1. 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 100 Remainder/ Factor Theorem End Behavior Zeros PolynomialsAsymptotes

2. Remainder/Factor Theorem 100 Main Get Answer Use the Remainder Theorem to find f(3) for f(x) = 4x 4 – 2x 3 – 10x 2 - 10 A. -10B. -60 C. 125D. 170

3. Use the Remainder Theorem to find f(3) for f(x) = 4x 4 – 2x 3 – 10x 2 - 10 A. -10B. -60 C. 125D. 170 Main Remainder/Factor Theorem 100

4. Main Get Answer Divide 2x 3 + 5x 2 – 7x – 1 by (2x+3) Remainder/Factor Theorem 200

5. Main Remainder/Factor Theorem 200 Divide 2x 3 + 5x 2 – 7x – 1 by (2x+3) x 2 + x – 5 + _14__ (2x+3)

6. Remainder/Factor Theorem 300 Main Get Answer Divide 3x 3 + 16x 2 + 21x + 22 by (x+4)

7. Remainder/Factor Theorem 300 Main 3x 2 + 4x + 5 + _2__ (x+4) Divide 3x 3 + 16x 2 + 21x + 22 by (x+4)

8. Remainder/Factor Theorem 400 If I were to look in the dictionary under the words “greatest” and “math teacher”, whose name would I see? Main Get Answer

9. Remainder/Factor Theorem 400 If I were to look in the dictionary under the words “greatest” and “math teacher”, whose name would I see? KAPLAN ! Come on guys, that was the easiest 400 points in the game! Main

10. Remainder/Factor Theorem 500 Main Get Answer Determine if (x – 2) is a factor of: f(x) = 4x 3 – 9x 2 – 3x + 12

11. Remainder/Factor Theorem 500 Main No, but you must prove it with synthetic division for your points! Determine if (x – 2) is a factor of: f(x) = 4x 3 – 9x 2 – 3x + 12

12. End Behavior 100 Main Get Answer Describe the end behavior of f(x) = -6x 17 + 5x 4 – 8x 2 + 10 As x  +, f(x)  ______ As x  -, f(x)  ______

13. End Behavior 100 Main Describe the end behavior of f(x) = -6x 17 + 5x 4 – 8x 2 + 10 As x  +, f(x)  ______ As x  -, f(x)  ______

14. End Behavior 200 Main Get Answer Describe the end behavior of f(x) = 6x 38 + 5x 3 – 8x + 11 As x  +, f(x)  ______ As x  -, f(x)  ______

15. End Behavior 200 Main Describe the end behavior of f(x) = 6x 38 + 5x 3 – 8x + 11 As x  +, f(x)  ______ As x  -, f(x)  ______

16. End Behavior 300 Main Get Answer Describe the end behavior of f(x) = -x 156 + x 3 – x As x  +, f(x)  ______ As x  -, f(x)  ______ Name one zero. ________

17. End Behavior 300 Main Describe the end behavior of f(x) = -x 156 + x 3 – x As x  +, f(x)  ______ As x  -, f(x)  ______ Name one zero. ________ x = 0

18. End Behavior 400 Main Get Answer Sketch the graph. How many turning points? What are the x-intercepts? f(x) = x 3 – 4x 2 + 4x

19. End Behavior 400 Main Sketch the graph. How many turning points? What are the x-intercepts? f(x) = x 3 – 4x 2 + 4x  (0, 0) and (2, 0) (factor and set factors to 0— What about multiplicity?)  Think about your ends.

20. End Behavior 500 Main Get Answer What is your favorite subject? a)Advanced b) Advanced Algebra Algebra c) Adv. Alg. d) Math – specifically Adv. Algebra

21. End Behavior 500 Main Easy choice! Of course no other subject was even a contender! What is your favorite subject? a)Advanced b) Advanced Algebra Algebra c) Adv. Alg. d) Math – specifically Adv. Algebra

22. Zeros 100 Main Get Answer If a factor has an EVEN exponent, what is special about the root when graphing? _____________________________ If a factor has an ODD exponent, what is special about the root when graphing? 

23. Zeros 100 Main If a factor has an EVEN exponent, what is special about the root when graphing? _____________________________ If a factor has an ODD exponent, what is special about the root when graphing?  It is a vertex! It crosses the x-axis!

24. Zeros 200 Main Get Answer Find all of the possible rational zeros of: f (x) = 3x 5 + 2x 4 – 7x 2 – 9x + 5

25. Zeros 200 Main Find all of the possible rational zeros of: f (x) = 3x 5 + 2x 4 – 7x 2 – 9x + 5       

26. Zeros 300 Main Get Answer Use the Zero Location Theorem to find the interval where there is a zero.. f (x) = x 3 – x – 2 x y 0 1 2 3 This is a clue!

27. Zeros 300 Main Use the Zero Location Theorem to find the interval where there is a zero.. f (x) = x 3 – x – 2 x y -2 -8 -1 -2 0 -2 1-2 24 322 Goes from negative to positive So we have a zero in between here!

28. Zeros 400 Main Get Answer Find the lowest integer upper bound zero for the polynomial: f (x) = x 3 + 3x 2 – 6x – 6

29. Zeros 400 Main Find the lowest integer upper bound zero for the polynomial: f (x) = x 3 + 3x 2 – 6x – 6 The possible zeros are: 1, 2, 3, 6       All positive so we have our upper bound! Upper bound is x = 2.

30. Zeros 500 Main Get Answer Find the highest integer lower bound zero for the polynomial: f (x) = -4x 4 + 12x 3 + 3x 2 – 12x – 7

31. Zeros 500 Main Find the highest integer lower bound zero for the polynomial: f (x) = -4x 4 + 12x 3 + 3x 2 – 12x + 7 The possible zeros are: 1, 7,,    The bottom row alternates in sign so we have found the lower bound! Lower bound is x = -1.  

32. Polynomials 100 Main Get Answer At most, how many roots does the following polynomial have? f(x) = 5x 4 – 2x 3 + x 2 - 7

33. Polynomials 100 Main At most, how many roots does the following polynomial have? f(x) = 5x 4 – 2x 3 + x 2 - 7 

34. Polynomials 200 Main Get Answer Find the polynomial of least degree given the roots: 1, -1, 3, -3

35. Polynomials 200 Main Find the polynomial of least degree given the roots: 1, -1, 3, -3 (x 2 – 1)(x 2 – 9) = x 4 – 10x 2 + 9

36. Polynomials 300 Main Get Answer What is the complex conjugate of (3 + 7i)?

37. Polynomials 300 Main What is the complex conjugate of (3 + 7i)? (3 – 7i)

38. Polynomials 400 MainGet Answer Find the reduced polynomial of f(x) = x 3 – 4x 2 – 6x - 36 if (x – 6) is a known factor.

39. Polynomials 400 Main Find the reduced polynomial of f(x) = x 3 – 4x 2 – 6x - 36 if (x – 6) is a known factor. You must divide! Reduced polynomial is: (x 2 + 2x + 6)

40. Polynomials 500 MainGet Answer Solve and sketch f(x) = (x – 4)(x 2 – 3x – 4)

41. Polynomials 500 Main Solve and sketch f(x) = (x – 4)(x 2 – 3x – 4) f (x) = (x-4)(x-4)(x+1) = (x-4) 2 (x+1) x = 4, -1

42. Asymptotes 100 MainGet Answer Name the asymptotes of: y = 6. 3x + 4

43. Asymptotes 100 Main Name the asymptotes of: y = 6. 3x + 4 Vertical asymptote is what makes the denominator zero! x = - 3. 4 Horizontal asymptote is the x-axis! y = 0.

44. Asymptotes 200 Main Get Answer Name the asymptotes of: y = 4x + 9. 12x - 4

45. Asymptotes 200 Main Name the asymptotes of: y = 4x + 9. 12x - 4 Vertical asymptote is what makes the denominator zero! x = 1. 3 Horizontal asymptote is the ratio of the x-coefficients! y = 1. 3.

46. Asymptotes 300 Main Get Answer Graph the hyperbola. Name the domain & range. Name the vertical and horizontal asymptotes. y = 3x. x + 2

47. Asymptotes 300 Main Graph the hyperbola. Name the domain & range. Name the vertical and horizontal asymptotes. y = 3x. x + 2 D: x = -2 R: y = 3 Vertical asymptote: x = -2 Horizontal asymptote : y = 3

48. Asymptotes 400 Main Get Answer Name the asymptotes and any holes. y = x 2 – 3x. x – 3

49. Asymptotes 400 Main Name the asymptotes and any holes. y = x 2 – 3x. x – 3 y = x(x – 3) (x – 3) Hole: (3, 3) V.A.: None H.A.: none S.A.: y = x

50. Asymptotes 500 Main Get Answer Graph the rational function. Name the domain. Name all asymptotes and any holes. y = x 2 – x – 12. x 2 – 2x – 8

51. Asymptotes 500 Main Graph the rational function. Name the domain. Name all asymptotes and any holes. y = x 2 – x – 12. x 2 – 2x – 8 = (x–4)(x+3). (x-4)(x+2) Hole: (4, 7/6 ) H.A.: y = 1 V.A.: x = -2 x-int: (3, 0) y-int: (0, 3/2) x y -1 2