1. Fundamental Theorem of Algebra
Unit 2 -- Polynomials
Fundamental Theorem of
Algebra

2. Zeros of a Polynomial Function
A Polynomial Function is usually written in function notation or in terms of x and y.
The Zeros of a Polynomial Function are the solutions to the equation you get when you set the polynomial equal to zero.

3. Factors, Roots, Zeros For our Polynomial Function:
The Factors are: (x + 5) & (x - 3)
The Roots/Solutions are: x = -5 and 3
The Zeros are at: (-5, 0) and (3, 0)

4. Roots & Zeros of Polynomials II
Finding the Roots/Zeros of Polynomials:
The Fundamental Theorem of Algebra
Descartes’ Rule of Signs
The Complex Conjugate Theorem

5. Fundamental Theorem Of Algebra
Every Polynomial Equation with a degree higher than zero has at least one root in the set of Complex Numbers.
A Polynomial Equation of the form P(x) = 0 of degree ‘n’ with complex coefficients has exactly ‘n’ Roots in the set of Complex Numbers.
COROLLARY:

6. Real/Imaginary Roots
If a polynomial has ‘n’ complex roots will its graph have ‘n’ x-intercepts?
By Fund Theorem of Alg Corollary, degree n = 3 means 3 roots in the complex system.
Factor and solve the 3 roots x = 0, x = 2 and x = -2.
3 x-intercepts

7. Real/Imaginary Roots
Just because a polynomial has ‘n’ complex roots doesn’t mean that they are all Real!
Again, by Fund Theorem of Alg Corollary, degree n = 3 means 3 roots in the complex system.
How many x-intercepts?
What are the other roots?

8. Descartes’ Rule of Signs
Arrange the terms of the polynomial P(x) in descending degree:
The number of times the coefficients of the terms of P(x) change sign = the number of Positive Real Roots (or less by any even number)
The number of times the coefficients of the terms of P(-x) change sign = the number of Negative Real Roots (or less by any even number)
In the examples that follow, use Descartes’ Rule of Signs to predict the number of + and - Real Roots!

9. Find Roots/Zeros of a Polynomial
We can find the Roots or Zeros of a polynomial by setting the polynomial equal to 0 and factoring.
Some are easier to factor than others!
The roots are: 0, -2, 2

10. Complex Conjugates Theorem
If complex roots/zeros are not Real, then they have an Imaginary component. Complex roots with Imaginary components always exist in Conjugate Pairs.
If a + bi (b ≠ 0) is a zero of a polynomial function, then its Conjugate, a - bi, is also a zero of the function.

11. Find Roots/Zeros of a Polynomial
If you know one imaginary root, you can use the Complex Conjugates Theorem to determine another root.
Ex: Find all the roots of
If one root is 4 - i.
By Complex Conjugate Theorem, we know 4 + i is another root.

12. Example (con’t) Ex: Find all the roots of
The root 4 – i has the factor [x - (4 - i)].
The root 4 + i has the factor [x - (4 + i)].
Multiply these factors:

13. Example (con’t) Ex: Find all the roots of If one root is 4 - i.
If the product of the two non-real factors is
then the third factor (that gives us the neg. real root) is the quotient of P(x) divided by :
The third root is x = -3

14. Complex Zeros; Fundamental Theorem of Algebra
Section 5.6 –
Complex Zeros; Fundamental Theorem of Algebra
Find the remaining complex zeros of the given polynomial functions

15. Complex Zeros; Fundamental Theorem of Algebra
Section 5.6 –
Complex Zeros; Fundamental Theorem of Algebra
Long Division

16. Complex Zeros; Fundamental Theorem of Algebra
Section 5.6 –
Complex Zeros; Fundamental Theorem of Algebra
Examples
4) A polynomial function of degree 4 has 2 with a zero multiplicity of 2 and 2 – i as it zeros. What is the function?