Loudoun County Public Schools / Overview



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) |

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|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)) |

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|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) |

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|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 |

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|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)). |

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|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)) |

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|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 |

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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!

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|1. (x + 1)3 |2. (2x4 − 8x2 − x) − (−5x4 − x + 5) |

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|3. (5d3 – 4d2 + 5) + (7d3 + 2d2 – 8d) |4. (x + 5i)(x – 5i) |

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|5. (2x + 3)(x – 2)(3x + 2) |6. (x + 4)(x2 + 2x – 3) |

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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 |

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|22) 125x3 - 27 |23) 2x4 + 128x |

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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)

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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.

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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: _______________

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