1. Warm - Up Perform the operation and write the result in standard form
Use the quadratic formula to solve the quadratic equation

2. Assignment p. 266 # 11, 12, 16, 24, 26 Announcements
Test Corrections Tomorrow After School
Notebook Quiz Next Week

3. 2.6 Fundamental Theorem of Algebra
How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function and find all zeros of polynomial functions, including complex zeros
How to find conjugate pairs of complex zeros
How to find zeros of polynomials by factoring

4. The Fundamental Theorem of Algebra
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.
Does it have to cross
the x-axis to be a zero?

5. Linear Factorization Theorem.
The highest degree of a polynomial is how many complex zeros we should have.
How many zeros for the following?
x3 + 2x 3
3x2 + 2x4 + x + 5 4
7x4 - 2x5 + x6 – 4x 6

6. Find all the zeros of the function and write the polynomial as a product of linear factors
Solve x3 + 6x – 7 = 0
Possible zeros:
P = ± 7, ±1
Q = ±1
(p/q) = ± 7, ±1
How do we find the possible rational zeros?
Factor Theorem – please test the four possibilities
X = 1

7. Synthetic Division to simplify x3 + 6x – 7, x = 16 -7 7
Now, what do we do once we have a quadratic equation?

8. Using the Quadratic Formula

9. So…what are the roots?
Using the factor theorem, synthetic division, and the quadratic formula, we have:
x3 + 6x – 7 =

10. Steps for finding the zeros
Use the Rational root test to find the possible rational zeros
Use the factor theorem to test the possible rational zeros
Use synthetic division until you find a quadratic equation
Use the quadratic formula to find the complex zeros

11. Example: find all the zeros
f(x) = x4 - 3x3 + x – 3
Use the Rational root test to find the possible rational zeros
P = ±3, ±1
Q =± 1
(p/q) = ±3, ±1

12. Find the zeros f(x) = x4 - 3x3 + x – 3 Which ones work? X = -1, 3
Use the factor theorem to test the possible rational zeros
(p/q) = ±3, ±1
f(x) = x4 - 3x3 + x – 3
Which ones work?
X = -1, 3

13. Use synthetic division until you find a quadratic equation
X = -1 1 -3 -1 4 -4 3

14. Use synthetic division until you find a quadratic equation
X = 3 1 -4 4 -3 3 -1

15. Use the quadratic formula to find the complex zeros

16. Again, what are our zeros?
f(x) = x4 - 3x3 + x – 3
Highest degree tells us there must be 4 zeros

17. Warm - Up
Find all the zeros of the function and write the polynomials as a product of linear factors

18. Announcements p. 266 # 38 – 44 even, 51 – 55 odd
Assignment
p. 266
# 38 – 44 even, 51 – 55 odd
Test Corrections today after school
Notebook Quiz Next Week

19. Conjugate Pair
whenever a + bi is a zero of f, a – bi is also a zero of f and vice versa.
Example:
Note that in Example 1, the two complex zeros were conjugates.

20. Name those conjugate pairs!!!

21. Find a polynomial function with integer coefficients that has the given zeros
Conjugates
Factor notation
Rearrange
FOIL
Combine like terms
Simplify

23. Find a fourth degree polynomial function with real coefficients that has 0, 1, and i as zeros.
How do we write the three initial zeros?
f(x) = x(x – 3)(x – i)
What is the fourth zero? (remember that complex numbers come in conjugates)
(x + i)
f(x) = x(x – 1)(x – i)(x + i)
= (x2 – x)(x2 – 1).
= x4 – x3 - x2 + x.

24. Find a polynomial function with integer coefficients that has the given zeros

25. Warm Up How do we find the conjugate pair of a complex number?
How do we multiply the factored form of complex numbers?

26. Announcements

27. Factoring a Polynomial
Factor f(x) = x4 –12x2– 13
a) factor without square roots
(x2 – 13)(x2 + 1)
b) factor with square roots.
c) factor completely

28. Finding the zeros of a polynomial
Find all zeros of f(x) = x4 – 4x x2 + 4x – 13
given that (2 + 3i) is a zero.
Knowing about conjugates, what must be another zero?
Since 2 + 3i is a zero, 2 – 3i is also a zero. That means that x2 – 4x + 13 is a factor of f(x).

29. Finding the zeros
Given x2 – 4x + 13 is a factor of f(x) = x4 – 4x3 + 12x2 + 4x – 13
What type of division could we use to find the other factors?
Long Division

30. Find the remaining zeros

31. …and so the zeros are? Zeros of f(x) = x4 – 4x3 + 12x2 + 4x – 13
After using complex conjugates, long division and factoring
All the zeros of f are
–1, 1, 2 + 3i, 2 – 3i.