3. Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.

4. COMPLEX NUMBER SYSTEM Natural Whole Integers RationalIrrational REAL NUMBER SYSTEM Imaginary Numbers

6. Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero. True for integral, rational, irrational and imaginary coefficients

7. Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero. True for integral, rational, irrational and imaginary coefficients

9. Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero. We are going to get integral, rational, irrational and imaginary ZEROS

11. Fundamental Theorem of Algebra Another Theorem

12. is a polynomial function of positive degree with complex coefficients, then P(x) has n linear factors.

14. Last Theorem Corrollary

15. Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

17. Find the Integral ZEROS only Determine the zeros of the polynomial

18. 13-3 NO Integral ZEROS

21. 13-3

22. 2 1 0 0 2 1 0 2 2 Determine the zeros of the polynomial

23. 2 1 0 0 2 1 0 22 Determine the zeros of the polynomial Stop Using Synthetic Division whenever the Depressed Polynomial is a Quadratic and Solve the Quadratic Equation Q(x) = 0 by Factoring, Completing the Square or the Quadratic Formula

25. 13-3 That’s why there was only ONE x-intercept

27. Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

29. Complex Conjugate Theorem If a complex number, a + bi, is a zero of a polynomial function with real coefficients, then its conjugate, a – bi, is also a zero of the polynomial

31. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 24-16 3 44-39 Leave more space between the numbers than you usually do

32. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i 24-16 3 44-39

33. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i 24-16 3 44-39

35. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i 24-16 3 44-39 -33 + 4i

36. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i 24-16 3 44-39

38. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 24-16 3 44 9 – 6i -35 – 6i -39

39. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 24-16 3 44 9 – 6i -39

41. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i 39 -39

42. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39

43. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39

44. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i Complex Conjugate Theorem

45. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i 3 9 – 6i 2

46. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i 9 – 6i 6 – 4i 32– 3

47. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

48. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

49. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

50. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3 Stop Using Synthetic Division whenever the Depressed Polynomial is a Quadratic and Solve the Quadratic Equation Q(x) = 0 by Factoring, Completing the Square or the Quadratic Formula

58. Determine all the zeros of the polynomial given one of the zeros. 3 + 2i 3 9 + 6i -7 + 6i -33 + 4i -9 + 4i -35 – 6i 0 24-16 3 44 39 9 – 6i -39 3 – 2i 9 – 6i 6 – 4i– 9 + 6i 032– 3

59. Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

63. Conjugate Radical Theorem If, is a zero of a polynomial function with rational coefficients, where a and b are rational, but is irrational, then the conjugate number is also a zero of the polynomial

65. Determine all the zeros of the polynomial given one of the zeros. 04 1 4–1–1 Leave more space between the numbers than you usually do

66. Determine all the zeros of the polynomial given one of the zeros. 1 04 1 4–1–1

67. 1 04 1 4–1–1

69. 1 04 1 4–1–1

70. 1 04 1 4–1–1

71. 1 04 1 4–1–1

73. 1 04 1 4–1–1 1 0

74. 1 04 1 4–1–1 1 0

75. 1 04 1 4–1–1 1 0

76. 1 04 1 4–1–1 1 0 Conjugate Radical Theorem

77. Determine all the zeros of the polynomial given one of the zeros. 1 04 1 4–1–1 1 0 10

78. 1 04 1 4–1–1 1 0 10 0 1

79. 1 04 1 4–1–1 1 0 10 0 1 0

80. 1 04 1 4–1–1 1 0 10 0 1 0

81. 1 04 1 4–1–1 1 0 10 0 1 0

82. 1 04 1 4–1–1 1 0 10 0 1 0 Stop Using Synthetic Division whenever the Depressed Polynomial is a Quadratic and Solve the Quadratic Equation Q(x) = 0 by Factoring, Completing the Square or the Quadratic Formula

83. Determine all the zeros of the polynomial given one of the zeros. 1 04 1 4–1–1 1 0 10 0 1 0

85. Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

86. Determine all the zeros of the polynomial given one of the zeros. Corollary: If P(x) is a polynomial function with complex coefficients and degree n, then P has exactly n zeros

89. Determine all remaining zeros of each polynomial given one of the zeros