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

7-5 Roots and Zeros 7-6 Rational Zero Theorem
Splash Screen.
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.
6.5 & 6.6 Theorems About Roots and the Fundamental Theorem of Algebra
2.5 Zeros of Polynomial Functions
Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.
LIAL HORNSBY SCHNEIDER
Copyright © 2007 Pearson Education, Inc. Slide 3-1.
1 5.6 Complex Zeros; Fundamental Theorem of Algebra In this section, we will study the following topics: Conjugate Pairs Theorem Finding a polynomial function.
Solving Polynomial Equations. Fundamental Theorem of Algebra Every polynomial equation of degree n has n roots!
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.
Precalculus Complex Zeros V. J. Motto. Introduction We have already seen that an nth-degree polynomial can have at most n real zeros. In the complex number.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 3 Polynomial and Rational Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Real Zeros of a Polynomial Function Objectives: Solve Polynomial Equations. Apply Descartes Rule Find a polynomial Equation given the zeros.
6.6 The Fundamental Theorem of Algebra
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.
Section 3.3 Theorems about Zeros of Polynomial Functions.
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.
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.
Zero of Polynomial Functions Factor Theorem Rational Zeros Theorem Number of Zeros Conjugate Zeros Theorem Finding Zeros of a Polynomial Function.
6.5 Theorems About Roots of Polynomial Equations
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.
POLYNOMIALS 9/20/2015. STANDARD FORM OF A POLYNOMIAL ax + bx + cx … mx + k nn-1n-2 The degree is ‘n’
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions.
Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.
Section 2.5. Objectives:  Use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function.  Find all zeros of polynomial.
Zeros (Solutions) Real Zeros Rational or Irrational Zeros Complex Zeros Complex Number and its Conjugate.
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.
College Algebra Chapter 3 Polynomial and Rational Functions
Chapter 3: Polynomial Functions
3.8 Complex Zeros; Fundamental Theorem of Algebra
The Rational Zero Theorem
Lesson 2.5 The Fundamental Theorem of Algebra
Finding Zeros of Polynomials
Copyright © 2014, 2010, 2007 Pearson Education, Inc.
Apply the Fundamental Theorem of Algebra
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
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 Use the Rational Zero Theorem to find possible rational zeros. Find zeros of a polynomial function. Solve polynomial equations. Use the Linear Factorization Theorem to find polynomials with given zeros. Use Descartes’ Rule of Signs. Objectives:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 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. 4 Example: Using the Rational Zero Theorem List all possible rational zeros of The constant term is –3 and the leading coefficient is 4. Factors of the constant term, –3: Factors of the leading coefficient, 4: Possible rational zeros are:

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 Example: Finding Zeros of a Polynomial Function Find all zeros of We begin by listing all possible rational zeros. Possible rational zeros = We now use synthetic division to see if we can find a rational zero among the four possible rational zeros.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of Possible rational zeros are 1, –1, 2, and –2. We will use synthetic division to test the possible rational zeros. Neither –2 nor –1 is a zero. We continue testing possible rational zeros.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of Possible rational zeros are 1, –1, 2, and –2. We will use synthetic division to test the possible rational zeros. We have found that –2 and –1 are not rational zeros. We continue testing with 1 and 2. We have found a rational zero at x = 2.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of We have found a rational zero at x = 2. The result of synthetic division is: This means that We now solve

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of We have found that We now solve The solution set is The zeros of are

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 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. 11 Example: Solving a Polynomial Equation Solve We begin by listing all possible rational roots: Possible rational roots = Possible rational roots are 1, –1, 13, and –13. We will use synthetic division to test the possible rational zeros.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Solving a Polynomial Equation (continued) Solve Possible rational roots are 1, –1, 13, and –13. We will use synthetic division to test the possible rational zeros. x = 1 is a root for this polynomial. We can rewrite the equation in factored form

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Solving a Polynomial Equation (continued) Solve We have found that x = 1 is a root for this polynomial. In factored form, the polynomial is We now solve We begin by listing all possible rational roots. Possible rational roots = Possible rational roots are 1, –1, 13, and –13. We will use synthetic division to test the possible rational zeros.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Solving a Polynomial Equation (continued) Solve Possible rational roots are 1, –1, 13, and –13. We will use synthetic division to test the possible rational zeros. Because –1 did not work for the original polynomial, it is not necessary to test that value. The factored form of this polynomial is x = 1 is a (repeated) root for this polynomial

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Solving a Polynomial Equation (continued) Solve The factored form of this polynomial is We will use the quadratic formula to solve The solution set of the original equation is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 16 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. 17 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. 18 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. Because i is a zero and the polynomial has real coefficients, the conjugate, –i, must also be a zero. We can now use the Linear Factorization Theorem.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 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. Applying the Linear Factorization Theorem, we found that The polynomial function is

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 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. 21 Descartes’ Rule of Signs (continued) Let be a polynomial with real coefficients. 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. 22 Example: Using Descartes’ Rule of Signs Determine the possible number of positive and negative real zeros of 1. To find possibilities for positive real zeros, count the number of sign changes in the equation for f(x). There are 4 variations in sign. The number of positive real zeros of f is either 4, 2, or 0.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Using Descartes’ Rule of Signs Determine the possible number of positive and negative real zeros of 2. To find possibilities for negative real zeros, count the number of sign changes in the equation for f(–x). There are no variations in sign. There are no negative real roots for f.