Download COMPLETE Unit 3 Packet

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
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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.
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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
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B) Check to make sure your answers match the original polynomial functions by multiplying the factors you
got in #1-4
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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 =
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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?
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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
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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 
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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
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Sketch the graph of each function based on the description of its zeros.
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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
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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
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19)
20)
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Finding EXACT Zeros
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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.
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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:
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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:
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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
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