1. 2.4 – Zeros of Polynomial Functions

2. Rational Zero Theorem:
If f is a polynomial function of the form
f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0
with degree n > 1, integer coefficients, and a0 ≠ 0, then every rational zero of f has the
form p/q, where p is all possible factors of
a0 and q is all possible factors of an.

3. Ex. 1 List all possible rational zeros of each function
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9

4. Ex. 1 List all possible rational zeros of each function
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1

5. Ex. 1 List all possible rational zeros of each function
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1 Test all possibilities using the Factor Theorem.

6. Ex. 1 List all possible rational zeros of each function
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1 Test all possibilities using the Factor Theorem. g(9) = (9)4 + 4(9)3 – 12(9) – 9 g(-9) = (-9)4 + 4(-9)3 – 12(-9) – 9 g(3) = (3)4 + 4(3)3 – 12(3) – 9 g(-3) = (-3)4 + 4(-3)3 – 12(-3) – 9 g(1) = (1)4 + 4(1)3 – 12(1) – 9 g(-1) = (-1)4 + 4(-1)3 – 12(-1) – 9

7. Ex. 1 List all possible rational zeros of each function
Ex. 1 List all possible rational zeros of each function. Then determine which, if any, are zeros. g(x) = x4 + 4x3 – 12x – 9 p = ±9, ±3, ±1 = ±9, ±3, ±1 (six possible) q ±1 Test all possibilities using the Factor Theorem. g(9) = (9)4 + 4(9)3 – 12(9) – 9 = 9360 g(-9) = (-9)4 + 4(-9)3 – 12(-9) – 9 = 3744 g(3) = (3)4 + 4(3)3 – 12(3) – 9 = 144 g(-3) = (-3)4 + 4(-3)3 – 12(-3) – 9 = 0 g(1) = (1)4 + 4(1)3 – 12(1) – 9 = -16 g(-1) = (-1)4 + 4(-1)3 – 12(-1) – 9 = 0

8. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0

9. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0
Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator.

10. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0
Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3

11. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0
Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3 *Divide to find remaining zero.

12. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0
Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3 *Divide to find remaining zero. -2| |

13. Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0
Ex. 2 Solve the equation. x4 + 2x3 – 7x2 – 20x – 12 = 0 *Find real zeros using graphing calculator. x = -2, x = -1, x = 3 *Divide to find remaining zero. -2| | 0 x3 – 7x – 6

14. -1| |

15. -1| | 0 x2 – x – 6

16. -1| 1 0 -7 -6 -1 1 6 1 -1 -6 | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0

17. -1| | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0 (x + 2)(x + 1)(x – 3)(x + 2) = 0

18. -1| | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0 (x + 2)(x + 1)(x – 3)(x + 2) = 0 (x + 1)(x – 3)(x + 2)2 = 0

19. -1| | 0 x2 – x – 6 So (x + 2)(x + 1)(x2 – x – 6) = 0 (x + 2)(x + 1)(x – 3)(x + 2) = 0 (x + 1)(x – 3)(x + 2)2 = 0 So x = -1, x = 3, and x = -2 (twice).

20. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48

21. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator.

22. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4

23. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4 *Divide to find remaining zero.

24. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4 *Divide to find remaining zero. -2| |

25. Ex. 3 Write each function as (a) the product of linear and irreducible factors and (b) the product of linear factors. Then (c) list all of its zeros. g(x) = x4 – 3x3 – 12x2 + 20x + 48 *Find real zeros on graphing calculator. x = -2, x = 3, x = 4 *Divide to find remaining zero. -2| | 0

26. 3| |

27. 3| | 0

28. 3| | 0 x2 – 2x – 8

29. 3| 1 -5 -2 24 3 -6 -24 1 -2 -8 | 0 x2 – 2x – 8 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8)

30. 3| | 0 x2 – 2x – 8 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4)

31. 3| | 0 x2 – 2x – 8 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4) = (x + 2)2(x – 3)(x – 4)

32. 3| | 0 x2 – 2x – 8 So g(x) = (x + 2)(x – 3)(x2 – 2x – 8) = (x + 2)(x – 3)(x + 2)(x – 4) (a) & (b) = (x + 2)2(x – 3)(x – 4) (c) x = -2 (twice), x = 3, x = 4

33. Ex. 4 Use the given zero to find all complex zeros of each function
Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function. h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x ; -5i

34. Ex. 4 Use the given zero to find all complex zeros of each function
Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function. h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x ; -5i -5i| |

35. Ex. 4 Use the given zero to find all complex zeros of each function
Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function.
h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x ; -5i
-5i|
-15i i i i
i i i i | 0

36. Ex. 4 Use the given zero to find all complex zeros of each function
Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function.
h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x ; -5i
-5i|
-15i i i i
i i i i | 0
5i| i i i i |

37. Ex. 4 Use the given zero to find all complex zeros of each function
Ex. 4 Use the given zero to find all complex zeros of each function. Then write the linear factorization of the function.
h(x) = 3x5 – 5x4 – 13x3 – 65x2 – 2200x ; -5i
-5i|
-15i i i i
i i i i | 0
5i| i i i i
+15i -25i i i
| 0

38. x3 – 5x2 – 88x + 60

39. 3 -5 -88 60 3x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator.

40. 3 -5 -88 60 3x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3

41. x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3 So x = -5i, x = 5i, x = 5, x = 6, x = 2/3

42. x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3 So x = -5i, x = 5i, x = 5, x = 6, x = 2/3 And (x + 5i)(x – 5i)(x – 5)(x – 6)(x – 2/3)

43. x3 – 5x2 – 88x + 60 *Find real zeros on graphing calculator. x = 5, x = 6, x = 2/3 So x = -5i, x = 5i, x = 5, x = 6, x = 2/3 And (x + 5i)(x – 5i)(x – 5)(x – 6)(x – 2/3) or (x + 5i)(x – 5i)(x – 5)(x – 6)(3x – 2)