Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 3.4 Zeros of Polynomial Functions.

Slides:

Advertisements

Similar presentations

The Rational Zero Theorem

Advertisements

4.4 Rational Root Theorem.

Notes 6.6 Fundamental Theorem of Algebra

Rational Root Theorem.

SECTION 3.6 COMPLEX ZEROS; COMPLEX ZEROS; FUNDAMENTAL THEOREM OF ALGEBRA FUNDAMENTAL THEOREM OF ALGEBRA.

Zeros of Polynomial Functions Section 2.5. Objectives Use the Factor Theorem to show that x-c is a factor a polynomial. Find all real zeros of a polynomial.

Warm-Up: January 9, 2012.

2.5 Zeros of Polynomial Functions

Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.

Copyright © 2007 Pearson Education, Inc. Slide 3-1.

Section 5.6 – Complex Zeros; Fundamental Theorem of Algebra Complex Numbers Standard form of a complex number is: a + bi. Every complex polynomial function.

The Rational Zero Theorem

Lesson 2.5 The Fundamental Theorem of Algebra. For f(x) where n > 0, there is at least one zero in the complex number system Complex → real and imaginary.

Warm up Use the Rational Root Theorem to determine the Roots of : x³ – 5x² + 8x – 6 = 0.

The Fundamental Theorem of Algebra And Zeros of Polynomials

Sullivan Algebra and Trigonometry: Section 5.6 Complex Zeros; Fundamental Theorem of Algebra Objectives Utilize the Conjugate Pairs Theorem to Find the.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Fundamental Theorem of Algebra ♦ Perform arithmetic operations on complex.

Academy Algebra II/Trig 5.5: The Real Zeros of a Polynomial Functions HW: p.387 (14, 27, 30, 31, 37, 38, 46, 51)

A3 3.4 Zeros of Polynomial Functions Homework: p eoo, odd.

Complex Zeros; Fundamental Theorem of Algebra

9.9 The Fundamental Theorem of Algebra

Zeros of Polynomial Functions Section 2.5 Page 312.

 Find a polynomial with specified zeros.  For a polynomial function with integer coefficients, find the rational zeros and the other zeros, if possible.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 3 Polynomial and Rational Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.

The Rational Root Theorem.  Is a useful way to find your initial guess when you are trying to find the zeroes (roots) of the polynomial.  THIS IS JUST.

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions.

6.6 The Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.4 Real Zeros of Polynomial Functions.

Do Now: Find all real zeros of the function.

Lesson 2.5, page 312 Zeros of Polynomial Functions Objective: To find a polynomial with specified zeros, rational zeros, and other zeros, and to use Descartes’

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

3.4 Zeros of Polynomial Functions. The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in.

Zeros of Polynomials 2.5.

Copyright © 2009 Pearson Education, Inc. CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions.

Section 4.4 Theorems about Zeros of Polynomial Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Copyright © 2011 Pearson Education, Inc. The Theory of Equations Section 3.3 Polynomial and Rational Functions.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

The Rational Zero Theorem The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all.

THE FUNDAMENTAL THEOREM OF ALGEBRA. Descartes’ Rule of Signs If f(x) is a polynomial function with real coefficients, then *The number of positive real.

UNIT 2, LESSON 5 DESCARTES LAW OF SIGNS. FINDING POLYNOMIALS WITH GIVEN ZEROS If we are given the zeros of a polynomial, we can generate the polynomial.

Zero of Polynomial Functions Factor Theorem Rational Zeros Theorem Number of Zeros Conjugate Zeros Theorem Finding Zeros of a Polynomial Function.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

7.5 Roots and Zeros Objectives: The student will be able to…

5.6 The Fundamental Theorem of Algebra. If P(x) is a polynomial of degree n where n > 1, then P(x) = 0 has exactly n roots, including multiple and complex.

2.5 The Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra – If f(x) is a polynomial of degree n, where.

Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.

Determine the number and type of roots for a polynomial equation

Algebra 2. Solve for x Algebra 2 (KEEP IN MIND THAT A COMPLEX NUMBER CAN BE REAL IF THE IMAGINARY PART OF THE COMPLEX ROOT IS ZERO!) Lesson 6-6 The Fundamental.

3.5 Complex Zeros & the Fundamental Theorem of Algebra.

College Algebra Chapter 3 Polynomial and Rational Functions

Descartes Rule of Signs Positive real zeros = Negative real zeros =

Chapter 3: Polynomial Functions

Section 6.6 The Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions

Rational Root and Complex Conjugates Theorem

3.8 Complex Zeros; Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions

The Rational Zero Theorem

Finding Zeros of Polynomials

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Zeros of a Polynomial Function

The Fundamental Theorem of Algebra And Zeros of Polynomials

The Rational Zero Theorem

Rational Root Theorem.

Chapter 3: Polynomial Functions

College Algebra Chapter 3 Polynomial and Rational Functions

6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor.

Presentation transcript:

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 The Rational Zero Theorem If has integer coefficients and (where is reduced to lowest terms) is a rational zero of f, then p is a factor of the constant term, a 0, and q is a factor of the leading coefficient, a n.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Example: Using the Rational Zero Theorem List all possible rational zeros of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Finding Zeros of a Polynomial Function Find all zeros of

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Properties of Roots of Polynomial Equations 1. If a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots. 2. If a + bi is a root of a polynomial equation with real coefficients then the imaginary number a – bi is also a root. Imaginary roots, if they exist, occur in conjugate pairs.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Solving a Polynomial Equation Solve

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where then the equation f(x) = 0 has at least one complex root.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 The Linear Factorization Theorem If where and then where c 1, c 2,..., c n are complex numbers (possibly real and not necessarily distinct). In words: An nth- degree polynomial can be expressed as the product of a nonzero constant and n linear factors, where each linear factor has a leading coefficient of 1.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Finding a Polynomial Function with Given Zeros Find a third-degree polynomial function f(x) with real coefficients that has –3 and i as zeros and such that f(1) = 8.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Descartes’ Rule of Signs Let be a polynomial with real coefficients. 1. The number of positive real zeros of f is either a. the same as the number of sign changes of f(x) or b. less than the number of sign changes of f(x) by a positive even integer. If f(x) has only one variation in sign, then f has exactly one positive real zero.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Descartes’ Rule of Signs (continued) 2. The number of negative real zeros of f is either a. The same as the number of sign changes in f(–x) or b. less than the number of sign changes in f(–x) by a positive even integer. If f(–x) has only one variation in sign, then f has exactly one negative real zero

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Using Descartes’ Rule of Signs Determine the possible number of positive and negative real zeros of