Algebra II. Slide 1 / 276. Slide 2 / 276. Slide 3 / 276. Polynomials: Operations and Functions. Table of Contents

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1 Slide 1 / 276 lgebra II Slide 2 / 276 Polynomials: Operations and Functions Table of ontents click on the topic to go to that section Slide 3 / 276 Properties of Exponents Review Operations with Polynomials Review Special inomial Products inomial Theorem Factoring Polynomials Review Dividing Polynomials Polynomial Functions nalyzing Graphs and Tables of Polynomial Functions Zeros and Roots of a Polynomial Function Writing Polynomials from its Given Zeros

2 Table of ontents click on the topic to go to that section Slide 3 () / 276 Properties of Exponents Review IMPORTNT TIP: Throughout this Operations with Polynomials unit, it is extremely Reviewimportant that you, as a teacher, emphasize correct Special inomial vocabulary Products and make sure students truly know the difference between inomial Theorem monomials and polynomials. Having Factoring Polynomials a solid understanding Review of rules that accompany each will give them a Dividing Polynomials strong foundation for future math Polynomial Functions classes. Teacher Notes nalyzing Graphs and Tables of Polynomial Functions Zeros and Roots of a Polynomial Function Writing Polynomials from its Given Zeros Slide 4 / 276 Properties of Exponents Review Return to Table of ontents This section is intended to be a brief review of this topic. For more detailed lessons and practice see lgebra 1. Slide 5 / 276 Goals and Objectives Students will be able to simplify complex expressions containing exponents.

3 Slide 6 / 276 Why do we need this? Exponents 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. Properties of Exponents Slide 7 / 276 Product of Powers Power of Powers Power of a product Negative exponent Power of 0 Quotient of Powers Slide 8 / 276

4 1 Simplify: 50m 6 q 8 15m 6 q 8 50m 8 q 15 Slide 9 / m 2 q 3 10m 4 q 5 D Solution not shown 1 Simplify:. 5m 2 q 3 10m 4 q 5 Slide 9 () / m 6 q 8 15m 6 q 8 50m 8 q 15 D Solution not shown Slide 10 / 276

5 Slide 10 () / 276 Slide 11 / Divide: D Solution not shown Slide 11 () / Divide: D Solution not shown

6 Slide 12 / Simplify: D Solution not shown Slide 12 () / Simplify: D D Solution not shown 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. Slide 13 / 276 Write with positive exponents: Write without a fraction:

7 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. Slide 13 () / 276 Write with positive exponents: Write without a fraction: 5 Simplify. The answer may be in either form. Slide 14 / 276 D Solution not shown 5 Simplify. The answer may be in either form. Slide 14 () / 276 D Solution not shown

8 Slide 15 / Simplify and write with positive exponents: D Solution not shown Slide 15 () / Simplify and write with positive exponents: D D Solution not shown When fractions are to a negative power, a short-cut is to invert the fraction and make the exponent positive. Slide 16 / 276 Try...

9 When fractions are to a negative power, a short-cut is to invert the fraction and make the exponent positive. Slide 16 () / 276 Try... Slide 17 / 276 Two more examples. Leave your answers with positive exponents. Slide 17 () / 276 Two more examples. Leave your answers with positive exponents. = =

10 Slide 18 / Simplify and write with positive exponents: D Solution not shown Slide 18 () / Simplify and write with positive exponents: D Solution not shown Slide 19 / 276

11 Slide 19 () / 276 Slide 20 / 276 Operations with Polynomials Review Return to Table of ontents This section is intended to be a brief review of this topic. For more detailed lessons and practice see lgebra 1. Slide 21 / 276 Goals and Objectives Students will be able to combine polynomial functions using operations of addition, subtraction, multiplication, and division.

12 Vocabulary Review Slide 22 / 276 monomial is an expression that is a number, a variable, or the product of a number and one or more variables with whole number exponents. polynomial is the sum of one or more monomials, each of which is a term of the polynomial. Put a circle around each term: Slide 23 / 276 Polynomials can be classified by the number of terms. The table below summarizes these classifications. Slide 24 / 276

13 Identify the degree of each polynomial: Slide 25 / 276 Identify the degree of each polynomial: Slide 25 () / =5th degree 5+1=6th degree Not a polynomial 4th degree Slide 26 / 276 Polynomials can also be classified by degree. The table below summarizes these classifications.

14 Polynomial Function Slide 27 / 276 polynomial function is a function in the form where n is a nonnegative integer and the coefficients are real numbers. The coefficient of the first term, a n, is the leading coefficient. polynomial function is in standard form when the terms are in order of degree from highest to lowest. Drag each relation to the correct box: Polynomial Functions Not Polynomial Functions Slide 28 / 276 f(x) = For extra practice, make up a few of your own! To add or subtract polynomials, simply distribute the + or - sign to each term in parentheses, and then combine the like terms from each polynomial. Slide 29 / 276 Examples: (2a 2 +3a - 9) + (a 2-6a +3) (2a 2 +3a - 9) - (a 2-6a +3) Watch your signs...forgetting to distribute the minus sign is one of the most common mistakes students make!!

15 losure: set is closed under an operation if when any two elements are combined with that operation, the result is also an element of the set. Slide 30 / 276 Is the set of all polynomials closed under - addition? - subtraction? Explain or justify your answer. losure: set is closed under an operation if when any two elements are combined with that operation, the result is also an element of the set. Yes to both. When you add or subtract polynomials, the answer will always be Is the set of all polynomials a closed polynomial. under Discuss with - addition? examples, and try to find a counterexample. - subtraction? Slide 30 () / 276 Explain or justify your answer. 9 Simplify Slide 31 / 276 D

16 9 Simplify Slide 31 () / 276 D Slide 32 / 276 Slide 32 () / 276

17 Slide 33 / 276 Slide 33 () / What is the perimeter of the following figure? (answers are in units, assume all angles are right) D x 2 +5x - 2 2x - 3 8x 2-3x x + 1 Slide 34 / 276

18 12 What is the perimeter of the following figure? (answers are in units, assume all angles are right) D x 2 +5x - 2 D 2x - 3 8x 2-3x x + 1 Slide 34 () / 276 Multiplying Polynomials Slide 35 / 276 To multiply a polynomial by a monomial, you use the distributive property of multiplication over addition together with the laws of exponents. Example: Simplify. -2x(5x 2-6x + 8) (-2x)(5x 2 ) + (-2x)(-6x) + (-2x)(8) -10x x x -10x x 2-16x 13 What is the area of the rectangle shown? Slide 36 / 276 D

19 13 What is the area of the rectangle shown? Slide 36 () / 276 D 14 Slide 37 / 276 D 14 Slide 37 () / 276 D

20 15 Find the area of a triangle (= 1 / 2bh) with a base of 5y and a height of 2y + 2. ll answers are in square units. Slide 38 / 276 D 15 Find the area of a triangle (= 1 / 2bh) with a base of 5y and a height of 2y + 2. ll answers are in square units. Slide 38 () / 276 D D ompare multiplication of polynomials with multiplication of integers. How are they alike and how are they different? Slide 39 / 276 Is the set of polynomials closed under multiplication?

21 ompare multiplication of polynomials with multiplication of integers. How are they alike and how are they different? Notice how the distributive property is used in both examples. Each term in the factor (2x + 2) must be multiplied by each term in the factor (2x 2 + 4x + 3), just like the value of each digit of 22 must be multiplied by each digit of 243. Teacher Notes [This object is a teacher notes pull tab] Slide 39 () / 276 Is the set of polynomials closed under multiplication? Slide 40 / 276 Discuss how we could check this result. = Slide 40 () / 276 Discuss how we could check this result. Teacher Notes = Encourage students to substitute a value for x in each expression, obtaining the same result.

22 To multiply a polynomial by a polynomial, distribute each term of the first polynomial to each term of the second. Then, add like terms. Slide 41 / 276 efore combining like terms, how many terms will there be in each product below? 3 terms x 5 terms 5 terms x 8 terms 100 terms x 99 terms To multiply a polynomial by a polynomial, distribute each term of the first polynomial to each term of the second. Then, add like terms. Slide 41 () / 276 efore combining like terms, how many terms will there be in each product below? 3 terms x 5 terms 5 terms x 8 terms 100 terms x 99 terms 15, 40, What is the total area of the rectangles shown? Slide 42 / 276 D

23 16 What is the total area of the rectangles shown? Slide 42 () / 276 D D 17 Slide 43 / 276 D 17 Slide 43 () / 276 D

24 18 Slide 44 / 276 D 18 Slide 44 () / 276 D Slide 45 / 276

25 Slide 45 () / 276 Slide 46 / 276 Example Part : town council plans to build a public parking lot. The outline below represents the proposed shape of the parking lot. Write an expression for the area, in square yards, of this proposed parking lot. Explain the reasoning you used to find the expression. From High School SS Flip ook Slide 46 () / 276 Example Part : *Let students work on Parts, and in town council their plans groups to and build share a public their work parking with the lot. The outline below represents class. This the proposed problem should shape take of more the time parking lot. Write an expression than typical for slides. the area, Give students in square the time yards, of this they need to complete the problems. proposed parking lot. Explain the reasoning you used to find the expression. Sample Response: Part Missing vertical dimension is 2x # 5 # (x # 5) = x. rea = x(x # 5) + x(2x + 15) = x 2 # 5x + 2x x = 3x x [This square object yards is a pull tab] From High School SS Flip ook

26 Example Part : The town council has plans to double the area of the parking lot in a few years. They create two plans to do this. The first plan increases the length of the base of the parking lot by p yards, as shown in the diagram below. Write an expression in terms of x to represent the value of p, in feet. Explain the reasoning you used to find the value of p. Slide 47 / 276 Example Part : The town council has plans to double the area of the parking lot in a few years. Part They create two plans to do this. The first plan increases the length of the base of the parking lot by p yards, as shown in the Doubled area = 6x diagram below. Write an x square yards. expression in terms of x to represent the rea of top left corner = value of p, in feet. Explain the 2 x# 5x square yards. reasoning you used to find the value of p. rea of lower portion with doubled area = 26x + 20x # (x 2 # 5x) = 5x x square yards Since the width remains x yards, the longest length must be (5x x) x = 5x + 25 yards long. So, y = 5x + 25 # (2x + 15) = 5x + 25 # 2x # 15 = 3x + 10 yards. Slide 47 () / 276 Example Part : The town council s second plan to double the area changes the shape of the parking lot to a rectangle, as shown in the diagram below. an the value of z be represented as a polynomial with integer coefficients? Justify your reasoning. Slide 48 / 276

27 Example Part : The town council s second plan to double the area changes the shape of the parking Part lot to a rectangle, as shown in the diagram below. If z is a polynomial with integer coefficients, the length of the rectangle, x z, would an the value of z be represented as a polynomial with integer be a factor of the doubled area. Likewise, x # 2 coefficients? Justify your reasoning. 5 would be a factor of the doubled area. ut 2x # 5 is not a factor of x x. So 2x z is not a factor either. Therefore, z cannot be represented as a polynomial with integer coefficients. Slide 48 () / Find the value of the constant a such that Slide 49 / D Find the value of the constant a such that Slide 49 () / D -6

28 Slide 50 / 276 Special inomial Products Return to Table of ontents Square of a Sum Slide 51 / 276 (a + b) 2 = (a + b)(a + b) = a 2 + 2ab + b 2 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 Slide 52 / 276 (a - b) 2 = (a - b)(a - b) = a 2-2ab + b 2 The square of a - b is the square of a minus twice the product of a and b plus the square of b. Example:

29 Product of a Sum and a Difference Slide 53 / 276 (a + b)(a - b) = a 2 + -ab + ab + -b 2 = Notice the sum of -ab and ab a 2 - b 2 equals 0. The product of a + b and a - b is the square of a minus the square of b. Example: 2 + = Slide 54 / 276 Practice the square of a sum by putting any monomials in for and. 2 - = Slide 55 / 276 Practice the square of a difference by putting any monomials in for and. How does this problem differ from the last? Study and memorize the patterns!! You will see them over and over again in many different ways.

30 + - = Slide 56 / 276 This very important product is called the difference of squares. Practice the product of a sum and a difference by putting any monomials in for and. How does this problem differ from the last two? 21 Slide 57 / 276 D 21 Slide 57 () / 276 D

31 22 Simplify: Slide 58 / 276 D 22 Simplify: Slide 58 () / 276 D D 23 Simplify: Slide 59 / 276 D

32 23 Simplify: Slide 59 () / 276 D 24 Multiply: Slide 60 / 276 D 24 Multiply: Slide 60 () / 276 D

33 Slide 61 / 276 hallenge: See if you can work backwards to simplify the given problem without a calculator. Slide 61 () / 276 hallenge: See if you can work backwards to simplify the given problem without a calculator. Rewrite as Problem is from: -PR Trina's Triangles lick for link for commentary and solution. lice and her friend Trina were having a conversation. Trina said "Pick any 2 integers. Find the sum of their squares, the difference of their squares and twice the product of the integers. These 3 numbers are the sides of a right triangle." Trina had tried this with several examples and it worked every time, but she wasn't sure this "trick" would always work. a. Investigate Trina's conjecture for several pairs of integers. Does it work? b. If it works, then give a precise statement of the conjecture, using variables to represent the chosen integers, and prove it. If not true, modify it so that it is true, and prove the new statement. c. Use Trina's trick to find an example of a right triangle in which all of the sides have integer length. all 3 sides are longer than 100 units, and the 3 side lengths do not have any common factors. Slide 62 / 276

34 Slide 63 / 276 inomial Theorem Return to Table of ontents The inomial Theorem is a formula used to generate the expansion of a binomial raised to any power. Slide 64 / 276 inomial Theorem ecause the formula itself is very complex, we will see in the following slides some procedures we can use to simplify raising a binomial to any power. Slide 65 / 276 What happens when you multiply a binomial by itself n times? Evaluate: n = 0 n = 1 n = 2 n = 3

35 Slide 65 () / 276 What happens when you multiply a binomial by itself n times? Evaluate: n = 0 n = 1 n = 2 n = 3 Slide 66 / 276 Let's try another one: Slide 67 / 276 Expand (x + y) 4 What will be the exponents in each term of (x + y) 5?

36 Slide 67 () / The exponent of x is 5 on the third term of the expansion of. Slide 68 / 276 True False 25 The exponent of x is 5 on the third term of the expansion of. Slide 68 () / 276 True False True

37 26 The exponents of y are decreasing in the expansion of Slide 69 / 276 True False 26 The exponents of y are decreasing in the expansion of Slide 69 () / 276 True False False 27 What is the exponent of a in the fourth term of? Slide 70 / 276

38 27 What is the exponent of a in the fourth term of? Slide 70 () / Slide 71 / 276 Pascal's Triangle Row 0 Row 4 Slide 72 / 276 To get the next row, we start and end with 1, then add the two numbers above the next terms. Fill in the next 2 rows... One way to find the coefficients when expanding a polynomial raised to the n th power is to use the n th row of Pascal's Triangle.

39 28 ll rows of Pascal's Triangle start and end with 1 Slide 73 / 276 True False 28 ll rows of Pascal's Triangle start and end with 1 Slide 73 () / 276 True False True 29 What number is in the 5th spot of the 6th row of Pascal's Triangle? Slide 74 / 276

40 29 What number is in the 5th spot of the 6th row of Pascal's Triangle? Slide 74 () / What number is in the 2nd spot of the 4th row of Pascal's Triangle? Slide 75 / What number is in the 2nd spot of the 4th row of Pascal's Triangle? Slide 75 () / 276 4

41 Now that we know how to find the exponents and the coefficients when expanding binomials, lets put it together. Slide 76 / 276 Expand Teacher Notes nother Example Expand: (In this example, 2a is in place of x, and 3b is in place of y.) Slide 77 / 276 nother Example Expand: (In this example, 2a is in place of x, and 3b is in place of y.) Since the exponent is 5, we are going to use the fifth row of Pascal's triangle as the coefficients. ombining this with the increasing and decreasing exponents, we get: Slide 77 () / 276

42 Now you try! Slide 78 / 276 Expand: Slide 78 () / What is the coefficient on the third term of the expansion of Slide 79 / 276

43 Slide 79 () / 276 Slide 80 / 276 Slide 80 () / 276

44 33 The binomial theorem can be used to expand Slide 81 / 276 True False 33 The binomial theorem can be used to expand Slide 81 () / 276 True False False Slide 82 / 276 Factoring Polynomials Review Return to Table of ontents

45 Factoring Polynomials Review Slide 83 / 276 The process of factoring involves breaking a product down into its factors. Here is a summary of factoring strategies: Slide 84 / 276 erry Method to factor Step 1: alculate ac. Step 2: Find a pair of numbers m and n, whose product is ac, and whose sum is b. Step 3: reate the product. Step 4: From each binomial in step 3, factor out and discard any common factor. The result is your factored form. Example: Example: Slide 85 / 276 Step 1: ac = -15 and b = -2 Step 2: find m and n whose product is -15 and sum is -2; so m = -5 and n = 3 Step 3: (ax + m)(ax + n) = (3x - 5)(3x + 3) Step 4: (3x + 3) = 3(x + 1) so discard the 3 Therefore, 3x 2-2x - 5 = (3x - 5)(x + 1)

46 More factoring review... Slide 86 / 276 (In this unit, sum or difference of cubes is not emphasized.) 34 Factor out the GF: 15m 3 n - 25m 2-15mn 3 Slide 87 / m(mn - 10m - n 3 ) 5m(3m 2 n - 5m - 3n 3 ) 5mn(3m 2-5m - 3n 2 ) D 5mn(3m 2-5m - 3n) E 15mn(mn - 10m - n 3 ) 34 Factor out the GF: 15m 3 n - 25m 2-15mn 3 Slide 87 () / m(mn - 10m - n 3 ) 5m(3m 2 n - 5m - 3n 3 ) 5mn(3m 2-5m - 3n 2 ) D 5mn(3m 2-5m - 3n) E 15mn(mn - 10m - n 3 )

47 35 Factor: x x + 25 Slide 88 / 276 (x - 5)(x - 5) (x - 5)(x + 5) (x + 15)(x + 10) D (x - 15)(x - 10) E Solution not shown 35 Factor: x x + 25 Slide 88 () / 276 (x - 5)(x - 5) (x - 5)(x + 5) (x + 15)(x + 10) D (x - 15)(x - 10) E Solution not shown E 36 Factor: mn + 3m - 4n 2-12n (n - 3)(m + 4n) (n - 3)(m - 4n) (n + 4)(m - n) D Not factorable E Solution not shown Slide 89 / 276

48 36 Factor: mn + 3m - 4n 2-12n (n - 3)(m + 4n) (n - 3)(m - 4n) (n + 4)(m - n) D Not factorable E Solution not shown E mn + 3m - 4n 2-12n = m(n + 3) - 4n(n + 3) = (n + 3)(m - 4n) Slide 89 () / Factor: (11m - 10n)(11m + 10m) (121m - n)(m + 100n) (11m - n)(11m + 100n) D Not factorable E Solution not shown 121m n 2 Slide 90 / Factor: (11m - 10n)(11m + 10m) (121m - n)(m + 100n) (11m - n)(11m + 100n) D Not factorable E Solution not shown 121m n 2 D Not factorable because it is a sum of squares with no GF Slide 90 () / 276

49 38 Factor: 121m 2-100n 2 Slide 91 / 276 (11m - 10n)(11m + 10n) (121m - n)(m + 100n) (11m - n)(11m + 100n) D Not factorable E Solution not shown 38 Factor: 121m 2-100n 2 Slide 91 () / 276 (11m - 10n)(11m + 10n) (121m - n)(m + 100n) (11m - n)(11m + 100n) D Not factorable E Solution not shown Slide 92 / Factor: 10x 2-11x + 3 (2x - 1)(5x - 3) (2x + 1)(5x + 3) (10x - 1)(x + 3) D (10x - 1)(x - 3) E Solution not shown

50 39 Factor: 10x 2-11x + 3 (2x - 1)(5x - 3) (2x + 1)(5x + 3) (10x - 1)(x + 3) D (10x - 1)(x - 3) E Solution not shown ac = 30, b = -11 so m = -5, n = -6 (10x - 5)(10x - 6) =(2x - 1)(5x - 3) Slide 92 () / Which expression is equivalent to 6x 3-5x 2 y - 24xy y 3? Slide 93 / 276 x 2 (6x - 5y) + 4y 2 (6x + 5y) x 2 (6x - 5y) + 4y 2 (6x - 5y) x 2 (6x - 5y) - 4y 2 (6x + 5y) D x 2 (6x - 5y) - 4y 2 (6x - 5y) From PR sample test 40 Which expression is equivalent to 6x 3-5x 2 y - 24xy y 3? Slide 93 () / 276 D x 2 (6x - 5y) + 4y 2 (6x + 5y) x 2 (6x - 5y) + 4y 2 (6x - 5y) x 2 (6x - 5y) - 4y 2 (6x + 5y) D x 2 (6x - 5y) - 4y 2 (6x - 5y) From PR sample test

51 41 Which expressions are factors of 6x 3-5x 2 y - 24xy y 3? Select all that apply. Slide 94 / 276 x 2 + y 2 6x - 5y 6x + 5y D x - 2y E x + 2y From PR sample test 41 Which expressions are factors of 6x 3-5x 2 y - 24xy y 3? Select all that apply. x 2 + y 2, D, E Slide 94 () / 276 6x - 5y 6x + 5y D x - 2y E x + 2y From PR sample test 42 The expression x 2 (x - y) 3 - y 2 (x - y) 3 can be written in the form (x - y) a (x +y), where a is a constant. What is the value of a? Slide 95 / 276 From PR sample test

52 42 The expression x 2 (x - y) 3 - y 2 (x - y) 3 can be written in the form (x - y) a (x +y), where a is a constant. What is the value of a? 4 Slide 95 () / 276 From PR sample test Write the expression x - xy 2 as the product of the greatest common factor and a binomial: Slide 96 / 276 Determine the complete factorization of x - xy 2 : From PR sample test Write the expression x - xy 2 as the product of the greatest common factor and a binomial: x(1 - y 2 ) Slide 96 () / 276 x(1 - y)(1 + y) Determine the complete factorization of x - xy 2 : From PR sample test

53 Slide 97 / 276 Dividing Polynomials Return to Table of ontents Division of Polynomials Slide 98 / 276 Here are 3 different ways to write the same quotient: Slide 99 / 276

54 Examples lick to Reveal Slide 100 / Simplify Slide 101 / 276 D 43 Simplify Slide 101 () / 276 D D

55 44 Simplify Slide 102 / 276 D 44 Simplify Slide 102 () / 276 D 45 The set of polynomials is closed under division. Slide 103 / 276 True False

56 45 The set of polynomials is closed under division. Slide 103 () / 276 True False False Slide 104 / 276 Slide 105 / 276

57 Slide 106 / 276 Slide 107 / 276 Slide 108 / 276

58 Slide 109 / 276 Slide 110 / 276 Slide 111 / 276

59 Slide 111 () / 276 Slide 112 / 276 Slide 112 () / 276

60 Slide 113 / 276 Slide 113 () / 276 Slide 114 / 276

61 Slide 114 () / 276 Slide 115 / 276 Slide 115 () / 276

62 46 Simplify. Slide 116 / 276 D 46 Simplify. Slide 116 () / 276 D 47 Simplify. Slide 117 / 276 D

63 47 Simplify. Slide 117 () / 276 D Slide 118 / 276 Slide 118 () / 276

64 Slide 119 / 276 Slide 119 () / 276 Slide 120 / 276

65 Slide 120 () / 276 Slide 121 / 276 Slide 121 () / 276

66 52 If f (1) = 0 for the function,, what is the value of a? Slide 122 / If f (1) = 0 for the function,, what is the value of a? Slide 122 () / 276 substitute 1 for x and solve for a. a = If f (3) = 27 for the function,, what is the value of a? Slide 123 / 276

67 53 If f (3) = 27 for the function,, what is the value of a? Slide 123 () / 276 a = 1 Slide 124 / 276 Polynomial Functions Return to Table of ontents Slide 125 / 276 Goals and Objectives Students will be able to sketch the graphs of polynomial functions, find the zeros, and become familiar with the shapes and characteristics of their graphs.

68 Slide 126 / 276 Why We Need This Polynomial functions are used to model a wide variety of real world phenomena. Finding the roots or zeros of a polynomial is one of algebra's most important problems, setting the stage for future math and science study. Graphs of Polynomial Functions Features: ontinuous curve (or straight line) Turns are rounded, not sharp Which are polynomials? Slide 127 / 276 The Shape of a Polynomial Function Slide 128 / 276 The degree of a polynomial function and the coefficient of the first term affect: the shape of the graph, the number of turning points (points where the graph changes direction), the end behavior, or direction of the graph as x approaches positive and negative infinity. If you have Geogebra on your computer, click below to go to an interactive webpage where you can explore graphs of polynomials.

69 Slide 129 / 276 Optional Spreadsheet ctivity Slide 130 / 276 See the spreadsheet activity on the unit page for this unit entitled "Exploration of the values of the terms of a polynomial". Explore the impact of each term by changing values of the coefficients in row 1. Take a look at the graphs below. These are some of the simplest polynomial functions, y = x n. Notice that when n is even, the graphs are similar. What do you notice about these graphs? Slide 131 / 276 What would you predict the graph of y = x 10 to look like? For discussion: despite appearances, how many points sit on the x-axis?

70 Take a look at the graphs below. These are some of the simplest polynomial functions, y = x n. Notice that when n is even, the graphs are similar. What do you notice about these graphs? Slide 131 () / 276 What would you predict the graph of y = x 10 to look like? For discussion: despite appearances, how many points sit on the x-axis? Slide 132 / 276 Notice the shape of the graph y = x n when n is odd. What do you notice as n increases? What do you predict the graph of y = x 21 would look like? Slide 132 () / 276 Notice the shape of the graph y = x n when n is odd. What do you notice as n increases? What do you predict the graph of y = x 21 would look like?

71 End behavior means what happens to the graph as x and as x -. What do you observe about end behavior? Slide 133 / 276 Polynomials of Even Degree Polynomials of Odd Degree End behavior means what happens to the graph as x and as x -. What do you observe about end behavior? Slide 133 () / 276 Polynomials of Even Degree Polynomials of Odd Degree Even degree - both ends are going in the same direction (both up or both down). Odd degree - one end is up and the other down. These are polynomials of even degree. Observations about end behavior? Slide 134 / 276 Positive Lead oefficient Negative Lead oefficient

72 These are polynomials of odd degree. Slide 135 / 276 Positive Lead oefficient Negative Lead oefficient Observations about end behavior? End ehavior of a Polynomial Slide 136 / 276 Lead coefficient is positive Lead coefficient is negative Left End Right EndLeft End Right End Polynomial of even degree Polynomial of odd degree End ehavior of a Polynomial Degree: even Degree: even Lead oefficient: positive Lead oefficient: negative Slide 137 / 276 s x, f(x) s x -, f(x) In other words, the function rises to the left and to the right. s x, f(x) - s x -, f(x) - In other words, the function falls to the left and to the right.

73 End ehavior of a Polynomial Degree: odd Degree: odd Lead oefficient: positive Lead oefficient: negative Slide 138 / 276 s s In other words, the function falls to the left and rises to the right. s s In other words, the function rises to the left and falls to the right. 54 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 139 / 276 D odd and positive odd and negative even and positive even and negative 54 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 139 () / 276 D odd and positive odd and negative even and positive even and negative D even and negative

74 55 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 140 / 276 D odd and positive odd and negative even and positive even and negative 55 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 140 () / 276 odd and positive odd and negative odd and positive even and positive D even and negative 56 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 141 / 276 D odd and positive odd and negative even and positive even and negative

75 56 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. odd and positive odd and negative even and positive Slide 141 () / 276 D even and positive even and negative 57 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 142 / 276 D odd and positive odd and negative even and positive even and negative 57 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Slide 142 () / 276 odd and positive odd and negative odd and negative even and positive D even and negative

76 Odd and Even Functions Slide 143 / 276 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 0 ) Examples: Odd function Even function Neither f(x)=3x 5-4x 3 + 2x h(x)=6x 4-2x g(x)= 3x 2 + 4x - 4 y = 5x y = x 2 y = 6x - 2 g(x)=7x 7 + 2x 3 f(x)=3x 10-7x 2 r(x)= 3x 5 +4x 3-2 Slide 144 / 276 Slide 144 () / 276

77 Slide 145 / 276 Slide 145 () / 276 Slide 146 / 276

78 n even function is symmetric about the y-axis. Slide 147 / 276 Definition of an Even Function 60 hoose all that apply to describe the graph. Slide 148 / 276 D E F Odd Degree Odd Function Even Degree Even Function Positive Lead oefficient Negative Lead oefficient 60 hoose all that apply to describe the graph. Slide 148 () / 276 D E F Odd Degree Odd Function Even Degree Even Function Positive Lead oefficient Negative Lead oefficient Odd- Degree Odd- Function E Positive Lead oefficient

79 61 hoose all that apply to describe the graph. Slide 149 / 276 D E F Odd Degree Odd Function Even Degree Even Function Positive Lead oefficient Negative Lead oefficient 61 hoose all that apply to describe the graph. Slide 149 () / 276 D E F Odd Degree Even- Degree D Even- Function Odd Function E Positive Lead oefficient Even Degree Even Function Positive Lead oefficient Negative Lead oefficient 62 hoose all that apply to describe the graph. Slide 150 / 276 D E F Odd Degree Odd Function Even Degree Even Function Positive Lead oefficient Negative Lead oefficient

80 62 hoose all that apply to describe the graph. Slide 150 () / 276 D E F Odd- Degree Odd Degree Odd- Function Odd Function F Negative Lead oefficient Even Degree Even Function Positive Lead oefficient Negative Lead oefficient 63 hoose all that apply to describe the graph. Slide 151 / 276 D E F Odd Degree Odd Function Even Degree Even Function Positive Lead oefficient Negative Lead oefficient 63 hoose all that apply to describe the graph. Slide 151 () / 276 D E F Odd Degree Odd- Degree Odd Function E Positive Lead oefficient Even Degree Even Function Positive Lead oefficient Negative Lead oefficient

81 64 hoose all that apply to describe the graph. Slide 152 / 276 D E F Odd Degree Odd Function Even Degree Even Function Positive Lead oefficient Negative Lead oefficient 64 hoose all that apply to describe the graph. Slide 152 () / 276 D E F Even- Degree Odd Degree D Even - Function F Negative Lead oefficient Odd Function Even Degree Even Function Positive Lead oefficient Negative Lead oefficient Zeros of a Polynomial "Zeros" are the points at which the polynomial intersects the x-axis. They are called "zeros" because at each point f (x) = 0. nother name for a zero is a root. Slide 153 / 276 polynomial function of degreen has at MOST n real zeros. n odd degree polynomial must have at least one real zero. (WHY?) Zeros

82 Relative Maxima and Minima polynomial function of degree n has at MOST n - 1 turning points, also called relative maxima and relative minima. These are points where the graph changes from increasing to decreasing, or from decreasing to increasing. Slide 154 / 276 Relative Maxima Relative Minima 65 How many zeros does the polynomial appear to have? Slide 155 / How many zeros does the polynomial appear to have? Slide 155 () / zeros

83 66 How many turning points does the polynomial appear to have? Slide 156 / How many turning points does the polynomial appear to have? Slide 156 () / turning points 67 How many zeros does the polynomial appear to have? Slide 157 / 276

84 67 How many zeros does the polynomial appear to have? Slide 157 () / zeros 68 How many turning points does the graph appear to have? How many of those are relative minima? Slide 158 / How many turning points does the graph appear to have? How many of those are relative minima? 3 turning points 2 relative min Slide 158 () / 276

85 69 How many zeros does the polynomial appear to have? Slide 159 / How many zeros does the polynomial appear to have? Slide 159 () / zeros 70 How many turning points does the polynomial appear to have? How many of those are relative maxima? Slide 160 / 276

86 70 How many turning points does the polynomial appear to have? How many of those are relative maxima? Slide 160 () / turning points 1 relative max 71 How many zeros does the polynomial appear to have? Slide 161 / How many zeros does the polynomial appear to have? Slide 161 () / 276 None

87 72 How many relative maxima does the graph appear to have? How many relative minima? Slide 162 / How many relative maxima does the graph appear to have? How many relative minima? Slide 162 () / relative max 2 relative min Slide 163 / 276 nalyzing Graphs and Tables of Polynomial Functions Return to Table of ontents

88 polynomial function can be sketched by creating a table, plotting the points, and then connecting the points with a smooth curve. Slide 164 / 276 x y Look at the first term to determine the end behavior of the graph. In this case, the coefficient is negative and the degree is odd, so the function rises to the left and falls to the right. How many zeros does this function appear to have? Slide 165 / 276 x y There is a zero at x = -1, a second between x = 1 and x = 2, and a third between x = 3 and x = 4. How can we recognize zeros given only a table? Slide 166 / 276 x y

89 Intermediate Value Theorem Slide 167 / 276 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, so there must be an x-value for which y = 0. The Intermediate Value Theorem justifies the statement that there is a zero between x = 1 and x = 2 and that there is another between x = 3 and x = 4. Slide 168 / 276 x y How many zeros of the continuous polynomial given can be found using the table? x y Slide 169 / 276

90 74 If the table represents a continuous function, between which two values of x can you find the smallest x-value at which a zero occurs? x y D 0 E 1 F 2 G 3 H Slide 170 / How many zeros of the continuous polynomial given can be found using the table? x y Slide 171 / ccording to the table, what is the least value of x at which a zero occurs on this continuous function? Slide 172 / D 0 E 1 F 2 G 3 H 4 x y

91 Relative Maxima and Relative Minima Slide 173 / 276 There are 2 relative maximum points at x = -1 and at x = 1. The relative maximum value appears to be -1 (the y-coordinate). There is a relative minimum at (0, -2). How do we recognize the relative maxima and minima from a table? Slide 174 / 276 In the table, as x goes from -3 to 1, f(x) is decreasing. s x goes from 1 to 3, f(x) is increasing. nd as x goes from 3 to 4, f(x) is decreasing. The relative maxima and minima occur when the direction changes from decreasing to increasing, or from increasing to decreasing. The y-coordinate indicates this change in direction as its value rises or falls. x f(x) Slide 175 / 276

92 Slide 176 / t approximately what x-values does a relative minimum occur? D 0 E 1 F 2 G 3 H 4 Slide 177 / t approximately what x-values does a relative minimum occur? D 0 E 1 F 2 G 3 H 4-1 E 1 Slide 177 () / 276

93 78 t about what x-values does a relative maximum occur? D 0 E 1 F 2 G 3 H 4 Slide 178 / t about what x-values does a relative maximum occur? D 0 E 1 F 2 G 3 H 4-2 F 2 Slide 178 () / t about what x-values does a relative minimum occur? x y -3 E F G D 0 H Slide 179 / 276

94 80 t about what x-values does a relative maximum occur? x y -3 E F G D 0 H Slide 180 / t about what x-values does a relative minimum occur? x y -3 E 1-2 F 2-1 G 3 D 0 H Slide 181 / t about what x-values does a relative maximum occur? x y -3 E 1-2 F 2-1 G 3 D 0 H Slide 182 / 276

95 Slide 183 / 276 Zeros and Roots of a Polynomial Function Return to Table of ontents Real Zeros of Polynomial Functions Slide 184 / 276 For a function f(x) and a real number a, if f (a) = 0, the following statements are equivalent: x = a is a zero of the function f(x). x = a is a solution of the equation f (x) = 0. (x - a) is a factor of the function f(x). (a, 0) is an x-intercept of the graph of f(x). The Fundamental Theorem of lgebra Slide 185 / 276 If f (x) is a polynomial of degree n, where n > 0, then f (x) = 0 has n zeros including multiples and imaginary zeros. n imaginary zero occurs when the solution to f (x) = 0 contains complex numbers. Imaginary zeros are not seen on the graph.

96 omplex Numbers Slide 186 / 276 omplex numbers will be studied in detail in the Radicals Unit. ut in order to fully understand polynomial functions, we need to know a little bit about complex numbers. Up until now, we have learned that there is no real number, x, such that x 2 = -1. However, there is such a number, known as the imaginary unit, i, which satisfies this equation and is defined as. The set of complex numbers is the set of numbers of the form a + bi, where a and b are real numbers. When a = 0, bi is called a pure imaginary number. The square root of any negative number is a complex number. For example, find a solution for x 2 = -9: Slide 187 / 276 Drag each number to the correct place in the diagram. Slide 188 / 276 omplex Numbers Real Imaginary 9+6i 2/3 3i i

97 Drag each number to the correct place in the diagram. Slide 188 () / 276 omplex Numbers Real Teacher Notes Imaginary The intersection of real and imaginary should be the empty set. [This object is a teacher notes pull tab] 9+6i 2/3 3i i The number of the zeros of a polynomial, both real and imaginary, is equal to the degree of the polynomial. Slide 189 / 276 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 in total - 4 real = 0 imaginary) This 5th degree polynomial has 5 zeros, but only 3 of them are real. Therefore, there must be two imaginary. Slide 190 / 276 (How do we know that this is a 5th degree polynomial?) Note: imaginary roots always come in pairs: if a + bi is a root, then a - bi is also a root. (These are called conjugates - more on that in later units.)

98 vertex on the x-axis indicates a multiple zero, meaning the zeroccurs two or more times. Slide 191 / zeros each This is a 4th-degree polynomial. It has two unique real zeros: -2 and 2. These two zeros are said to have a multiplicity of two, which means they each occur twice. There are 4 real zeros and therefore no imaginary zeros for this function. What do you think are the zeros and their multiplicity for this function? Slide 192 / 276 What do you think are the zeros and their multiplicity for this function? Slide 192 () / 276 Zeros are -2 with multiplicity of 2 and 2 with multiplicity of 4.

99 Notice the function for this graph. x - 1 is a factor two times, and x = 1 is a zero twice. x + 2 is a factor two times, and x = -2 is a zero twice. Therefore, 1 and -2 are zeros with multiplicity of 2. x + 3 is a factor once, and x = 3 is a zero with multiplicity of 1. Slide 193 / How many real zeros does the 4th-degree polynomial graphed have? Slide 194 / D 3 E 4 F 5 83 How many real zeros does the 4th-degree polynomial graphed have? Slide 194 () / D 3 E 4 F 5 E 4

100 84 Do any of the zeros have a multiplicity of 2? Slide 195 / 276 Yes No 84 Do any of the zeros have a multiplicity of 2? Slide 195 () / 276 Yes No No 85 How many imaginary zeros does this 7th degree polynomial have? Slide 196 / D 3 E 4 F 5

101 85 How many imaginary zeros does this 7th degree polynomial have? Slide 196 () / D 3 E 4 F How many real zeros does the 3rd degree polynomial have? Slide 197 / D 3 E 4 F 5 86 How many real zeros does the 3rd degree polynomial have? Slide 197 () / D 3 E 4 F 5 D 3

102 87 Do any of the zeros have a multiplicity of 2? Slide 198 / 276 Yes No 87 Do any of the zeros have a multiplicity of 2? Slide 198 () / 276 Yes No Yes 88 How many imaginary zeros does the 5th degree polynomial have? Slide 199 / D 3 E 4 F 5

103 88 How many imaginary zeros does the 5th degree polynomial have? Slide 199 () / D 3 E 4 F 5 89 How many imaginary zeros does this 4 th -degree polynomial have? Slide 200 / D 3 E 4 F 5 89 How many imaginary zeros does this 4 th -degree polynomial have? Slide 200 () / D 3 E 4 F 5 2

104 90 How many real zeros does the 6th degree polynomial have? Slide 201 / D 3 E 4 F 6 90 How many real zeros does the 6th degree polynomial have? Slide 201 () / D 3 E 4 F 6 F 6 91 Do any of the zeros have a multiplicity of 2? Slide 202 / 276 Yes No

105 91 Do any of the zeros have a multiplicity of 2? Slide 202 () / 276 Yes No Yes 92 How many imaginary zeros does the 6th degree polynomial have? Slide 203 / D 3 E 4 F 5 92 How many imaginary zeros does the 6th degree polynomial have? Slide 203 () / D 3 E 4 F 5 2

106 Finding the Zeros from an Equation in Factored Form: Slide 204 / 276 Recall the Zero Product Property. If the product of two or more quantities or factors equals 0, then at least one of the quantities must equal 0. Slide 205 / 276 So, if, then the zeros of are 0 and -1. So, if, then the zeros of are and. Slide 206 / 276

107 Slide 206 () / 276 Slide 207 / 276 Slide 207 () / 276

108 Slide 208 / 276 Slide 208 () / 276 Find the zeros, including multiplicities, of the following polynomial. Slide 209 / 276 or or or or Don't forget the ±!!

109 Find the zeros, including multiplicities, of the following polynomial. Slide 209 () / 276 or or or or Don't forget the ±!! This polynomial has five distinct real zeros: -6, -4, -2, 2, and 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 9 th degree polynomial. Slide 210 / 276 Slide 211 / 276

110 Slide 211 () / 276 Slide 212 / 276 Slide 212 () / 276

111 Slide 213 / 276 Slide 213 () / 276 Slide 214 / 276

112 Slide 214 () / 276 Slide 215 / 276 Slide 215 () / 276

113 Slide 216 / 276 Slide 216 () / 276 Slide 217 / 276

114 Slide 217 () / 276 Slide 218 / 276 Slide 218 () / 276

115 Slide 219 / 276 Slide 219 () / 276 Slide 220 / 276

116 Slide 220 () / 276 Find the zeros, showing the multiplicities, of the following polynomial. Slide 221 / 276 To find the zeros, you must first write the polynomial in factored form. or or or or This polynomial has two distinct real zeros: 0 and 1. This is a 3 rd degree polynomial, so there are 3 zeros (count 1 twice). 1 has a multiplicity of 2. 0 has a multiplicity of 1. There are no imaginary zeros. Find the zeros, including multiplicities, of the following polynomial. Slide 222 / 276 or or or This polynomial has 4 zeros. There are two distinct real zeros:, both with a multiplicity of 1. There are two imaginary zeros:, both with a multiplicity of 1.

117 105 How many zeros does the polynomial function have? 0 Slide 223 / D 3 E How many zeros does the polynomial function have? 0 Slide 223 () / D 3 D E How many REL zeros does the polynomial equation have? Slide 224 / D 3 E 4

118 106 How many REL zeros does the polynomial equation have? Slide 224 () / D D 3 E What are the zeros and their multiplicities of the polynomial function? Slide 225 / 276 x = -2, mulitplicity of 1 x = -2, multiplicity of 2 x = 3, multiplicity of 1 D x = 3, multiplicity of 2 E x = 0, multiplicity of 1 F x = 0, multiplicity of What are the zeros and their multiplicities of the polynomial function? Slide 225 () / 276 x = -2, mulitplicity x = -2, multiplicity of 1 1 x = -2, multiplicity x = 3, multiplicity of 2 1 x = 3, E multiplicity x = 0, multiplicity of 1 1 D x = 3, multiplicity of 2 E x = 0, multiplicity of 1 F x = 0, multiplicity of 2

119 108 Find the solutions of the following polynomial equation, including multiplicities. Slide 226 / 276 x = 0, multiplicity of 1 x = 3, multiplicity of 1 x = 0, multiplicity of 2 D x = 3, multiplicity of Find the solutions of the following polynomial equation, including multiplicities. Slide 226 () / 276 x = 0, multiplicity of 1 iplicity 1 x = 3, multiplicity of 1 x = 0, multiplicity of 2 iplicity 2 D x = 3, multiplicity of 2 a pull tab] 109 Find the zeros of the polynomial equation, including multiplicities: Slide 227 / 276 x = 2, multiplicity 1 x = 2, multiplicity 2 x = -i, multiplicity 1 D x = i, multiplicity 1 E x = -i, multiplcity 2 F x = i, multiplicity 2

120 109 Find the zeros of the polynomial equation, including multiplicities: Slide 227 () / 276 x = 2, multiplicity 1 x = 2, multiplicity 1 x = 2, multiplicity 2 x = -i, multiplicity 1 x = -i, multiplicity 1 D x = i, multiplicity 1 D x = i, multiplicity 1 E x = -i, multiplcity 2 F x = i, multiplicity Find the zeros of the polynomial equation, including multiplicities: Slide 228 / 276 2, multiplicity of 1 2, multiplicity of 2-2, multiplicity of 1 D -2, multiplicity of 2 E, multiplicity of 1 F -, multiplicity of Find the zeros of the polynomial equation, including multiplicities: Slide 228 () / 276 x = -2, multiplicity 1 2, multiplicity of 1 E x =, multiplicity 1 2, multiplicity of 2 F x = -, multiplicity 1-2, multiplicity of 1 D -2, multiplicity of 2 E, multiplicity of 1 F -, multiplicity of 1

121 Find the zeros, showing the multiplicities, of the following polynomial. Slide 229 / 276 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?? We are going to need to do some long division, but by what do we divide? Slide 230 / 276 The Remainder Theorem told us that for a function, f (x), if we divide f (x) by x - a, then the remainder is f (a). If the remainder is 0, then x - a if a factor of f (x). In other words, if f (a) = 0, then x - a is a factor of f (x). So how do we figure out what a should be???? We could use guess and check, but how can we narrow down the choices? Let The Rational Zeros Theorem: Slide 231 / 276 with integer coefficients. There is a limited number of possible roots or zeros. Integer zeros must be factors of the constant term, a 0. Rational zeros can be found by writing and simplifying fractions where the numerator is an integer factor of a 0 and the denominator is an integer fraction of a n.

122 RTIONL ZEROS THEOREM Make list of POTENTIL rational zeros and test them out. Slide 232 / 276 Potential List: Hint: To check for zeros, first try the smaller integers -- they are easier to work with. Using the Remainder Theorem, we find that 1 is a zero: Slide 233 / 276 therefore (x -1) is a factor of the polynomial. Use POLYNOMIL 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 no imaginary zeros. Using the Remainder Theorem, we find that 1 is a zero: Slide 233 () / 276 therefore (x -1) is a factor *When of you the find polynomial. a distinct zero, Use POLYNOMIL DIVISION write the zero to factor in factored out. form and then complete polynomial division. Teacher Notes *This is a good time to introduce synthetic division as a means to shorten the written work. or or or or This polynomial has three distinct real zeros: -2, -1/3, and 1, each with a multiplicity of 1. There are no imaginary zeros.

123 Find the zeros using the Rational Zeros Theorem, showing the multiplicities, of the following polynomial. Slide 234 / 276 Potential List: ± ±1 Hint: since all of the signs in the polynomial are +, only negative numbers will work. Try -3: -3 is a distinct zero, therefore (x + 3) is a factor. Use POLYNOMIL DIVISION to factor out. Slide 235 / 276 or or or or This polynomial has two distinct real zeros: -3, and has a multiplicity of 2 (there are 2 factors of x + 3). -1 has a multiplicity of 1. There are no imaginary zeros. 111 Which of the following is a zero of? Slide 236 / 276 x = -1 x = 1 x = 7 D x = -7

124 111 Which of the following is a zero of? Slide 236 () / 276 x = -1 x = 1 x = 7 D x = -7, 112 Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem Slide 237 / 276 x = 1, multiplicity 1 x = 1, mulitplicity 2 x = 1, multiplicity 3 D x = -3, multiplicity 1 E x = -3, multiplicity 2 F x = -3, multiplicity Find the zeros of the polynomial equation, including multiplicities, using the Rational Zeros Theorem Slide 237 () / 276 x = 1, multiplicity 1 x = 1, mulitplicity 2 x = 1, multiplicity 3 D x = -3, multiplicity 1 x = 1, multiplicity 2 D x = -3, multiplicity 1 E x = -3, multiplicity 2 F x = -3, multiplicity 3