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

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.

Zeros of Polynomial Functions

Roots & Zeros of Polynomials II

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.

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.

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.

5.7 Apply the Fundamental Theorem of Algebra

9.9 The Fundamental Theorem of Algebra

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

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

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 :

General Results for Polynomial Equations In this section, we will state five general theorems about polynomial equations, some of which we have already.

Section 3.3 Theorems about Zeros of Polynomial Functions.

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

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.

Complex Zeros and the Fundamental Theorem of Algebra.

2.3 Real and Non Real Roots of a Polynomial Polynomial Identities Secondary Math 3.

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

1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Start Up Day 13 Find all zeros of the given function:

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.

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.

1/27/2016 Math 2 Honors - Santowski 1 Lesson 21 – Roots of Polynomial Functions Math 2 Honors - Santowski.

Section 4.6 Complex Zeros; Fundamental Theorem of Algebra.

Roots & Zeros of Polynomials II Finding the Solutions (Roots/Zeros) of Polynomials: The Fundamental Theorem of Algebra The Complex Conjugate Theorem.

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

Today in Pre-Calculus Notes: –Fundamental Theorem of Algebra –Complex Zeros Homework Go over quiz.

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.

Precalculus Lesson 2.5 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.

Zeros of Polynomial Fns Algebra III, Sec. 2.5 Objective Determine the number of rational and real zeros of polynomial functions, and find the zeros.

Chapter 2 – Polynomial and Rational Functions 2.5 – The Fundamental Theorem of Algebra.

3.5 Complex Zeros & the Fundamental Theorem of Algebra.

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

Fundamental Theorem of Algebra

Conjugate Pairs Let f (x) is a polynomial function that has real

Complex Zeros and the Fundamental Theorem of Algebra

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

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

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

Lesson 2.5 The Fundamental Theorem of Algebra

4.5 The Fundamental Theorem of Algebra (1 of 2)

The Fundamental Theorem of Algebra (Section 2-5)

Lesson 2.5 The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

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

4.5 The Fundamental Theorem of Algebra (1 of 2)

Fundamental Theorem of Algebra

(1) Find all of the zeros of f.

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

Pre-AP Pre-Calculus Chapter 2, Section 5

Fundamental Theorem of Algebra

Presentation transcript:

Section 2-6 Finding Complex Zeros

Section 2-6 Fundamental Theorem of Algebra Fundamental Theorem of Algebra Linear Factorization Theorem Linear Factorization Theorem Complex conjugate zeros Complex conjugate zeros Finding a poly from given zeros Finding a poly from given zeros Factoring a poly with complex zeros Factoring a poly with complex zeros Factoring a poly with real coefficients Factoring a poly with real coefficients

Fundamental Theorem of Algebra a poly function of degree n has n complex zeros a poly function of degree n has n complex zeros some of these zeros may be real and some may be nonreal (contain i) some of these zeros may be real and some may be nonreal (contain i) the real zeros represent x-intercepts, but the nonreal zeros do not show in the graph the real zeros represent x-intercepts, but the nonreal zeros do not show in the graph

Fundamental Theorem of Algebra the graph of the function shows that there is only one real zero, so the other two zeros must be nonreal the graph of the function shows that there is only one real zero, so the other two zeros must be nonreal

Linear Factorization Theorem if a poly has degree n then it has n linear factors if a poly has degree n then it has n linear factors if z 1, z 2,..., z n are the complex zeros and a is the leading coefficient then f (x) factors into if z 1, z 2,..., z n are the complex zeros and a is the leading coefficient then f (x) factors into

Complex Conjugate Zeros if a + bi is a zero of f (x) then, then a – bi is also a zero of f (x) if a + bi is a zero of f (x) then, then a – bi is also a zero of f (x)

Ex. Write a poly in standard form with real coefficients whose zeros include -3, 4, and 2 - i

Find all the zeros and write the linear factorization

Factoring with Real Coefficients the linear factorization theorem explains how to factor poly’s, but sometime the factors do not have real coefficients the linear factorization theorem explains how to factor poly’s, but sometime the factors do not have real coefficients it’s also possible to factor a poly using only linear factors and irreducible quadratic factor, all having real coefficients it’s also possible to factor a poly using only linear factors and irreducible quadratic factor, all having real coefficients