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

Slides:

Advertisements

Similar presentations

4.4 Rational Root Theorem.

Advertisements

Notes 6.6 Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

Rational Root Theorem.

Complex Numbers The imaginary number i is defined as so that Complex numbers are in the form a + bi where a is called the real part and bi is the imaginary.

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

Factoring with Real Number Coefficients

Zeros of Polynomial Functions

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

Copyright © Cengage Learning. All rights reserved. 2 Polynomial and Rational Functions.

 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 :

2.5 ZEROS OF POLYNOMIAL FUNCTIONS Copyright © Cengage Learning. All rights reserved.

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.

Section 3.3 Theorems about Zeros of Polynomial Functions.

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.

Remainder Theorem If f(x) is divided by x – r, then the remainder is equal to f(r). We can find f(r) using Synthetic Division.

Complex Zeros and the Fundamental Theorem of Algebra.

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:

Copyright © 2011 Pearson, Inc. 2.5 Complex Zeros and the Fundamental Theorem of Algebra.

Section 4.6 Complex Zeros; Fundamental Theorem of Algebra.

Bellwork Perform the operation and write the result in standard from ( a + bi)

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

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.

Warm-Up 2/26 1. J. Rigor: You will learn how to find the real and complex zeros of polynomial functions. Relevance: You will be able to use graphs and.

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.

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

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

7.4 The Fundamental Theorem of Algebra. Fundamental Theorem of Algebra: Every polynomial function of positive degree with complex coefficients has at.

Remainder Theorem Let f(x) be an nth degree polynomial. If f(x) is divided by x – k, then the remainder is equal to f(k). We can find f(k) using Synthetic.

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

The Fundamental Theorem of Algebra and Complete Factorization

4.1 Continued: Synthetic Division

Roots and Zeros 5.7.

Section 6.6 The Fundamental Theorem of Algebra

Notes Over 3.4 The Rational Zero Test

FST Chapter 7 Review Questions.

Complex Zeros and the Fundamental Theorem of Algebra

Copyright © Cengage Learning. All rights reserved.

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

Rational Zero Theorem Rational Zero Th’m: If the polynomial

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 Fundamental Theorem of Algebra (Section 2-5)

Lesson 2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

Lesson: _____ Section 2.5 The Fundamental Theorem of Algebra

Warm-up: Find all real solutions of the equation X4 – 3x2 + 2 = 0

Rational Root Theorem.

4.6 - Fundamental Theorem of Algebra

Polynomial and Rational Functions

Fundamental Theorem of Algebra

(1) Find all of the zeros of f.

(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

Packet #9 Zeros of Polynomials

Remainder Theorem If P(x) is divided by x – r, then the remainder is equal to P(r). We can find P(r) using Synthetic Division.

Fundamental Theorem of Algebra

Copyright © Cengage Learning. All rights reserved.

Presentation transcript:

Conjugate Pairs Let f (x) is a polynomial function that has real coefficients. If a + bi, where b ≠ 0, is a zero of the function, the conjugate a – bi is also a zero of the function. Ex 1: Find a fourth degree polynomial with real coefficients that has -1, -1, and 3i as zeros.

Prime/Irreducible Factors Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros. A quadratic factor with no real zeros is said to be prime or irreducible over the reals.

Ex 2: Write the polynomial f (x) = x4 – x2 – 20 a. As the product of factors irreducible over the rationals. b. As a product of linear factors. c. As the product of linear factors and quadratic factors that are irreducible over the reals.

Ex 3: Find all the zeros of f (x) = x4 – 3x3 + 6x2 + 2x – 60 given that 1 + 3i is a zero of f .

Practice Assignment: Section 2.5B (pg 141) #35 – 52