1. The Fundamental Theorem of Algebra (Section 2-5)

2. The Fundamental Theorem of Algebra Linear Factorization Theorem
If f(x) is a polynomial function of degree n, where n > 0, then f has at least one zero in the complex number system.
Linear Factorization Theorem
This leads to the conclusion that if f(x) is a polynomial function of degree n, where n > 0, then f has exactly n linear factors f(x) = an(x-c1)(x-c2)…(x-cn), where c1, c2,…cn are complex zeros.
Remember that the zeros are part of the real or complex numbers and can be repeated.

3. 1 2 3 4 Hints to find zeros Use any given (Synthetic division) Factor
Possible rational zeros p/q
Descartes’ Rule of Signs 3
Quadratic formula 4

4. Find all the zeros of the function.
Example 1 f(x) = (x – 3)(x + 1)

5. Find all the zeros of the function.
Example 2 g(x) = (x + 4)(x + 3i)(x – 3i)

6. Example 3 Confirm that f(x) = x3 + 4x has exactly 3 zeros.

7. Example 4
Confirm that f(x) = x3 + 4x2 + 9x + 36 has exactly 3 zeros (one of which is x = -4).

8. Find all the zeros of the function and write the polynomial as a product of linear factors.
Example 5 f(x) =x2 + 10x + 23

9. Find all the zeros of the function and write the polynomial as a product of linear factors.
Example 6 f(x) =3x3 – 2x2 + 75x – 50

10. Find all the zeros of the function and write the polynomial as a product of linear factors.
Example 7 f(x) =x5 + x3 + 2x2 - 12x + 8

11. (a) Find all the zeros of the function, (b) write the polynomial as a product of linear factors and (c) Use your factorization to determine the x-intercepts of the graph of the function.
Example 8 f(x) =x3 + 10x2 + 33x + 34

12. HW #29 pg 144-145 (1-35 every other odd)

13. Conjugate Pairs Theorem
If f is a polynomial function with real coefficients, then whenever a+bi is a zero, a-bi is also a zero of f.

14. Find a polynomial function with real coefficients that has the given zeros.
Example , -1, 2 + 5i

15. Find a polynomial function with real coefficients that has the given zeros.
Example , -2, 2i

16. Use the given zero to find all the zeros of the function.
Example 11 f(x) = x4 – 3x3 + 6x2 + 2x – 60; 1 – 3i

17. Use the given zero to find all the zeros of the function.
Example 12 f(x) =4x3 +23x2 + 34x – i

18. HW #30 pg (37-41 odd, odd)