1. The Fundamental Theorem of Algebra
Honors Precalculus
February 27, 2018
Mr. 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 how to find it.
Use the theorem to prove a root exists…

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