1. Section 5.6 Complex Zeros; Fundamental Theorem of Algebra

6. Example 1 A polynomial of degree 5 whose coefficients are real numbers has the zeros -2, -3i, and 2+4i. Find the remaining two zeros.

7. By the conjugate pairs theorem, if x = -3i and x = 2+4i then x = 3i and x = 2-4i

8. Example 2: Find the remaining zeros of f. Degree 5; zeros: 1, i, 2i The degree indicates the number of zeros a polynomial has. We have three zeros, so we are missing two!!! By the Conjugate Pairs Theorem Remaining zeros: -i, -2i

10. Example 3 Find a polynomial f of degree 4 whose coefficients are real numbers and that has the zeros 1, 1, 4+i.

11. By the Conjugate Pairs Theorem

15. Example 4: Use the given zero to find the remaining zeros of each function. Note:

16. Zeros: 2i, -2i, 4

17. Example 5

18. Note: The resulting quadratic equation can not be factored since there is no number that multiplied gives you five and at the same time added gives you two.

19. We have to use completing the square or the quadratic formula Quadratic Formula Substitute

20. Simplify and solve! Complex zeros: Real zeros: Remember the problem only asks for the complex zeros!