3.4 Zeros of Polynomial Functions: Real, Rational, and Complex

Slides:

Advertisements

Similar presentations

Notes 6.6 Fundamental Theorem of Algebra

Advertisements

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

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

Solving Polynomial Equations. Fundamental Theorem of Algebra Every polynomial equation of degree n has n roots!

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.

6.7 – Using the Fundamental Theorem of Algebra

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

Objectives Fundamental Theorem of Algebra 6-6

5.7 Apply the Fundamental Theorem of Algebra

Lesson 4-1 Polynomial Functions.

7.5.1 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.

Rational Root and Complex Conjugates Theorem. Rational Root Theorem Used to find possible rational roots (solutions) to a polynomial Possible Roots :

1 Using the Fundamental Theorem of Algebra.  Talk about #56 & #58 from homework!!!  56 = has -1 as an answer twice  58 = when you go to solve x 2 +

Do Now: Find all real zeros of the function.

Section 3.3 Theorems about Zeros of Polynomial Functions.

Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.

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

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

Complex Zeros and the Fundamental Theorem of Algebra.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1.

Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the.

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

The Fundamental Theorem of Algebra It’s in Sec. 2.6a!!! Homework: p odd, all.

Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 5.6 Complex Zeros; Fundamental Theorem of Algebra.

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.

Section 4.6 Complex Zeros; Fundamental Theorem of Algebra.

Holt McDougal Algebra Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least.

7.5 Roots and Zeros Objectives:

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.

Solve polynomial equations with complex solutions by using the Fundamental Theorem of Algebra. 5-6 THE FUNDAMENTAL THEOREM OF ALGEBRA.

Every polynomial P(x) of degree n>0 has at least one zero in the complex number system. N Zeros Theorem Every polynomial P(x) of degree n>0 can be expressed.

PreCalculus Section 2.7 Apply general theorems about polynomials Fundamental Theorem of Algebra In the complex number system, a polynomial of degree n.

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.

Fundamental Theorem of Algebra

College Algebra Chapter 3 Polynomial and Rational Functions

Rational Root Theorem and Fundamental Theorem of Algebra

3.4 Zeros of Polynomial Functions

Chapter 3: Polynomial Functions

Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the.

Section 6.6 The Fundamental Theorem of Algebra

Notes Over 3.4 The Rational Zero Test

Chapter 3: Polynomial Functions

Complex Zeros and the Fundamental Theorem of Algebra

Copyright © Cengage Learning. All rights reserved.

Warm-up Multiply the factors and write in standard form.

Fundamental Theorem of Algebra

Review & 6.6 The Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions

Rational Root and Complex Conjugates Theorem

Rational Root Theorem and Fundamental Theorem of Algebra

Lesson 7.2: Finding Complex Solutions of Polynomial Equations

3.8 Complex Zeros; Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions

4.5 The Fundamental Theorem of Algebra (1 of 2)

5.7 Apply the Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Chapter 3: Polynomial Functions

3.3 Zeros of Polynomials.

Fundamental Theorem of Algebra

4.5 The Fundamental Theorem of Algebra (1 of 2)

Fundamental Theorem of Algebra

6.7 Using the Fundamental Theorem of Algebra

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.

Fundamental Theorem of Algebra

Copyright © Cengage Learning. All rights reserved.

Presentation transcript:

3.4 Zeros of Polynomial Functions: Real, Rational, and Complex Complex Zeros and the Fundamental Theorem of Algebra It can be shown that if a + bi is a zero of a polynomial function with real coefficients, then so is its complex conjugate, a – bi. Conjugate Zeros Theorem If P(x) is a polynomial having only real coefficients, and if a + bi is a zero of P, then the conjugate a – bi is also a zero of P.

3.4 Topics in the Theory of Polynomial Functions (II) Example Find a polynomial having zeros 3 and 2 + i that satisfies the requirement P(–2) = 4. Solution Since 2 + i is a zero, so is 2 – i. A general solution is Since P(–2) = 4, we have

3.4 Zeros of a Polynomial Function Example Find all complex zeros of given that 1 – i is a zero. Solution Number of Zeros Theorem A function defined by a polynomial of degree n has at most n distinct complex zeros.

3.4 Zeros of a Polynomial Function Using the Conjugate Zeros Theorem, 1 + i is also a zero. The zeros of x2 – 5x + 6 are 2 and 3. Thus, and has four zeros: 1 – i, 1 + i, 2, and 3.

3.4 The Rational Zeros Theorem Example List all possible rational zeros. Use a graph to eliminate some of the possible zeros listed in part (a). Find all rational zeros and factor P(x). The Rational Zeros Theorem

3.4 The Rational Zeros Theorem Solution (a) From the graph, the zeros are no less than –2 and no greater than 1. Also, –1 is clearly not a zero since the graph does not intersect the x-axis at the point (-1,0).

3.4 The Rational Zeros Theorem (c) Show that 1 and –2 are zeros. Solving the equation 6x2 + x – 1 = 0, we get x = –1/2, 1/3.