 etermine if the graph represents an odd-degree or an even degree polynomial,and if the lead coefficient is positive or negative. Slide 56 / 57 odd and positive even and positive odd and negative even and negative 5 Write the polynomial function of lowest degree using the zeros from the given graphs, including any multiplicities. Slide 57 / 57