HOMEWORK PACKET ~ UNIT 5 “POLYNOMIAL FUNCTIONS”

Name______Date ______

Day 1 Homework

Factoring Polynomials by GCF, Sum/Difference of Cubes, Operations with Polynomials

  1. REVIEW – Factor all of these completely

1. 2.

3. 4.

II. FACTOR

5. x3 – 27 / 6. x3 + 27
7. x3 – 64 / 8. 2x3 + 16
9. / 10.
11. / 12.

III. Complete the table

Function / Polynomial:
YES or NO / Standard form of the polynomial / Degree of polynomial / Leading Coefficient of the polynomial / Constant
13. y = -3x2 + 8x4 - 2 - 5x
14.
15.
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)
23. (x - 3)3 / 24. (2x + 5)3

Name______Date ______

Day 2 Homework: Function Composition

1. f(x) = 3x – 5 g(x) = 2x + 1
Find f o g [this means f(g(x)] / 2. f(x) = x – 2 g(x) = x2 + 3
Find g o f / 3. f(x) = 5x – 6 g(x) = 3x
Find f(g(x))
4. f(x) = 2x2 + 11 g(x) = 4x
Find g(f(x)). / 5. f(x) = x2 – 9 g(x) = 3x2
Find f o g (x) / 6. f(x) = 5x2 g(x) = 6x – 2
Find f o f (x) and find g o g (x)
7. f(x) = -4x + 6 g(x) = 5x – 1
Find f o g (2) [this means f(g(2)] / 8. f(x) = 2x2 g(x) = x2 + 7
Find f(g(-1)) and find g(f(-2)) / 9. f(x) = -3x + 5 g(x) = -2x
Find f o g
10. f(x) = x – 3 g(x) = x2
Find g o f / 11. f(x) = 6x – 5 g(x) = -4x
Find f(g(x)). / 12. f(x) = x2 – 9 g(x) = 7x + 1
Find g(f(x)).
13. f(x) = x2 + 10 g(x) = 2x2
Find f(g(x)) / 14. f(x) = 3x2 g(x) = 4x – 3
Find f o f and find g o g / 15. f(x) = 4x – 6 g(x) = 3x – 2
Find f(g(3))
16. f(x) = x2 + 3x g(x) = x2 + 6
Find f(g(-2)). Also find g(f(-2)). / 17. f(x) = g(x) =
Find f o g and find g o f / 18. k(x) = h(x) =
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. 2.

#3-5 Solve each of the following by the zero product property (factor).

3. 3x2 – 27 = 04. 4x2 – 9 = 05. 2x2 – 2x – 12 = 0

II. PRACTICE

#6-9 Graph each of the following. Identify the Point of Inflection and describe the transformations.

6. 7.

Point of inflection______Point of inflection______

Horizontal shift: ______Horizontal shift:______

Vertical shift: ______Vertical shift: ______

Reflection: ______Reflection: ______

Stretch/shrink: ______Stretch/shrink: ______

8. 9.

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) ______

  1. 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 behaviorof the graph of the polynomial function.

5. 6.

Degree ______Degree ______

Leading Coefficient ______Leading Coefficient ______

#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.

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.

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 5Homework: Evaluating Polynomials with Synthetic Substitution
and The Fundamental Theorem of Algebra

I. REVIEW – Simplify

1. 2.

Solve by the Zero Product Property (factor)

3. 3x2 – 27 = 04. 4x2– 9 = 05. 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 SubstitutionSynthetic Substitution

7. f(x) = 8x4 + 12x3 + 6x2 – 5x + 9 for x = -2
Direct SubstitutionSynthetic Substitution

8. f(x) = x3 + 8x2 – 7x + 35 for x = –6
Direct SubstitutionSynthetic Substitution

9. f(x) = –8x3 + 14x – 35 for x = 4
Direct SubstitutionSynthetic Substitution

10. f(x) = –2x4 + 3x3 – 8x + 13 for x = 2
Direct SubstitutionSynthetic 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, 317. -5, -1, 2

18. 2, -i, i19. 2, -3i

20. 4, , 21. 3,

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:______

  1. Determine the number of turning points ______
  1. Identify all relative minimum/maximum points.

______

  1. Identify all absolute minimum/maximum points.

______

  1. Over what intervals is f(x) Decreasing______
  1. Over what intervals is f(x) Increasing______
  1. Describe the end behavior of the graph:

As, As ,

Name______Date ______

Day 6 Homework: Applying the Remainder and Factor Theorems

  1. REVIEW – Multiply
  1. 2.
  1. 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 DIVISIONLONG DIVISION

3. (x3 3x2 + 8x 5)  (x 1)

4. (x4 7x2 + 9x 10)  (x 2)

5. (2x4x3 + 4)  (x + 1)

  1. (2x4 11x3 + 15x2 + 6x 18)  (x 3)
  1. Factor the following polynomials completely using synthetic division and factoring.
  1. f(x) = x3 3x2 16x 12; given that (x 6) is a factor
  1. f(x) = x3 12x2 + 12x + 80; given that (x – 10) is a factor
  1. f(x) = x3 18x2 + 95x 126; given that (x – 9) is a factor
  1. f(x) = x3x2 21x + 45; given that (x + 5) is a factor
  1. f(x) = x3 + 2x2 20x + 24; given that 6 is a zero
  1. f(x) = 15x3 119x2 l0x + 16; given that 8 is a zero
  1. 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 -615. f(x) = x4 -7x2 +12

16. f (x) = x3 -9x2 +20x -1217. 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 + 1Degree _____ Leading Coefficient _____Constant ____

11. g(x) = 9x4 + x – 7Degree _____ 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 – 9as , as ,

17. f(x) = 6x5 + 7x4 – 8xas , as ,

18. f(x) = -8x3 – 9x2 + x – 4 as , as ,

19. f(x) = x4 + 16as , as ,

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. (fᵒg)(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 + 4834. 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)