Download Unit 5 Home Work Packet ~ Polynomial Functions

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.
16x 2  y 2
2.
3.
8 x 2  50 y 2
4.
8 x 2  50 y 2
3x3  10 x 2  8 x
II. FACTOR
5. x3 – 27
6. x3 + 27
7. x3 – 64
8. 2x3 + 16
9.
8 x3  27
11. 64 x  1
3
10.
8 x3  27
12. 8m  64n
3
3
III. Complete the table
Polynomial:
YES or NO
Function
Standard form of the
polynomial
Degree of
polynomial
Leading
Coefficient of
the polynomial
13. y = -3x2 + 8x4 - 2 5x
1
14. y  x3  x 2  7
15. f ( x)  3x  8 x  5 x
5
8
16. g(x) = - 7x2 + 7x
-3
IV. Add, subtract, multiply or divide to simplify. Write your answer in standard form.
17.
(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)
24.
(2x + 5)3
23. (x - 3)3
Constant
Name________________________
Date ____________
Day 2 Homework: Function Composition
1. f(x) = 3x – 5
Find f o g
4. f(x) = 2x2 + 11
g(x) = 2x + 1
[this means f(g(x)]
g(x) = 4x
Find g(f(x)).
7. f(x) = -4x + 6
Find f o g (2)
2. f(x) = x – 2
g(x) = x2 + 3
Find g o f
5. f(x) = x2 – 9
[this means f(g(2)]
8. f(x) = 2x2
g(x) = 3x
Find f(g(x))
g(x) = 3x2
Find f o g (x)
g(x) = 5x – 1
3. f(x) = 5x – 6
6. f(x) = 5x2
Find f o f (x)
g(x) = x2 + 7
Find f(g(-1)) and find g(f(-2))
9. f(x) = -3x + 5
Find f o g
g(x) = 6x – 2
and find g o g (x)
g(x) = -2x
10. f(x) = x – 3
g(x) = x2
Find g o f
13. f(x) = x2 + 10
g(x) = -4x
Find f(g(x)).
g(x) = 2x2
Find f(g(x))
16. f(x) = x2 + 3x
11. f(x) = 6x – 5
g(x) = x2 + 6
g(x) = 4x – 3
and find g o g
17. f(x) = 2x  4
Find f(g(-2)). Also find g(f(-2)).
Find f o g
g(x) = 7x + 1
Find g(f(x)).
14. f(x) = 3x2
Find f o f
12. f(x) = x2 – 9
g(x) =
1
x2
2
and find g o f
15. f(x) = 4x – 6
g(x) = 3x – 2
Find f(g(3))
18. k(x) = 2 x  6
Find k o h
h(x) = x  4
and find h o k
Name________________________
Date _____________
Day 3 Homework: Graphing Cubic Functions
I. REVIEW
#1 & 2 Simplify each of the following.
1.
1
4
2 x  3x
1
3
2.
6
4
3 1
2 6
(x y z )
#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. f  x     x  1
7. y 
3
 y

1 3
x 1
2
Point of inflection_________
 y
Point of inflection_______
Horizontal shift: ___________

Horizontal shift:_________


x





Vertical shift: _____________

Reflection: _______________


x



Stretch/shrink: ____________





Reflection: ____________

Stretch/shrink: _________

8. y   x  1  2
9. g  x   2  x  3  1
3
3
 y
 y
Point of inflection_________


Point of inflection_______


Horizontal shift: ___________
x
x





Vertical shift: __________


Vertical shift: _____________

Reflection: _______________

Stretch/shrink: ____________









Horizontal shift:_________
Vertical shift: __________
Reflection: ____________
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)  ______
As x    f(x)  ______
and as x  +  f(x)  ______
As x    f(x) ______
and as x  +  f(x)  ______
As x    f(x)  ______
and as x  +  f(x)  ______
2. f(x) = 9x7 + 2x8 + 10
Degree:
Leading coefficient:
3. f(x) = - 95x50 + 407x2013x80
Degree:
Leading coefficient:
4. f(x) = 2017x57  1998x46 + 1999x23
Degree:
Leading coefficient:
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 __________
As x    then f ( x)  _____
As x    then f ( x)  _____
As x    then f ( x)  _____
As x    then f ( x)  _____
#7-9 Use your calculator to match the graph with its function.
7. f(x) = 2x 4  3x 2  2
A.
8. f(x) = 2x 6  6x 4 + 4x 2  2
B.
9. f(x) = 2x 4 + 3x 2  2
C.
#10 & 11 Given the graph, complete the following.
10.
f ( x) = x3 - 4 x 2 - 3x + 18
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.
g ( x) = x 3 - 7 x - 6
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.
1
4
2 x  3x
1
3
2.
6
4
3 1
2 6
(x y z )
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,  5 ,
5
21. 3, 1  2
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, x   f ( x)  _____
As x   , f ( x)  _____
Name________________________
Day 6 Homework: Applying the Remainder and Factor Theorems
I.
REVIEW – Multiply
1.
( x  5)( x  5)
Date ________
2. ( x  2i )( x  2i )
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
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)
LONG DIVISION
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.
Sketch the graph.
Pick a zero that matches a value from the list above.
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.
Sketch the graph.
Pick a zero that matches a value from the list above.
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.
Leading Coefficient: (positive or negative) ___________________
End Behavior: As x  , f ( x)  ____
9 y
8
7
6
5
4
3
2
1
-1
-9-8 -7 -6 -5-4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6 7 8 9
As x  , f ( x)  ____
Identify the real zeros of the graph:
___________________________________________
x
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 x  , f ( x)  ____
As x  , f ( x)  ____
Identify the real zeros of the graph:
9 y
8
7
6
5
4
3
2
1
-1
-9-8 -7 -6 -5-4 -3 -2 -1
-2
-3
-4
-5
-6
-7
-8
-9
1 2 3 4 5 6 7 8 9
___________________________________________
Identify the factors based on your zeros listed above.
___________________________________
x
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: _______________
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   or  
16. f(x) = -7x2 + 4x – 9
as x   , f ( x)  _____
as x   , f ( x)  _____
17. f(x) = 6x5 + 7x4 – 8x
as x   , f ( x)  _____
as x   , f ( x)  _____
18. f(x) = -8x3 – 9x2 + x – 4
as x   , f ( x)  _____
as x   , f ( x)  _____
as x   , f ( x)  _____
as x   , f ( x)  _____
19.
f(x) = x4 + 16
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)