Loudoun County Public Schools
HOMEWORK PACKET ~ UNIT 5 “POLYNOMIAL FUNCTIONS”
Name________________________ Date ____________
Day 1 Homework
Factoring Polynomials by GCF, Sum/Difference of Cubes, Operations with Polynomials
I. REVIEW – Factor all of these completely
1. [pic] 2. [pic]
3. [pic] 4. [pic]
II. FACTOR
| 5. x3 – 27 |6. x3 + 27 |
| 7. x3 – 64 | 8. 2x3 + 16 |
| 9. [pic] | 10. [pic] |
| 11. [pic] | 12. [pic] |
III. Complete the table
|Function |Polynomial: |Standard form of the polynomial |Degree of |Leading Coefficient of |Constant |
| |YES or NO | |polynomial |the polynomial | |
|13. y = -3x2 + 8x4 - 2 - 5x | | | | | |
|14. [pic] | | | | | |
|15. [pic] | | | | | |
|16. g(x) = - 7x2 + 7x -3 | | | | | |
IV. Add, subtract, multiply or divide to simplify. Write your answer in standard form.
|17. [pic] (7q ( 3q3) + (16 ( 8q3 + 5q2 ( q) |18. ((4z4 + 6z ( 9) + (11 ( z3 + 3z2 + z4) |
|19. (l0v4 ( 2v2 + 6v3 ( 7) ( (9 ( v + 2v4) |20. (4x5 + 3x4 ( 5x + l) ( (x3 + 2x4 ( x5 + 1) |
| | |
|21. 2x3(5x ( 1) |22. (n + 5)(2n2 ( n ( 7) |
|23. (x - 3)3 |24. (2x + 5)3 |
Name________________________ Date ____________
Day 2 Homework: Function Composition
|1. f(x) = 3x – 5 g(x) = 2x + 1 |2. f(x) = x – 2 g(x) = x2 + 3 |3. f(x) = 5x – 6 g(x) = 3x |
|Find f o g [this means f(g(x)] |Find g o f |Find f(g(x)) |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|4. f(x) = 2x2 + 11 g(x) = 4x |5. f(x) = x2 – 9 g(x) = 3x2 |6. f(x) = 5x2 g(x) = 6x – 2 |
|Find g(f(x)). |Find f o g (x) |Find f o f (x) and find g o g (x) |
| | | |
| | | |
| | | |
| | | |
| | | |
|7. f(x) = -4x + 6 g(x) = 5x – 1 |8. f(x) = 2x2 g(x) = x2 + 7 |9. f(x) = -3x + 5 g(x) = -2x |
|Find f o g (2) [this means f(g(2)] |Find f(g(-1)) and find g(f(-2)) |Find f o g |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|10. f(x) = x – 3 g(x) = x2 | 11. f(x) = 6x – 5 g(x) = -4x |12. f(x) = x2 – 9 g(x) = 7x + 1 |
|Find g o f |Find f(g(x)). |Find g(f(x)). |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|13. f(x) = x2 + 10 g(x) = 2x2 |14. f(x) = 3x2 g(x) = 4x – 3 |15. f(x) = 4x – 6 g(x) = 3x – 2 |
|Find f(g(x)) |Find f o f and find g o g |Find f(g(3)) |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
|16. f(x) = x2 + 3x g(x) = x2 + 6 |17. f(x) = [pic] g(x) = [pic] |18. k(x) = [pic] h(x) = [pic] |
|Find f(g(-2)). Also find g(f(-2)). |Find f o g and find g o f |Find k o h and find h o k |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
Name________________________ Date _____________
Day 3 Homework: Graphing Cubic Functions
I. REVIEW
#1 & 2 Simplify each of the following.
1. [pic] 2. [pic]
#3-5 Solve each of the following by the zero product property (factor).
3. 3x2 – 27 = 0 4. 4x2 – 9 = 0 5. 2x2 – 2x – 12 = 0
II. PRACTICE
#6-9 Graph each of the following. Identify the Point of Inflection and describe the transformations.
6. [pic] 7. [pic]
Point of inflection_________ Point of inflection_______
Horizontal shift: ___________ Horizontal shift:_________
Vertical shift: _____________ Vertical shift: __________
Reflection: _______________ Reflection: ____________
Stretch/shrink: ____________ Stretch/shrink: _________
8. [pic] 9. [pic]
Point of inflection_________ Point of inflection_______
Horizontal shift: ___________ Horizontal shift:_________
Vertical shift: _____________ Vertical shift: __________
Reflection: _______________ Reflection: ____________
Stretch/shrink: ____________ Stretch/shrink: _________
Name________________________ Date ________
Day 4 Homework: End Behavior, Zeros and Multiplicity
State the degree of each polynomial and the leading coefficient. Describe the end behavior of the graph of the polynomial function.
1. f(x) = 2x5 ( 7x2 ( 4x
Degree: Leading coefficient: As x ( ( ( f(x) ( ______ and as x ( + ( f(x) ( ______
2. f(x) = 9x7 + 2x8 + 10
Degree: Leading coefficient: As x ( ( ( f(x) ( ______ and as x ( + ( f(x) ( ______
3. f(x) = - 95x50 + 407x(2013x80
Degree: Leading coefficient: As x ( ( ( f(x) (______ and as x ( + ( f(x) ( ______
4. f(x) = 2017x57 ( 1998x46 + 1999x23
Degree: Leading coefficient: As x ( ( ( f(x) ( ______ and as x ( + ( f(x) ( ______
Describe the degree (even or odd) and leading coefficient (positive or negative) of the polynomial function. Then describe the end behavior of the graph of the polynomial function.
5. 6.
Degree _______ Degree _______
Leading Coefficient __________ Leading Coefficient __________
[pic] [pic]
#7-9 Use your calculator to match the graph with its function.
7. f(x) = 2x4 ( 3x2 ( 2 8. f(x) = 2x6 ( 6x4 + 4x2 ( 2 9. f(x) = (2x4 + 3x2 ( 2
A. B. C.
#10 & 11 Given the graph, complete the following.
10. [pic][pic]
a. Degree __________
b. Lead coefficient: positive or negative
c. Zeros ______________ Multiplicity? _____________
d. Factors __________________________
e. Number of turning points _____________________
f. Identify all relative minimum/maximum points.
_______________________________________
g. Identify all absolute minimum/maximum points.
_______________________________________
h. Increasing intervals _________________
i. Decreasing intervals _________________
j. y-intercept ________________________
//11. [pic][pic]
a. Degree __________
b. Lead coefficient: positive or negative
c. Zeros ______________ Multiplicity? _____________
d. Factors __________________________
e. Number of turning points _____________________
f. Identify all relative minimum/maximum points.
_______________________________________
g. Identify all absolute minimum/maximum points.
_______________________________________
h. Increasing intervals _________________
i. Decreasing intervals _________________
j. y-intercept ________________________
Name________________________ Date ________
Day 5 Homework: Evaluating Polynomials with Synthetic Substitution
and The Fundamental Theorem of Algebra
I. REVIEW – Simplify
1. [pic] 2. [pic]
Solve by the Zero Product Property (factor)
3. 3x2 – 27 = 0 4. 4x2 – 9 = 0 5. 2x2 – 2x – 12 = 0
II. Evaluate the polynomial for the given value of x in TWO ways – by direct substitution and by synthetic substitution
6. f(x) = 5x3 – 2x2 – 8x + 16 for x = 3
Direct Substitution Synthetic Substitution
7. f(x) = 8x4 + 12x3 + 6x2 – 5x + 9 for x = -2
Direct Substitution Synthetic Substitution
8. f(x) = x3 + 8x2 – 7x + 35 for x = –6
Direct Substitution Synthetic Substitution
9. f(x) = –8x3 + 14x – 35 for x = 4
Direct Substitution Synthetic Substitution
10. f(x) = –2x4 + 3x3 – 8x + 13 for x = 2
Direct Substitution Synthetic Substitution
#11-15 Identify the total number of zeros and maximum number of turning points.
11. f(x) = 5x3 – 2x2 – 8x + 16 ZEROS:_________ Max. # of turning points: ___________
12. f(x) = 8x4 + 12x3 + 6x2 – 5x + 9 ZEROS:_________ Max. # of turning points: ___________
13. f(x) = x7 + 8x4 – 7x + 35 ZEROS:_________ Max. # of turning points: ___________
14. f(x) = –8x3 + 14x – 35 ZEROS:_________ Max. # of turning points: ___________
15. f(x) = –2x6 + 3x3 – 8x + 13 ZEROS:_________ Max. # of turning points: ___________
III. Write a polynomial function in standard form of least degree that has a leading coefficient of 1 and the given zeros. (Remember, imaginary and irrational solutions always come in pairs! You may have to find the other half of the pair!)
16. -2, 1, 3 17. -5, -1, 2
18. 2, -i, i 19. 2, -3i
20. 4, [pic], [pic] 21. 3, [pic]
22. Graph f(x) = 3x4 + x3 - 10x2 + 2x + 7 using your calculator. Sketch its graph below.
Determine the total number of zeros for the polynomial __________
# of real zeros:__________ # of imaginary zeros:____________
a. Determine the number of turning points _________________________
b. Identify all relative minimum/maximum points.
_____________________________________________
c. Identify all absolute minimum/maximum points.
_____________________________________________
d. Over what intervals is f(x) Decreasing___________________________________________
e. Over what intervals is f(x) Increasing___________________________________________
f. Describe the end behavior of the graph:
As, [pic][pic] As [pic], [pic]
Name________________________ Date ________
Day 6 Homework: Applying the Remainder and Factor Theorems
I. REVIEW – Multiply
1. [pic] 2. [pic]
II. PRACTICE – Use SYNTHETIC DIVISION and LONG DIVISION to divide the polynomials. Be sure to write your answer in the form of a polynomial and a remainder.
SYNTHETIC DIVISION LONG DIVISION
3. (x3 ( 3x2 + 8x ( 5) ( (x ( 1)
4. (x4 ( 7x2 + 9x ( 10) ( (x ( 2)
5. (2x4 ( x3 + 4) ( (x + 1)
6. (2x4 ( 11x3 + 15x2 + 6x ( 18) ( (x ( 3)
III. Factor the following polynomials completely using synthetic division and factoring.
7. f(x) = x3 ( 3x2 ( 16x ( 12; given that (x ( 6) is a factor
8. f(x) = x3 ( 12x2 + 12x + 80; given that (x – 10) is a factor
9. f(x) = x3 ( 18x2 + 95x ( 126; given that (x – 9) is a factor
10. f(x) = x3 ( x2 ( 21x + 45; given that (x + 5) is a factor
11. f(x) = x3 + 2x2 - 20x + 24; given that (6 is a zero
12. f(x) = 15x3 ( 119x2 ( l0x + 16; given that 8 is a zero
13. f(x) = 2x3 + 3x2 ( 39x ( 20; given that 4 is a zero
Name________________________ Date ___________
Day 7 Homework: Finding All Rational Zeros
#1-4 Use the function, g(x) = x3 – 5x + 4 to answer the questions.
1. List all possible rational zeros of g(x):
2. Graph g(x) on your calculator. Pick a zero that matches a value from the list above.
Sketch the graph. Test that zero using synthetic division:
3. Solve the depressed polynomial, using your method of choice (factoring, quadratic formula, or
completing the square).
4. List all of the zeros of g(x):
#5-9 Use the function, h(x) = 2x3 + 2x2 – 8x – 8 to answer the questions.
5. List all possible rational zeros of h(x):
6. Graph h(x) on your calculator. Pick a zero that matches a value from the list above.
Sketch the graph. Test that zero using synthetic division:
7. Solve the depressed polynomial, using your method of choice (factoring, quadratic formula, or
completing the square).
8. List all of the zeros of h(x):
9. Solve 0 = 2x3 + 2x2 – 8x – 8 by factoring.
(Hint: How do you factor a polynomial with 4 terms?)
Does this answer match your answer from #8?
10. a) Is it possible for a cubic function to have more than three real zeros?
Explain. (Your explanation can include a picture).
b) Is it possible for a cubic function to have no real zeros?
Explain. (Your explanation can include a picture).
11. How many possible solutions (real or imaginary) are guaranteed by the Fundamental Theorem of Algebra for the following equation? y = x5 – 3x4 – 5x3 + 15x2 + 4x – 12
Maximum number of turning points?
12. List all possible rational zeros for the function in question 11:
13. Find all the solutions for the function in question 11.
(Hint: They are ALL real! You will have to do synthetic division 3 times using zeros
from your graph before you have a quadratic to solve.)
#14-18: Use your calculator and synthetic division to find all possible solutions. Remember, complex numbers are also solutions.
14. y = x3 -6x2 +11x -6 15. f(x) = x4 -7x2 +12
16. f (x) = x3 -9x2 +20x -12 17. y = x5 -7x4 +10x3 + 44x2 -24x
Name________________________ Date ________
Day 8 Homework: REVIEW
Analyzing Graphs
Answer the questions based on the given graph.
Operations & Substitution
Simplify each expression. Show work!
| | |
|1. (x + 1)3 |2. (2x4 − 8x2 − x) − (−5x4 − x + 5) |
| | |
| | |
| | |
| | |
| | |
|3. (5d3 – 4d2 + 5) + (7d3 + 2d2 – 8d) |4. (x + 5i)(x – 5i) |
| | |
| | |
| | |
| | |
| | |
| | |
|5. (2x + 3)(x – 2)(3x + 2) |6. (x + 4)(x2 + 2x – 3) |
| | |
| | |
| | |
| | |
| | |
| | |
7. Evaluate the polynomial function using Direct Substitution.
f(x) = -3x3 + x2 – 12x – 5 when x = -2
8. Evaluate the polynomial function using Synthetic Substitution.
F(X) = x4 + 2x3 + 5x - 8 for f(-4)
9. Write a polynomial function in standard form that has real coefficients, the given zeros, and a leading
coefficient of 1.
Zeros: 2, 4, -3i
Recognizing and "Reading" Polynomials
Identify the degree, leading coefficient, and constant of the polynomial. (State the numerical value.)
10. f(x) = 6x5 – 4x3 + 1 Degree _____ Leading Coefficient _____ Constant ____
11. g(x) = 9x4 + x – 7 Degree _____ Leading Coefficient _____ Constant ____
12. f(x) = -2x2 – 3x4 + 5x – 9x3 + 5 Degree _____ Leading Coefficient _____ Constant ____
Tell if each of the following are polynomials? If no, explain why not!
(Yes or No) Explanation
13. 4x2 + 2x + x2 + 3 _________ _____________________
14. 3x + 1 _________ _____________________
15. 5x + 2x½ +3 _________ _____________________
Describe the end behavior of the graph. Use [pic] or [pic]
16. f(x) = -7x2 + 4x – 9 as [pic], [pic] as [pic], [pic]
17. f(x) = 6x5 + 7x4 – 8x as [pic], [pic] as [pic], [pic]
18. f(x) = -8x3 – 9x2 + x – 4 as [pic], [pic] as [pic], [pic]
19. f(x) = x4 + 16 as [pic], [pic] as [pic], [pic]
Factoring
Factor the following completely. If not possible, write PRIME.
|20) 2x3 + 16 |21) 25x2 – 81 |
| | |
| | |
|22) 125x3 - 27 |23) 2x4 + 128x |
| | |
| | |
Composition of Functions
For Problems 24-29, let: f(x) = 2x-2, g(x) = 3x, h(x) = x2 +1
24. f(g(-3)) 25. f(h(7)) 26. (g(f(x))
27. f(x+1) 28. h(g(x)) 29. (f ᵒg)(x)
Synthetic Substitution
Evaluate the function at the given value, then determine if the value given is a solution.
30. f(x) = x4 + 2x3 – 13x2 + 15x + 22, x = -5
31. f(x) = x5 + x4 – 15x3 – 19x2 – 6x + 1, x = 4
32. g(x) = -5x5 + 11x4 + 9x3 + 11x2 – 8x + 4, x = 3
Finding Solutions
(33-34) Use your calculator and synthetic division to find all solutions of the given equations. Remember, complex numbers are also solutions.
33. y = x4 - 2x3 – 19x2 + 32x + 48 34. y = x5 + 4x4 – 10x3 + 82x2 – 375x - 450
Use long division to simplify the polynomial, then find all of the zeros of f(x).
35. f(x) = (12x3 + 2 + 11x + 20x2) ÷ (2x + 1)
-----------------------
Leading Coefficient: (positive or negative) ___________________
End Behavior: As [pic]
As [pic]
Identify the real zeros of the graph:
___________________________________________
Identify the factors based on your zeros listed above.
___________________________________
Circle the turning points on the graph. Determine if they are relative maximums or minimums, absolute maximums or minimums.
Determine the intervals where the polynomials are
Increasing: ______________________________
Decreasing: _____________________________
Determine the domain and range of the polynomial.
Domain: _________
Range: _______________
Leading Coefficient: (positive or negative) ___________________
End Behavior: As [pic]
As [pic]
Identify the real zeros of the graph:
___________________________________________
Identify the factors based on your zeros listed above.
___________________________________
Circle the turning points[pic]12|ƒ?”•£¤»Ûö÷ø óßË´Ë ´Ë‰weSeSDe4-h{I?hDÇ5?OJQJ\?^JhDÇ5?CJOJQJ^JaJ#hJchr–5?CJOJQJ^JaJ#hJchJc5?CJOJQJ^JaJ#hJchÍ[?]ä5?CJOJQJ^JaJ,hJchJc5?6?CJOJQJ\?]?^JaJ&hÀ¯5?6?CJOJQJ\?]?^JaJ,h{I?hDÇ5?6?CJOJQJ\?]?^JaJ&hJc5?6 on the graph. Determine if they are relative maximums or minimums, absolute maximums or minimums.
Determine the intervals where the polynomials are
Increasing: ______________________________
Decreasing: _____________________________
Determine the domain and range of the polynomial.
Domain: _____________
Range: _______________
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
Related searches
- baltimore county public schools calendar
- baltimore county public schools calendar 2019 2020
- gadsden county public schools employment
- baltimore county public schools 2019 2020
- baltimore county public schools application
- baltimore county public schools employ
- broward county public schools 2019 2020 calendar
- gadsden county public schools employm
- gadsden county public schools fl
- jefferson county public schools parent portal
- baltimore county public schools careers
- baltimore county public schools lunch