1. 5.6 Complex Zeros; Fundamental Theorem of Algebra
MAT SPRING 2009
5.6 Complex Zeros; Fundamental Theorem of Algebra
In this section, we will study the following topics:
Conjugate Pairs Theorem
Finding a polynomial function given specified zeros
Finding complex zeros of a polynomial

2. Review: Complex Numbers (pp 109-114)
The set of COMPLEX NUMBERS includes all real and imaginary numbers.
The imaginary unit, i, is defined as
Thus, we have
Complex numbers are of the form:

3. Complex Zeros of a Polynomial Function
In section 5.1, we saw that an nth-degree polynomial could have AT MOST n real zeros. In this section, we will do better than that.
The following result is derived from the all-important FUNDAMENTAL THEOREM OF ALGEBRA.
In the complex number system, a polynomial of degree n (n ≥ 1) has exactly n complex zeros (though not necessarily distinct*).

4. Complex Zeros of a Polynomial Function
The zeros may be all real, all imaginary, or a combination.
It depends upon the degree of the polynomial and the individual function.
*Note that the number of zeros includes repeated zeros. In other words, a double zero counts as two zeros, a triple zero counts as three zeros, etc.
For example, the cubic polynomial function f(x) = (x – 2)3 has a triple zero at x = 2.

5. Complex Zeros of a Polynomial Function
Now that we can determine the number of zeros, we need to actually find those zeros.
We will use the same techniques as we did in sect. 5.5 to find the real zeros for higher-degree polynomials:
Use the Rational Zero Theorem and the graph to locate one or more rational zeros
Use synthetic division to find the depressed equation (keep dividing until the depressed equation is quadratic)
Use factoring, extracting the roots, or Quadratic Formula to solve the resulting quadratic equation

7. Complex Zeros of a Polynomial Function
Example#1*
Solution:
By the F.T. of A., we know that f has ____ complex zeros.
Start by listing all of the potential rational zeros:

8. Complex Zeros of a Polynomial Function
Example #1 (continued)
We can now either sub each of the possible (+) rational zeros into the polynomial or use synthetic division, until we find an actual rational zero.  OR
We can use our graphing calculators to help us locate one rational zero and then verify it is an actual zero using synthetic division. 
From the graph it looks like the rational zero is about ________.
Use synthetic division to verify that _______ is actually a zero.

9. Complex Zeros of a Polynomial Function
Example #1* (cont)
The remainder is ____, therefore x = ________ is a zero.
The depressed equation is _____________ = 0
Solve this quadratic equation for x to find the remaining two zeros.
So, the zeros of f(x) are ___________________________.

10. Complex Zeros of a Polynomial Function
Example #1 (continued)
The factored form of f(x) is

11. Complex Zeros of a Polynomial Function
Example #2

12. Complex Zeros of a Polynomial Function
Example #2 (cont)

13. Complex Zeros of a Polynomial Function
Example #3

14. Complex Zeros of a Polynomial Function
Example #3 (cont)

15. Conjugate Pairs Theorem
We can work backwards to find a polynomial with specified zeros.
But first……
Conjugate Pairs Theorem
Let f(x) be a polynomial whose coefficients are REAL numbers.
If is a zero of the function, the CONJUGATE is ALSO a zero of the function.
Complex zeros occur in conjugate pairs!!
e.g. If you know that – 3 + i is a zero of a given polynomial function (with real coefficients), you also know that _________ is a zero.

16. Finding a Polynomial Function with Given Zeros
MAT SPRING 2009
Finding a Polynomial Function with Given Zeros
Example #1*
Find a cubic polynomial with real coefficients that has zeros -1 and 6 + 5i
Solution:
We know that _______________ is also a zero.
Write each zero in factored form.
Distribute the negative to remove inner parentheses.
Multiply the trinomials.
Multiply the binomial x trinomial.

17. Finding a Polynomial Function with Given Zeros
Example #2
Find a quartic polynomial with integer coefficients that has zeros

19. Using Complex Conjugates to Find All Zeros
Here’s a neat example of how you can use the fact that complex zeros are conjugates to find all of the zeros of a polynomial function.
Example*

20. Using Complex Conjugates to Find All Zeros
MAT SPRING 2009
Using Complex Conjugates to Find All Zeros
Solution:
1. Use the fact that _________ is a zero and write each complex zero in factored form. Multiply the factors.
2. Divide this product into f to obtain the remaining factor.
So, the zeros of f(x) are ___________________________

21. MAT SPRING 2009
End of Section 5.6