The Fundamental Theorem of Algebra

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Presentation transcript:

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

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

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?

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…

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

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)

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…

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