Section 4.6 Complex Zeros; Fundamental Theorem of Algebra.

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.

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.

OBJECTIVES: 1. USE THE FUNDAMENTAL THEOREM OF ALGEBRA 2. FIND COMPLEX CONJUGATE ZEROS. 3. FIND THE NUMBER OF ZEROS OF A POLYNOMIAL. 4. GIVE THE COMPLETE.

The Fundamental Theorem of Algebra Section 4.6 beginning on page 198.

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

5.7 Apply the Fundamental Theorem of Algebra

Complex Zeros; Fundamental Theorem of Algebra

9.9 The Fundamental Theorem of Algebra

Zeros of Polynomial Functions Section 2.5 Page 312.

PRE-AP PRE- CALCULUS CHAPTER 2, SECTION 5 Complex Zeros and the fundamental Theorem of Algebra.

6.6 The Fundamental Theorem of Algebra

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

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

COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

Section 3.3 Theorems about Zeros of Polynomial Functions.

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

Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number.

Chapter 2 Polynomial and Rational Functions. Warm Up

2.5 The Fundamental Theorem of Algebra Students will use the fundamental theorem of algebra to determine the number of zeros of a polynomial. Students.

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

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.

Complex Zeros and the Fundamental Theorem of Algebra.

3.6 Complex Zereos. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra says that every polynomial with complex coefficients must have.

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

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

1/27/ Fundamental Theorem of Algebra. Intro Find all zeros for each of the following: Multiplicity – When more than one zero occurs at the.

Copyright © 2011 Pearson, Inc. 2.5 Complex Zeros and the 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.

2015/16 TI-Smartview 2.5 The Fundamental Theorem of Algebra.

The Fundamental Theorem of Algebra Intro - Chapter 4.6.

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

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.

Section 6: Fundamental Theorem of Algebra Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with.

REMAINDER THEOREM HONORS ADVANCED ALGEBRA PRESENTATION 2-7.

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.

Conjugate Pairs Theorem Every complex polynomial function of degree n  1 has exactly n complex zeros, some of which may repeat. 1) A polynomial function.

Section 2-6 Finding Complex Zeros. Section 2-6 Fundamental Theorem of Algebra Fundamental Theorem of Algebra Linear Factorization Theorem Linear Factorization.

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.

“Is it better to be feared or respected? And I say, is it too much to ask for both?”

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra and Complete Factorization

Roots and Zeros 5.7.

Notes Over 3.4 The Rational Zero Test

Complex Zeros and the Fundamental Theorem of Algebra

Complex Zeros Objective: Students will be able to calculate and describe what complex zeros are.

The Fundamental Theorem of Algebra

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

Fundamental Theorem of Algebra

Rational Root and Complex Conjugates Theorem

Rational Root and Complex Conjugates Theorem

Lesson 7.2: Finding Complex Solutions of Polynomial Equations

3.8 Complex Zeros; Fundamental Theorem of Algebra

Lesson 2.5 The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Zeros of a Polynomial Function

The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Lesson: _____ Section 2.5 The Fundamental Theorem of Algebra

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

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Writing Polynomial Functions

1) Find f(g(x)) and g(f(x) to show that f(x) and g(x) are inverses

Packet #9 Zeros of Polynomials

Pre-AP Pre-Calculus Chapter 2, Section 5

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Presentation transcript:

Section 4.6 Complex Zeros; Fundamental Theorem of Algebra

A polynomial of degree 5 whose coefficients are real numbers has the zeros –2, –3i, and 2 + 4i. Find the remaining two zeros. Since f has coefficients that are real numbers, complex zeros appear as conjugate pairs. It follows that 3i, the conjugate of –3i, and 2 – 4i, the conjugate of 2 + 4i, are the two remaining zeros.

Find a polynomial f of degree 4 whose coefficients are real numbers and that has the zeros 1, 1, 4 + i. Since 4 + i is a zero, by the Conjugate Pairs Theorem, 4 – i is also a zero. By the factor theorem:

Step 1: The degree of f is 4 so there will be 4 complex zeros. Step 2: