1. The Fundamental Theorem of Algebra
Honors Precalculus
February 16-17, 2017
Mrs. Agnew

2. Essential Question Essential Vocabulary
How can we use the Fundamental Theorem of Algebra to help find the zeros of a polynomial?
Essential Vocabulary
Fundamental Theorem of Algebra
Zero of a polynomial
Descartes’ Rule of Signs

3. Zeros of Polynomials What are the zeros of a polynomial?
How do you find the zeros of a polynomial?
How are the zeros of a polynomial and its graph related?
How are the zeros of a polynomial and the degree of a polynomial related?

4. Imaginary & Irrational Roots
If a polynomial P has real coefficients…
(1) if a + bi is a zero of P, a – bi is also a zero.
(2) if a + √b is a zero of P, a – √b is also a zero.
In other words, imaginary zeros and irrational zeros “come in conjugate pairs” (given real coefficients).
Examples…

5. Fundamental Theorem
The Fundamental Theorem of Algebra states that a polynomial of degree n with real number coefficients has n linear factors
In other words, polynomial of degree n has n zeros/roots
Counting multiple zeros
Stewart GP: Page 307 #40 – 52 (Even)
Applications: page 271 #76, 78, 79

6. Descartes’ Rule of Signs
Describes possible number of positive real zeros, negative real zeros, and imaginary zeros
Each “row” must add to degree of polynomial
GP: Page 288 #54, 56, 66 – 70 (E)

7. Intermediate Value Theorem
Let f denote a continuous function. If a < b, and if f(a) and f(b) are of opposite sign, then f has at least one zero between a and b.
Notice the theorem is an EXISTENCE theorem… it guarantees the existence of a zero but does not tell you have to find it.
Use the theorem to prove a root exists…

8. Homework: 2/20/14
page 211 #95, 96, 101, 103 page 215 #7, 13, 17, 19, 33, 39 – 43