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Name____________ Period _____ Pre-Calculus Unit Three Polynomial Functions Enrichment Packet Due: Thursday, Oct 20 (A day) Tuesday, Oct 25 (B day) Elise Calhoun [email protected] Tutoring Times: W, Th, F 7:45 – 8:20 M, W, Th 4:00 – 4:30 1 Date Objective Activities Finding Maximum Volume 1. Building a Box Activity 2. EA “Sandy Box, Candy Box” pages 1, 3, 4 3. Handout “Popcorn Box” Project Directions 1. EA ”Bending & Ending” 2. EA “Tables, Graphs, Equations” 3. EA “Synthetic Substitution” EA = Enrichment Activity Day One A: 9/28 B: 9/29 Day Two Finding Functions to Model Data A: 9/30 B: 10/3 Day Three A: 10/4 B: 10/5 Day Four Finding the Factors of Polynomials Graphically and Algebraically Using Polynomial Long Division A: 10/6 B: 10/7 1. DUE: WS “Pump Up the Volume & …” 2. Quiz: Unit 3 #1: Maximizing Volume, Characteristics of Polynomials, and Synthetic Substitution 3.EA “Function Factors” (Nearpod Activity) 4.EA ”Factoring Practice” 5.EA “Graphs and Factors” 1. DUE: WebAssign Unit 3 #1 2. EA “The Missing Factor” pages 1-3 Homework WS = Worksheet Work on Popcorn Box Project due… A: Tuesday 10/11 B: Wednesday 10/12 WS “Pump Up the Volume”, “Synthetic Substitution Practice”, and “Polynomials” – due… A: Tuesday 10/4 B: Wednesday 10/5 WebAssign Unit 3 #1 due… A: Thursday 10/6 B: Friday 10/7 WebAssign Unit 3 #2 due… A: Monday 10/13 B: Tuesday 10/17 Popcorn Box Project due next class! 10/10 Day Five A: 10/11 B: 10/12 Day Six School Holiday Explore Properties of Imaginary Numbers and How They Relate to the Zeroes of a Polynomial Find the Exact Zeroes of a Polynomial A: 10/13 1.DUE: Popcorn Box Project 2.EZ “Number Classification” 3.EA“Imaginary Numbers” 4.EA “The Complex Nature of Things” 1. DUE: WebAssign Unit 3#2 2. Finish EA “The Complex Nature of Things” 3. EA “Finding Exact Zeroes” WebAssign Unit 3 #3 due… A: Wednesday 10/18 B: Thursday 10/19 10/14 Homecoming B: 10/17 Day Seven Solving Inequalities A: 10/18 B: 10/19 Day Seven A: 10/20 Test Unit 3: Polynomial Functions 10/21 NO School 10/24 NO School 1. DUE: WebAssign Unit 3#3 2. Quiz Unit 3 #2: Factors and Zeroes of Polynomials 2. EA: “Solving Inequalitites” 3. Review Unit 3 1.Due: Unit 3 Enrichment Packet 2.Test: Unit 3 Study for Unit 3 Test Last day for retakes in Unit 2 B: 10/25 2 3 4 In each case below, enter the given data into the lists in the calculator (STAT, 1:Edit) so that height values are in L1 and volumes are in L2. Then, generate a linear regression equation (STAT, CALC, LinReg (ax +b)). Sketch it over the scatter plot. Record and describe the values of “r” that result. 5 6 Bending and Ending Terminology: DEGREE NUMBER OF BENDS: END BEHAVIOR: 7 Using Bending and Ending KEY, we just discussed in class, answer the following questions: 8 9 10 11 Function Factors Using the Function Factors KEY that we just discussed in class, answer the following questions: 1. Describe the relationship between the number of factors in the function and its degree: 2. What is the general relationship between the factor of a function and the x-intercept related to the factor? 3. Graphically, what is the difference between an x-intercept caused by a single factor, (x-3) and one that is generated by a factor that is squared, ( x 3) ? 2 4. Only two of the preceding functions have graphs that point down on the right (or approach increases). Which ones are they? Why? as x 5. True or False: The number of x-intercepts on a polynomial function is the same as the function’s degree. Explain your answer. 12 6. Without graphing, tell the x-intercepts of each function described below: A) f ( x) ( x 8)( x 6)( x 12) B) f ( x) x( x 1)2 ( x 3.7)( x 22.8) 7. Without the aid of a calculator, sketch the graph of each function given below: A) f ( x) ( x 2)( x 1)( x 4) B) f ( x) x( x 3)( x 4)2 C) f ( x) ( x 4)( x 2)( x 1) 8. Write the “factored” function for each graph below: Factoring Practice, Algebraically 13 Factor each of the following to solve: (1.) 2 x3 16 x 0 (2.) x2 8x 15 0 (3.) 4 x2 25 0 (4.) (5.) x3 4 x2 x 4 0 (5.) 3x3 27 x Use the Quadratic Formula to solve the following: x (7.) 3x2 5x 1 0 2 y2 5 y 3 0 b b2 4ac 2a (8.) 4 x2 12 x 9 0 14 B) Check to make sure your answers match the original polynomial functions by multiplying the factors you got in #1-4 15 16 Use polynomial division (and factoring) to find the factors of each of the following: 3. Try dividing #3 and #4 using SYNTHETIC Division: Factors of 2𝑥 3 + 17𝑥 2 + 31𝑥 − 20 = 4. Factors of 2𝑥 4 − 3𝑥 3 − 14𝑥 2 + 19𝑥 − 12 = 5. Factors of 2𝑥 4 − 9𝑥 3 − 2𝑥 2 + 14𝑥 − 15 = 17 6. Is x + 1 a factor of 𝑥 2 + 3𝑥 + 5? Why or why not? How are synthetic division and synthetic substitution the same? How are they different? 18 Number Classification 19 i Imaginary Numbers Definintion: “i” number whose square is -1 square root of a -1 is “i” i 2 1 i non-negative real number, x x 1 x i x 1 i Square roots of negative numbers can be simplified using “i”: If x is a non-negative real number, then x 1 x 1 x i x When you have a negative number under a radical, you MUST take the negative out as “i” Sample Problems: Simplify the following complex numbers by writing them in terms of i. A) 49 = B) 121 = C) 17 42 = D) 3 = E) 12 36 = 2 F) 5 250 10 Addition/Subtraction of Complex Numbers Real parts combine. Imaginary parts combine. Answers must be in a + bi form. EXAMPLES: a. 8 9i 12 11i b. 8 9i 12 11i 20 Multiplication of Complex Numbers Distribute and collect like terms. Remember that “i” cannot have a power and ALL i 2 MUST be changed to (-1). Answers must be in a + bi form. EXAMPLES: a. 4 2i 4 2i b. 6 2i 5 3i Practice: Simplify the following problems. Write answers in a + bi form. 2.) 85 169 5.) 3 14i 9 5i 6.) 5 2i 9 144 6 10.) 1.) 9.) 81 2 2 160 2 3.) 7.) 16 49 5i 11.) 2 3 75 8 4.) 8.) 12.) 5 6 2i 6 2i 1 8 2 21 22 Sketch the graph of each function based on the description of its zeros. 23 24 15) Write a cubic polynomial with zeros of 5, 7, 7 16) Write a cubic polynomial with zeros 1 and 3i 17) Write a second degree polynomial with zeros 5 2i and 5 2i 25 FACTORING REVIEW: Remember: x2 4 = , so how would you factor x2 4 ? If the factors are (x + 3)(x + 3), which is x 3 , what was the polynomial that got factored? 2 18) Factor each of the following: a) 9 x 2 16 b) x2 18x 81 c) x3 64 x d) x4 12 x2 36 e) x3 2 x2 4 x 8 f) x 4 256 26 19) 20) 27 Finding EXACT Zeros 28 A) Use a calculator to find any rational zeros of this function. p q factors( p) : factors(q) : All possible RATIONAL factors from factors( p) : factors(q) Using the TABLE, scroll through all x-values from ___________ to _____________, looking for the x-values that have a y-value of 0 (because those are the RATIONAL zeros, x-intercepts, roots, solutions). Your 2 RATIONAL roots are: _______________________, since you should have 4 roots in all, B)****you must now use synthetic substitution and factoring/quadratic formula to find the remaining/other roots (irrational and complex/imaginary roots): The polynomials that are the result of your synthetic substitution are called depressed polynomials All EXACT for roots of 4 x4 7 x3 49 x2 27 x 9 are __________________________________ Now find the EXACT roots for each of the following: 1) f ( x) x3 4 x2 11x 30 **how many zeros will this polynomial have? A) Use a calculator to find any rational zeros of this function. 29 2) f ( x) 2 x3 21x2 52x 30 A) Use a calculator to find any rational zeros of this function. B) Use synthetic substitution to find the (depressed) polynomials. C) From the QUADRATIC factor, find the other irrational or non-real zeros: 3) f ( x) x4 5x3 19x2 62x 24 A) Use a calculator to find any rational zeros of this function. B) Use synthetic substitution to find the (depressed) polynomials. C) From the QUADRATIC factor, find the other irrational or non-real zeros: 30 4) 3x4 20 x3 42 x2 44 x 16 A) Use a calculator to find any rational zeros of this function. B) Use synthetic substitution to find the (depressed) polynomials. C) From the QUADRATIC factor, find the other irrational or non-real zeros: 31 Solving Inequalities To determine the intervals on which the values of a polynomial are entirely negative or positive, use the following steps: 1. Find all real zeros of the polynomial, and arrange the zeros in increasing order. These zeros are the key numbers of the polynomial. 2. Use the key numbers to determine the test intervals. 3. Choose one representative x-value in each test interval and evaluate the polynomial at that value or use a graphing device to check. Solve the polynomial inequalities: (1) 2𝑥 2 + 5𝑥 > 12 (4) 𝑥 4 (𝑥 − 3) ≤ 0 (2) (𝑥 + 2)2 < 25 (5) 2𝑥 3 − 3𝑥 2 − 32𝑥 + 48 > 0 (3) 𝑥 3 − 4𝑥 ≥ 0 (6) 2𝑥 3 + 3𝑥 2 < 11𝑥 + 6 32