Slide 39 (Answer) / 276 Compare 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 Teacher Notes 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. [This object is a teacher notes pull tab] Is the set of polynomials closed under multiplication? Slide 40 / 276 Discuss how we could check this result. = Slide 40 (Answer) / 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. [This object is a pull tab]
Slide 41 / 276 To multiply a polynomial by a polynomial, distribute each term of the first polynomial to each term of the second. Then, add like terms. Before 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 Slide 41 (Answer) / 276 To multiply a polynomial by a polynomial, distribute each term of the first polynomial to each term of the second. Then, add like terms. Before combining like terms, how many terms will there be in each product below? Answer 15, 40, 9900 3 terms x 5 terms 5 terms x 8 terms 100 terms x 99 terms [This object is a pull tab] Slide 42 / 276 16 What is the total area of the rectangles shown? A B C D
Slide 42 (Answer) / 276 16 What is the total area of the rectangles shown? A D B Answer C D [This object is a pull tab] Slide 43 / 276 17 A B C D Slide 43 (Answer) / 276 17 A B Answer B C D [This object is a pull tab]
Slide 44 / 276 18 A B C D Slide 44 (Answer) / 276 18 A B Answer C C D [This object is a pull tab] Slide 45 / 276
Slide 45 (Answer) / 276 Slide 46 / 276 Example Part A: A 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 CCSS Flip Book Slide 46 (Answer) / 276 Example Part A: *Let students work on Parts A,B and C in A town council plans to build a public parking lot. The outline their groups and share their work with the below represents the proposed shape of the parking lot. class. This problem should take more time Write an expression for the area, in square yards, of this than typical slides. Give students the time they need to complete the problems. proposed parking lot. Explain the reasoning you used to find the Answer expression. Sample Response: Part A Missing vertical dimension is 2x # 5 # (x # 5) = x. Area = x(x # 5) + x(2x + 15) = x 2 # 5x + 2x 2 + 15x [This object is a pull tab] = 3x 2 + 10x square yards From High School CCSS Flip Book
Slide 47 / 276 Example Part B: 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 (Answer) / 276 Example Part B: 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 Part B the length of the base of the parking lot by p yards, as shown in the Doubled area = 6x 2 + 20x square yards. 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 Area of top left corner = x 2 # 5x square yards. of p . Answer Area of lower portion with doubled area = 6x 2 + 20x # (x 2 # 5x) = 5x 2 + 25x square yards Since the width remains x yards, the longest length must be (5x 2 + 25x) ÷ x = 5x + 25 yards long. So, y = 5x + 25 # (2x + 15) = 5x + 25 # 2x # 15 = 3x + 10 yards. [This object is a pull tab] Slide 48 / 276 Example Part C: 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. Can the value of z be represented as a polynomial with integer coefficients? Justify your reasoning.
Slide 48 (Answer) / 276 Example Part C: The town council’s second plan to double the area changes the Part C shape of the parking lot to a rectangle, as shown in the diagram below. If z is a polynomial with integer coefficients, x + 15 + z , would the length of the rectangle, 2 Can the value of z be represented as a polynomial with integer x # Answer be a factor of the doubled area. Likewise, 2 coefficients? Justify your reasoning. 5 would be a factor of the doubled area. But 2 x # 5 is not a factor of 6 x x . 2 + 20 So 2x + 15 + z is not a factor either. Therefore, z cannot be represented as a polynomial with integer coefficients. [This object is a pull tab] Slide 49 / 276 20 Find the value of the constant a such that A 2 B 4 C 6 D -6 Slide 49 (Answer) / 276 20 Find the value of the constant a such that A 2 B 4 C 6 Answer A D -6 [This object is a pull tab]
Slide 50 / 276 Special Binomial Products Return to Table of Contents Slide 51 / 276 Square of a Sum (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: Slide 52 / 276 Square of a Difference (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:
Slide 53 / 276 Product of a Sum and a Difference (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: Slide 54 / 276 2 + = 2 2 +2 + Practice the square of a sum by putting any monomials in for and . Slide 55 / 276 2 - = 2 2 - 2 + 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.
Slide 56 / 276 + - = 2 2 - 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? Slide 57 / 276 21 A B C D Slide 57 (Answer) / 276 21 A B Answer C B D [This object is a pull tab]
Slide 58 / 276 22 Simplify: A B C D Slide 58 (Answer) / 276 22 Simplify: A Answer D B C D [This object is a pull tab] Slide 59 / 276 23 Simplify: A B C D
Slide 59 (Answer) / 276 23 Simplify: A Answer C B C D [This object is a pull tab] Slide 60 / 276 24 Multiply: A B C D Slide 60 (Answer) / 276 24 Multiply: A Answer A B C D [This object is a pull tab]
Slide 61 / 276 Challenge: See if you can work backwards to simplify the given problem without a calculator. Slide 61 (Answer) / 276 Challenge: See if you can work backwards to simplify the given problem without a calculator. Rewrite as Answer [This object is a pull tab] Slide 62 / 276 Problem is from: A-APR Trina's Triangles Click for link for commentary and solution. Alice 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 63 / 276 Binomial Theorem Return to Table of Contents Slide 64 / 276 The Binomial Theorem is a formula used to generate the expansion of a binomial raised to any power. Binomial Theorem Because 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
Slide 65 (Answer) / 276 What happens when you multiply a binomial by itself n times? Evaluate: n = 0 n = 1 Answer n = 2 n = 3 [This object is a pull tab] Slide 66 / 276 Slide 67 / 276 Let's try another one: Expand (x + y) 4 What will be the exponents in each term of (x + y) 5 ?
Slide 67 (Answer) / 276 Slide 68 / 276 25 The exponent of x is 5 on the third term of the expansion of . True False Slide 68 (Answer) / 276 25 The exponent of x is 5 on the third term of the expansion of . Answer True True False [This object is a pull tab]
Slide 69 / 276 26 The exponents of y are decreasing in the expansion of True False Slide 69 (Answer) / 276 26 The exponents of y are decreasing in the expansion of True Answer False False [This object is a pull tab] Slide 70 / 276 27 What is the exponent of a in the fourth term of ?
Slide 70 (Answer) / 276 27 What is the exponent of a in the fourth term of ? Answer 7 [This object is a pull tab] Slide 71 / 276 Slide 72 / 276 Pascal's Triangle 1 Row 0 1 1 1 2 1 1 3 3 1 1 4 6 4 1 Row 4 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.
Slide 73 / 276 28 All rows of Pascal's Triangle start and end with 1 True False Slide 73 (Answer) / 276 28 All rows of Pascal's Triangle start and end with 1 True Answer True False [This object is a pull tab] Slide 74 / 276 29 What number is in the 5th spot of the 6th row of Pascal's Triangle?
Slide 74 (Answer) / 276 29 What number is in the 5th spot of the 6th row of Pascal's Triangle? Answer 15 [This object is a pull tab] Slide 75 / 276 30 What number is in the 2nd spot of the 4th row of Pascal's Triangle? Slide 75 (Answer) / 276 30 What number is in the 2nd spot of the 4th row of Pascal's Triangle? Answer 4 [This object is a pull tab]
Slide 76 / 276 Now that we know how to find the exponents and the coefficients when expanding binomials, lets put it together. Teacher Notes Expand Slide 77 / 276 Another Example (In this example, 2 a is in place of x , and 3 b is in place of y .) Expand: Slide 77 (Answer) / 276 Another Example (In this example, 2 a is in place of x , and 3 b is in place of y .) Expand: Since the exponent is 5, we are going to use the fifth row of Pascal's triangle as the coefficients. Combining this with the increasing and decreasing exponents, we get: [This object is a pull tab]
Slide 78 / 276 Now you try! Expand: Slide 78 (Answer) / 276 Slide 79 / 276 31 What is the coefficient on the third term of the expansion of
Slide 79 (Answer) / 276 Slide 80 / 276 Slide 80 (Answer) / 276
Slide 81 / 276 33 The binomial theorem can be used to expand True False Slide 81 (Answer) / 276 33 The binomial theorem can be used to expand False Answer True False [This object is a pull tab] Slide 82 / 276 Factoring Polynomials Review Return to Table of Contents
Slide 83 / 276 Factoring Polynomials Review The process of factoring involves breaking a product down into its factors. Here is a summary of factoring strategies: Slide 84 / 276 Berry Method to factor Step 1: Calculate ac . Step 2: Find a pair of numbers m and n , whose product is ac , and whose sum is b . Step 3: Create the product . Step 4: From each binomial in step 3, factor out and discard any common factor. The result is your factored form. Example: Slide 85 / 276 Example: 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 ) = (3 x - 5)(3 x + 3) Step 4: (3 x + 3) = 3( x + 1) so discard the 3 Therefore, 3 x 2 - 2 x - 5 = (3 x - 5)( x + 1)
Slide 86 / 276 More factoring review.... (In this unit, sum or difference of cubes is not emphasized.) Slide 87 / 276 34 Factor out the GCF: 15m 3 n - 25m 2 - 15mn 3 A 15m(mn - 10m - n 3 ) B 5m(3m 2 n - 5m - 3n 3 ) C 5mn(3m 2 - 5m - 3n 2 ) D 5mn(3m 2 - 5m - 3n) E 15mn(mn - 10m - n 3 ) Slide 87 (Answer) / 276 34 Factor out the GCF: 15m 3 n - 25m 2 - 15mn 3 A 15m(mn - 10m - n 3 ) B Answer B 5m(3m 2 n - 5m - 3n 3 ) C 5mn(3m 2 - 5m - 3n 2 ) D 5mn(3m 2 - 5m - 3n) [This object is a pull tab] E 15mn(mn - 10m - n 3 )
Slide 88 / 276 35 Factor: x 2 + 10x + 25 A (x - 5)(x - 5) B (x - 5)(x + 5) C (x + 15)(x + 10) D (x - 15)(x - 10) E Solution not shown Slide 88 (Answer) / 276 35 Factor: x 2 + 10x + 25 A (x - 5)(x - 5) Answer E B (x - 5)(x + 5) C (x + 15)(x + 10) [This object is a pull tab] D (x - 15)(x - 10) E Solution not shown Slide 89 / 276 36 Factor: mn + 3m - 4n 2 - 12n A (n - 3)(m + 4n) B (n - 3)(m - 4n) C (n + 4)(m - n) D Not factorable E Solution not shown
Slide 89 (Answer) / 276 36 Factor: mn + 3m - 4n 2 - 12n A (n - 3)(m + 4n) E B (n - 3)(m - 4n) Answer mn + 3m - 4n 2 - 12n C (n + 4)(m - n) = m(n + 3) - 4n(n + 3) D Not = (n + 3)(m - 4n) factorable E Solution not [This object is a pull tab] shown Slide 90 / 276 37 Factor: 121m 2 + 100n 2 A (11m - 10n)(11m + 10m) B (121m - n)(m + 100n) C (11m - n)(11m + 100n) D Not factorable E Solution not shown Slide 90 (Answer) / 276 37 Factor: 121m 2 + 100n 2 A (11m - 10n)(11m + 10m) D B (121m - n)(m + 100n) Answer Not factorable C (11m - n)(11m + 100n) because it is a D Not factorable sum of squares with no GCF E Solution not [This object is a pull tab] shown
Slide 91 / 276 38 Factor: 121m 2 - 100n 2 A (11m - 10n)(11m + 10n) B (121m - n)(m + 100n) C (11m - n)(11m + 100n) D Not factorable E Solution not shown Slide 91 (Answer) / 276 38 Factor: 121m 2 - 100n 2 A (11m - 10n)(11m + 10n) B (121m - n)(m + 100n) A Answer C (11m - n)(11m + 100n) D Not factorable E Solution not shown [This object is a pull tab] Slide 92 / 276 39 Factor: 10x 2 - 11x + 3 A (2x - 1)(5x - 3) B (2x + 1)(5x + 3) C (10x - 1)(x + 3) D (10x - 1)(x - 3) E Solution not shown
Slide 92 (Answer) / 276 39 Factor: 10x 2 - 11x + 3 A A (2x - 1)(5x - 3) Answer B (2x + 1)(5x + 3) ac = 30, b = -11 C (10x - 1)(x + 3) so m = -5, n = -6 (10x - 5)(10x - 6) D (10x - 1)(x - 3) =(2x - 1)(5x - 3) E Solution not shown [This object is a pull tab] Slide 93 / 276 40 Which expression is equivalent to 6 x 3 - 5 x 2 y - 24 xy 2 + 20 y 3 ? A x 2 (6x - 5 y ) + 4 y 2 (6x + 5 y ) B x 2 (6x - 5 y ) + 4 y 2 (6x - 5 y ) C x 2 (6x - 5 y ) - 4 y 2 (6x + 5 y ) D x 2 (6x - 5 y ) - 4 y 2 (6x - 5 y ) From PARCC sample test Slide 93 (Answer) / 276 40 Which expression is equivalent to 6 x 3 - 5 x 2 y - 24 xy 2 + 20 y 3 ? D Answer A x 2 (6x - 5 y ) + 4 y 2 (6x + 5 y ) B x 2 (6x - 5 y ) + 4 y 2 (6x - 5 y ) [This object is a pull tab] C x 2 (6x - 5 y ) - 4 y 2 (6x + 5 y ) D x 2 (6x - 5 y ) - 4 y 2 (6x - 5 y ) From PARCC sample test
Slide 94 / 276 41 Which expressions are factors of 6 x 3 - 5 x 2 y - 24 xy 2 + 20 y 3 ? Select all that apply. A x 2 + y 2 B 6 x - 5 y C 6 x + 5 y D x - 2 y E x + 2 y From PARCC sample test Slide 94 (Answer) / 276 41 Which expressions are factors of 6 x 3 - 5 x 2 y - 24 xy 2 + 20 y 3 ? Select all that apply. Answer B, D, E A x 2 + y 2 B 6 x - 5 y [This object is a pull tab] C 6 x + 5 y D x - 2 y E x + 2 y From PARCC sample test Slide 95 / 276 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? From PARCC sample test
Slide 95 (Answer) / 276 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? Answer 4 [This object is a pull tab] From PARCC sample test Slide 96 / 276 Write the expression x - xy 2 as the product of the greatest common factor and a binomial: Determine the complete factorization of x - xy 2 : From PARCC sample test Slide 96 (Answer) / 276 Write the expression x - xy 2 as the product of the greatest common factor and a binomial: x (1 - y 2 ) Answer x (1 - y )(1 + y ) Determine the complete factorization of x - xy 2 : [This object is a pull tab] From PARCC sample test
Slide 97 / 276 Dividing Polynomials Return to Table of Contents Slide 98 / 276 Division of Polynomials Here are 3 different ways to write the same quotient: Slide 99 / 276
Slide 100 / 276 Examples Click to Reveal Answer Slide 101 / 276 43 Simplify A B C D Slide 101 (Answer) / 276 43 Simplify A Answer D B C [This object is a pull tab] D
Slide 102 / 276 44 Simplify A B C D Slide 102 (Answer) / 276 44 Simplify A Answer A B C [This object is a pull tab] D Slide 103 / 276 45 The set of polynomials is closed under division. True False
Slide 103 (Answer) / 276 45 The set of polynomials is closed under division. True Answer False False [This object is a pull tab] Slide 104 / 276 Slide 105 / 276
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Slide 111 (Answer) / 276 Slide 112 / 276 Slide 112 (Answer) / 276
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Slide 116 / 276 46 Simplify. A B C D Slide 116 (Answer) / 276 46 Simplify. A Answer B B C D [This object is a pull tab] Slide 117 / 276 47 Simplify. A B C D
Slide 117 (Answer) / 276 47 Simplify. A Answer B B C D [This object is a pull tab] Slide 118 / 276 Slide 118 (Answer) / 276
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Slide 122 / 276 52 If f (1) = 0 for the function, , what is the value of a ? Slide 122 (Answer) / 276 52 If f (1) = 0 for the function, , what is the value of a ? substitute 1 for x and Answer solve for a. a = 0. [This object is a pull tab] Slide 123 / 276 53 If f (3) = 27 for the function, , what is the value of a ?
Slide 123 (Answer) / 276 53 If f (3) = 27 for the function, , what is the value of a ? Answer a = 1 [This object is a pull tab] Slide 124 / 276 Polynomial Functions Return to Table of Contents 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.
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. Slide 127 / 276 Graphs of Polynomial Functions Features: · Continuous curve (or straight line) · Turns are rounded, not sharp Which are polynomials? Slide 128 / 276 The Shape of a Polynomial Function 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.
Slide 129 / 276 Slide 130 / 276 Optional Spreadsheet Activity 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. Slide 131 / 276 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? 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 131 (Answer) / 276 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? What would you Answer predict the graph of y = x 10 to look like? [This object is a pull tab] 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 (Answer) / 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? Answer [This object is a pull tab]
Slide 133 / 276 ∞ End behavior means what happens to the graph as x → and as ∞ x → - . What do you observe about end behavior? Polynomials of Even Degree Polynomials of Odd Degree Slide 133 (Answer) / 276 ∞ End behavior means what happens to the graph as x → and as ∞ x → - . What do you observe about end behavior? Polynomials of Even Degree Polynomials of Odd Degree Even degree - both ends are going in the same Answer direction (both up or both down). Odd degree - one end is up and the other down. [This object is a pull tab] Slide 134 / 276 These are polynomials of even degree. Observations about end behavior? Positive Lead Coefficient Negative Lead Coefficient
Slide 135 / 276 These are polynomials of odd degree. Positive Lead Coefficient Negative Lead Coefficient Observations about end behavior? Slide 136 / 276 End Behavior of a Polynomial Lead coefficient Lead coefficient is positive is negative Left End Right End Left End Right End Polynomial of even degree Polynomial of odd degree Slide 137 / 276 End Behavior of a Polynomial Degree: even Degree: even Lead Coefficient: positive Lead Coefficient: negative As x → ∞, f(x) → ∞ As x → ∞, f(x) → -∞ As x → -∞, f(x) → ∞ As x → -∞, f(x) → -∞ In other words, the function In other words, the function rises to the left and to the falls to the left and to the right. right.
Slide 138 / 276 End Behavior of a Polynomial Degree: odd Degree: odd Lead Coefficient: positive Lead Coefficient: negative As As As As In other words, the function In other words, the function falls to the left and rises to rises to the left and falls to the right. the right. Slide 139 / 276 54 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. A odd and positive B odd and negative C even and positive D even and negative Slide 139 (Answer) / 276 54 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Answer D even and negative A odd and positive B odd and negative C even and [This object is a pull tab] positive D even and negative
Slide 140 / 276 55 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. A odd and positive B odd and negative C even and positive D even and negative Slide 140 (Answer) / 276 55 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. A odd and positive Answer A odd and positive B odd and negative C even and positive D even and negative [This object is a pull tab] Slide 141 / 276 56 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. A odd and positive B odd and negative C even and positive D even and negative
Slide 141 (Answer) / 276 56 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. Answer C even and positive A odd and positive B odd and negative C even and positive D even and negative [This object is a pull tab] Slide 142 / 276 57 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. A odd and positive B odd and negative C even and positive D even and negative Slide 142 (Answer) / 276 57 Determine if the graph represents a polynomial of odd or even degree, and if the lead coefficient is positive or negative. A odd and positive Answer B odd and negative B odd and negative C even and positive D even and negative [This object is a pull tab]
Slide 143 / 276 Odd and Even Functions 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 = 7x 0 ) Examples: Odd function Even function Neither f(x)=3x 5 - 4x 3 + 2x h(x)=6x 4 - 2x 2 + 3 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 (Answer) / 276
Slide 145 / 276 Slide 145 (Answer) / 276 Slide 146 / 276
Slide 147 / 276 An even function is symmetric about the y-axis. Definition of an Even Function Slide 148 / 276 60 Choose all that apply to describe the graph. Odd Degree A Odd Function B Even Degree C Even Function D E Positive Lead Coefficient F Negative Lead Coefficient Slide 148 (Answer) / 276 60 Choose all that apply to describe the graph. Odd Degree A Answer A Odd- Degree B Odd- Function Odd Function B E Positive Lead Coefficient C Even Degree Even Function D [This object is a pull tab] Positive Lead Coefficient E Negative Lead Coefficient F
Slide 149 / 276 61 Choose all that apply to describe the graph. Odd Degree A B Odd Function Even Degree C Even Function D Positive Lead Coefficient E Negative Lead Coefficient F Slide 149 (Answer) / 276 61 Choose all that apply to describe the graph. Odd Degree Answer C Even- Degree A D Even- Function Odd Function B E Positive Lead Coefficient Even Degree C Even Function D [This object is a pull tab] E Positive Lead Coefficient F Negative Lead Coefficient Slide 150 / 276 62 Choose all that apply to describe the graph. Odd Degree A Odd Function B C Even Degree Even Function D Positive Lead Coefficient E Negative Lead Coefficient F
Slide 150 (Answer) / 276 62 Choose all that apply to describe the graph. A Odd- Degree Answer Odd Degree A B Odd- Function F Negative Lead Coefficient B Odd Function Even Degree C Even Function D [This object is a pull tab] Positive Lead Coefficient E Negative Lead Coefficient F Slide 151 / 276 63 Choose all that apply to describe the graph. Odd Degree A Odd Function B Even Degree C Even Function D E Positive Lead Coefficient F Negative Lead Coefficient Slide 151 (Answer) / 276 63 Choose all that apply to describe the graph. A Odd- Degree Answer Odd Degree A Odd Function E Positive Lead Coefficient B C Even Degree Even Function D [This object is a pull tab] Positive Lead Coefficient E Negative Lead Coefficient F
Slide 152 / 276 64 Choose all that apply to describe the graph. Odd Degree A B Odd Function Even Degree C Even Function D Positive Lead Coefficient E Negative Lead Coefficient F Slide 152 (Answer) / 276 64 Choose all that apply to describe the graph. C Even- Degree Answer Odd Degree A D Even - Function F Negative Lead Coefficient Odd Function B Even Degree C Even Function D [This object is a pull tab] E Positive Lead Coefficient F Negative Lead Coefficient Zeros of a Polynomial Slide 153 / 276 "Zeros" are the points at which the polynomial intersects the x-axis. They are called "zeros" because at each point f (x) = 0. Another name for a zero is a root. A polynomial function of degree n has at MOST n real zeros. An odd degree polynomial must have at least one real zero. (WHY?) Zeros
Slide 154 / 276 Relative Maxima and Minima A 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. Relative Maxima Relative Minima Slide 155 / 276 65 How many zeros does the polynomial appear to have? Slide 155 (Answer) / 276 65 How many zeros does the polynomial appear to have? Answer 5 zeros [This object is a pull tab]
Slide 156 / 276 66 How many turning points does the polynomial appear to have? Slide 156 (Answer) / 276 66 How many turning points does the polynomial appear to have? Answer 4 turning points [This object is a pull tab] Slide 157 / 276 67 How many zeros does the polynomial appear to have?
Slide 157 (Answer) / 276 67 How many zeros does the polynomial appear to have? Answer 4 zeros [This object is a pull tab] Slide 158 / 276 68 How many turning points does the graph appear to have? How many of those are relative minima? Slide 158 (Answer) / 276 68 How many turning points does the graph appear to have? How many of those are relative minima? Answer 3 turning points 2 relative min [This object is a pull tab]
Slide 159 / 276 69 How many zeros does the polynomial appear to have? Slide 159 (Answer) / 276 69 How many zeros does the polynomial appear to have? Answer 3 zeros [This object is a pull tab] Slide 160 / 276 70 How many turning points does the polynomial appear to have? How many of those are relative maxima?
Slide 160 (Answer) / 276 70 How many turning points does the polynomial appear to have? How many of those are relative maxima? Answer 2 turning points 1 relative max [This object is a pull tab] Slide 161 / 276 71 How many zeros does the polynomial appear to have? Slide 161 (Answer) / 276 71 How many zeros does the polynomial appear to have? Answer None [This object is a pull tab]
Slide 162 / 276 72 How many relative maxima does the graph appear to have? How many relative minima? Slide 162 (Answer) / 276 72 How many relative maxima does the graph appear to have? How many relative minima? 3 relative max Answer 2 relative min [This object is a pull tab] Slide 163 / 276 Analyzing Graphs and Tables of Polynomial Functions Return to Table of Contents
Slide 164 / 276 A polynomial function can be sketched by creating a table, plotting the points, and then connecting the points with a smooth curve. x y -3 58 -2 19 -1 0 0 -5 1 -2 2 3 3 4 4 -5 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. Slide 165 / 276 How many zeros does this function appear to have? Answer x y -3 58 -2 19 -1 0 0 -5 1 -2 2 3 3 4 4 -5 Slide 166 / 276 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? Answer x y -3 58 -2 19 -1 0 0 -5 1 -2 2 3 3 4 4 -5
Slide 167 / 276 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, so there must be an x-value for which y = 0. Slide 168 / 276 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. x y -3 58 -2 19 -1 0 0 -5 1 -2 2 3 3 4 4 -5 Slide 169 / 276 73 How many zeros of the continuous polynomial given can be found using the table? x y -3 -12 -2 -4 Answer -1 1 0 3 1 0 2 -2 3 4 4 -5
Slide 170 / 276 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 -3 A -3 -12 -2 -4 -2 B Answer -1 1 -1 C 0 3 D 0 1 0 1 E 2 -2 3 4 2 F 4 -5 3 G 4 H Slide 171 / 276 75 How many zeros of the continuous polynomial given can be found using the table? x y -3 2 Answer -2 0 -1 5 0 2 1 -3 2 4 3 4 4 -5 Slide 172 / 276 76 According to the table, what is the least value of x at which a zero occurs on this continuous function? x y -3 A -3 2 B -2 -2 0 Answer -1 C -1 5 0 D 0 2 1 -3 1 E 2 4 2 F 3 4 G 3 4 -5 4 H
Slide 173 / 276 Relative Maxima and Relative Minima 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). Slide 174 / 276 How do we recognize the relative ma xima and minima from a table? x f(x) In the table, as x goes from -3 to 1, f(x) is x goes from 1 to 3, f(x) is decreasing. As -3 5 s x goes from 3 to 4, f(x) increasing. And a is decreasing. -2 1 -1 -1 The relative maxima and minima 0 -4 occur when the direction changes 1 -5 from decreasing to increasing, or from increasing to decreasing. 2 -2 The y-coordinate indicates this 3 2 change in direction as its value rises 4 0 or falls. Slide 175 / 276
Slide 176 / 276 Slide 177 / 276 77 At approximately what x-values does a relative minimum occur? -3 1 A E -2 2 B F -1 3 C G D 0 H 4 Slide 177 (Answer) / 276 77 At approximately what x-values does a relative minimum occur? -3 1 A E Answer -2 2 C -1 B F E 1 -1 3 C G 0 4 D H [This object is a pull tab]
Slide 178 / 276 78 At about what x-values does a relative maximum occur? A -3 E 1 -2 2 B F -1 3 C G 0 4 D H Slide 178 (Answer) / 276 78 At about what x-values does a relative maximum occur? -3 1 A E Answer B -2 F 2 -2 2 B F -1 3 C G D 0 H 4 [This object is a pull tab] Slide 179 / 276 79 At about what x-values does a relative minimum occur? x y -3 1 A E -3 5 -2 2 B F Answer -2 1 -1 3 -1 -1 C G 0 -4 0 4 D H 1 -5 2 -2 3 2 4 0
Slide 180 / 276 80 At about what x-values does a relative maximum occur? x y A -3 E 1 -3 5 Answer -2 2 B F -2 1 -1 3 -1 -1 C G 0 -4 0 4 D H 1 -5 2 -2 3 2 4 0 Slide 181 / 276 81 At about what x-values does a relative minimum occur? x y -3 1 A E -3 2 -2 2 B F Answer -2 0 -1 3 C G -1 5 D 0 H 4 0 2 1 -3 2 4 3 4 4 -5 Slide 182 / 276 82 At about what x-values does a relative maximum occur? x y -3 1 A E -3 2 -2 2 B F Answer -2 0 -1 3 C G -1 5 0 4 D H 0 2 1 -3 2 4 3 5 4 -5
Slide 183 / 276 Zeros and Roots of a Polynomial Function Return to Table of Contents Slide 184 / 276 Real Zeros of Polynomial Functions 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) . Slide 185 / 276 The Fundamental Theorem of Algebra If f ( x ) is a polynomial of degree n , where n > 0, then f ( x ) = 0 has n zeros including multiples and imaginary zeros. An imaginary zero occurs when the solution to f ( x ) = 0 contains complex numbers . Imaginary zeros are not seen on the graph.
Slide 186 / 276 Complex Numbers Complex numbers will be studied in detail in the Radicals Unit. But 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 . Slide 187 / 276 The square root of any negative number is a complex number. For example, find a solution for x 2 = -9: Slide 188 / 276 Drag each number to the correct place in the diagram. Complex Numbers Real Imaginary 2 - 4i 3i -0.765 9+6i -11 2/3
Slide 188 (Answer) / 276 Drag each number to the correct place in the diagram. Complex Numbers Real Teacher Notes Imaginary The intersection of real and imaginary should be the empty set. [This object is a teacher notes pull tab] 2 - 4i 3i -0.765 9+6i -11 2/3 Slide 189 / 276 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 in total - 4 real = 0 imaginary) Slide 190 / 276 This 5th degree polynomial has 5 zeros, but only 3 of them are real. Therefore, there must be two imaginary. (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.)
Slide 191 / 276 A vertex on the x-axis indicates a multiple zero, meaning the zero occurs two or more times. This is a 4th-degree polynomial. It has two unique real zeros: -2 2 zeros and 2. These two each 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. Slide 192 / 276 What do you think are the zeros and their multiplicity for this function? Slide 192 (Answer) / 276 What do you think are the zeros and their multiplicity for this function? Answer Zeros are -2 with multiplicity of 2 and 2 with multiplicity of 4. [This object is a pull tab]
Slide 193 / 276 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 194 / 276 83 How many real zeros does the 4th-degree polynomial graphed have? 0 A B 1 2 C 3 D 4 E 5 F Slide 194 (Answer) / 276 83 How many real zeros does the 4th-degree polynomial graphed have? 0 Answer A E 4 1 B 2 C 3 D [This object is a pull tab] E 4 5 F
Slide 195 / 276 84 Do any of the zeros have a multiplicity of 2 ? Yes No Slide 195 (Answer) / 276 84 Do any of the zeros have a multiplicity of 2 ? Yes Answer No No [This object is a pull tab] Slide 196 / 276 85 How many imaginary zeros does this 7th degree polynomial have? 0 A 1 B 2 C 3 D E 4 5 F
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