Review for Chapter 3 Test Name: ______________________ Polynomial Functions Circle the Polynomial Functions.

1. y = x2 2. y = 3x5 + 2x2 + x1/2 + 6

3. y = –2x3 – x–2 + x + 1 4. y = (x + 4) (x – 3) (x + 1)

Find the End Behavior. Function Degree Lead Coeff. End Behavior

5 f (x )= 4x 3 − 2x 2 −3x +6 x →−∞, f(x)→ _____

x →∞, f(x)→ _____

6 g(x )= x 2(x −5)(x +3) x →−∞, f(x)→ _____

x →∞, f(x)→ _____

7 g(x )= −6(x +1)2(x −3) x →−∞, f(x)→ _____

x →∞, f(x)→ _____

8 f (x )= −3x 4 + 4x 3 +7 x →−∞, f(x)→ _____

x →∞, f(x)→ _____ Find all intercepts. 9. Find ALL the intercepts of: f(x) = (x – 2) (x + 3) (x – 1)2 ________ ________ ________ ________

10. Find ALL the intercepts of: f(x) = x2 (x + 5) 2 (x + 2) ________ ________ ________ ________

Given the information provided, create a polynomial function in STANDARD FORM. 11. Zeros: 3 (multiplicity 2) Degree: 4 –1 (multiplicity 2) Function: ____________________________________________

12. Zeros: 3, –1, 2i Degree: 4 Function: ____________________________________________ Determine the zeros, multiplicity at each zero, and whether the graph bounces or crosses at each zero. 13. g(x )= x(5x − 2)2(x +18) Zeros: _____ _____ _____ _____ Mult: B/C: *there may be fewer than 4 zeros

14. f (x )= x 2(x +3)(x − 4)3 Zeros: _____ _____ _____ _____ Mult: B/C: *there may be fewer than 4 zeros

15. Find all the REAL zeros of f(x). f (x )= 3x(x 2 + 4)2(x 2 −1) Real Zeros: _________________

16. Find ALL zeros of f(x). f (x )= −2(x +7)3(x 2 − 2)(x 2 + 4) All Zeros: _________________

Answer the questions below. Show all work. 17. f(x) = x3 – 2x2 – 21x – 18 Is 6 a zero?

18. f(x) = x3 – 9x2 + 8x + 60 Is x = 5 a solution?

19. f(x) = 2x3 – 9x2 – 5 Does f(2) = 0?

20. f(x) = x3 – x2 – 21x – 45 Is (x + 3) is a factor?

Finding zeros of f(x). 21. Given x = –2 is a solution of f(x) = x3 + 2x2 + 5x + 10, find all solutions.

22. f(x) = x3 – 9x2 + 25x – 25. Find all zeros, provided that one zero is 5.

23. f(x) = x3 – 10x2 + 29x – 26. Find all zeros given that (x – 2) is a factor of f(x).

24. List the possible rational zeros of the polynomial function. f(x) = x4 – x3 + 3x2 + 4x – 15 Possible Rational Zeros: _________________________________

25. List the possible rational zeros of the polynomial function. f(x) = 3x3 + 4x2 + x – 10 Possible Rational Zeros: _________________________________

26. Find ALL zeros of the polynomial function.

f(x) = 2x3 + 3x2 – 39x – 20

27. Find the REAL zeros of the polynomial function.

f(x) = 2x4 + x3 + 3x2 + 2x – 2

28. Find the real zeros of f(x) = 4x3 – 12x2 – x + 15.

29. Find the quotient and remainder for (–8x3 + 50x – 45) and (2x – 3).

30. Is (2x – 3) a factor of (–8x3 + 50x – 45)? Explain.

31. Function Vertical

Asymptote Horizontal Asymptote

Slant Asymptote

Hole

A f(x) =

2x −13x

Yes or No ( , )

B f(x) =

5xx 2 − 4

Yes or No ( , )

C f(x) =

x 4 + x 3 − 2(x −3)(x +1)

Yes or No ( , )

D f(x) =

xx 2 −1

Yes or No ( , )

E f(x) =

x 2 + 4x − 5

Yes or No ( , )

F f(x) =

2(x −1)(x +3)(x + 6)(x + 3)

Yes or No ( , )

G f(x) =

(x +3)(x − 2)(x − 2)(x + 4)

Yes or No ( , )

H f(x) =

(x −5)(x +3)2

(x +1)(x + 2) Yes or No ( , )

Find the slant asymptote.

32. f(x) =

x 3 + 4x 2 −6x +5x 2 +3

33. f(x) =

2x 2 +5x −124x + 4

34. f (x)= x2 − 4

x2 + x− 2 Removable Discontinuity: _____________

VA:

Domain:

HA:

Range:

Table

Graph

35. f (x) = x2 + 4x + 3

x + 2 Removable Discontinuity: _____________

VA:

Domain:

HA:

Range:

Table

Graph

Solve the Inequalities. 37. x3 + 2x2 – 3x > 0 38. x4 > 1

40. x + 12

x ≤ 7 41.

1x − 2

≤ 23x −9

Word Problems 42. The perimeter of a rectangle is 46 feet. Express its area A as a function of the width of a side.

43. A projectile is launched upward from the ground. It’s height, s, in feet above the ground is given by the equation s = 64t – 16t2. After how many seconds in the air will it hit the ground?

44. An open box is to be constructed from a rectangular piece of sheet metal whose length is twice its width by removing a square of side 1 foot from each corner and turning up the edges. If the box is to hold 4 cubic feet, what should be the dimensions of the sheet metal?

Question Answer A Answer B 1 Polynomial or not: f(x) = 2x4 + 4x2 + x-1 Yes No

2 Polynomial or not: f(x) = x3 + x1/2 Yes No

3 End Behavior: f(x) = –3x4 – 2x + 5 x → −∞, f (x )→ −∞

x →∞, f (x )→ −∞ x → −∞, f (x )→ −∞

x →∞, f (x )→∞

4 End Behavior: f(x) = 2x5 + 3x2 – 2x + 1 x → −∞, f (x )→∞

x →∞, f (x )→∞ x → −∞, f (x )→ −∞

x →∞, f (x )→∞

5 Find Real Zeros: f(x) = 3 (x + 2) (x – 4) (0, -2) (0, 4) (-2, 0) (4, 0)

6 Find Real Zeros: f(x) = x (x2 + 4) (x + 1)2 x = 0, –1 x = 0, ± 2, –1

7 Find All Zeros: f(x) = –2x (x2 + 6) 2 (x – 7) x = 0, ±i 6 , 7 x = 0, ± 6 , 7

8 Multiplicity of 3: f(x) = (x – 2) 2 (x – 3) Multiplicity = 3 Multiplicity = 1

9 Bounce/Cross at 4: f(x) = x3 (x – 4)2 Bounce Cross

10 Bounce/Cross at 0: f(x) = x3 (x – 1)2 Bounce Cross

11 Function with zeros at 3 and 1 f(x) = (x – 3) (x – 1) f(x) = (x + 3) (x + 1)

12 Given –2 + 3i is a zero, another zero is: –2 – 3i 2 – 3i

13 Given 5 is a zero, another zero is: −1+ 5 − 5

14 Given (x – 4), when using synthetic div, what goes on the outside of the “L”? 4 –4

15 Given f(2) = 0, when using synthetic div, what goes on the outside of the “L”? 2 –2

16 Given –3 is a zero, when using synthetic div, what goes on the outside of the “L”? 3 –3

17 (x – c) is a factor of a polynomial if… If the remainder is c If the remainder is 0

18 f(x) =

3xx 2 − 25

, VA is… x = 5 x = ± 5

19 f(x) =

x −3(x + 4)(x −3)

, VA is… x = –4 x = –4, 3

20 f(x) =

8x + 22x 2 −1

, HA is… y = 0 y = 4

21 f(x) =

x 2 + 2x −53x 2 + 4

, HA is… y = 1 y =

13

22 f(x) =

x 4 + 2x 3 + x −3x 2 − 4

Slant Asymptote No Slant Asympote

23 f(x) =

x −3(x + 4)(x −3)

, Find the hole (3, 0) (3,

17

)

24 Possible rational zeros: f(x) = 2x4 + x – 5 ± 1,2,5{ }

± 1, 1

2 , 5, 52{ }

1. Describe the steps to finding all real zeros (given nothing) 2. Describe the Factor Theorem. 3. What is the Rational Root Theorem? When do you use it?