SECTION 3.4 Complex Zeros and the Fundamental Theorem of Algebra.

Slides:

Advertisements

Similar presentations

Notes 6.6 Fundamental Theorem of Algebra

Advertisements

2.5 complex 0’s & the fundamental theorem of algebra

If you need to hear it and go through it with me, go to the “LINKS” section of my webpage and open it up there!!

5.3 Complex Numbers; Quadratic Equations with a Negative Discriminant.

Section 7.8 Complex Numbers  The imaginary number i  Simplifying square roots of negative numbers  Complex Numbers, and their Form  The Arithmetic.

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.

Warm up Use the Rational Root Theorem to determine the Roots of : x³ – 5x² + 8x – 6 = 0.

1.3 Complex Number System.

Honors Topics.  You learned how to factor the difference of two perfect squares:  Example:  But what if the quadratic is ? You learned that it was.

Sullivan Algebra and Trigonometry: Section 1.3 Quadratic Equations in the Complex Number System Objectives Add, Subtract, Multiply, and Divide Complex.

Objectives Define and use imaginary and complex numbers.

Complex Numbers and Roots

5.7 Apply the Fundamental Theorem of Algebra

Objectives Define and use imaginary and complex numbers.

Complex Zeros; Fundamental Theorem of Algebra

9.9 The Fundamental Theorem of Algebra

Objectives Define and use imaginary and complex numbers.

Zeros of Polynomial Functions Section 2.5 Page 312.

The Rational Root Theorem.  Is a useful way to find your initial guess when you are trying to find the zeroes (roots) of the polynomial.  THIS IS JUST.

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 :

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 +

Ch 2.5: The Fundamental Theorem of Algebra

Solving Quadratic Equations

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

2.7 Apply the Fundamental Theorem of Algebra Polynomials Quiz: Tomorrow (over factoring and Long/Synthetic Division) Polynomials Test: Friday.

Section 3.3 Theorems about Zeros of Polynomial Functions.

Essential Questions How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots?

Complex Zeros and the Fundamental Theorem of Algebra.

Holt McDougal Algebra Complex Numbers and Roots 2-5 Complex Numbers and Roots Holt Algebra 2 Warm Up Warm Up Lesson Presentation Lesson Presentation.

Imaginary Number: POWERS of i: Is there a pattern? Ex:

Remainder and Factor Theorems

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

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.

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

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

How do I use the imaginary unit i to write complex numbers?

5.9 Complex Numbers Alg 2. Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i.

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.

Complex Numbers Any number in form a+bi, where a and b are real numbers and i is imaginary. What is an imaginary number?

3.5 Complex Zeros & the Fundamental Theorem of Algebra.

Section 2.5 – Quadratic Equations

College Algebra Chapter 3 Polynomial and Rational Functions

Objectives Define and use imaginary and complex numbers.

Roots and Zeros 5.7.

Section 6.6 The Fundamental Theorem of Algebra

When solving #30 and #34 on page 156, you must “complete the square.”

The Fundamental Theorem of Algebra

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

Copyright © Cengage Learning. All rights reserved.

Complex Numbers.

Rational Root and Complex Conjugates Theorem

Finding Real Roots of Polynomial Equations

(Sections 4-5 pt. 1 & 2, 4-6 pt. 1, 4-7, 4-8 pt. 1 & 2)

Lesson 2.5 The Fundamental Theorem of Algebra

SECTION 9-3 : SOLVING QUADRATIC EQUATIONS

Roots, Radicals, and Complex Numbers

3.2 Complex Numbers.

Lesson: _____ Section 2.5 The Fundamental Theorem of Algebra

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

College Algebra Chapter 1 Equations and Inequalities

Rational Root Theorem.

Fundamental Theorem of Algebra

5.4 Complex Numbers.

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

Complex Numbers and Roots

Objective SWBAT solve polynomial equations in factored form.

Fundamental Theorem of Algebra Roots/Zeros/X-Intercepts

Presentation transcript:

SECTION 3.4 Complex Zeros and the Fundamental Theorem of Algebra

I NTRODUCTION Consider the polynomial p(x) = x The zeros of p are the solutions to x = 0, or x 2 = -1 This equation has no real solutions, but you may recall from Intermediate Algebra that we can formally extract the square roots of both sides to get i is the so-called imaginary unit

I MAGINARY U NIT The imaginary unit i satisfies the two following properties If c is a real number with c 0 then

C OMPLEX NUMBERS A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit

E XAMPLE Perform the indicated operations and simplify. Write your final answer in the form a + bi. 1. (1 - 2 i ) - (3 + 4 i ) 2. (1 - 2 i )(3 + 4 i ) 3. (1 - 2 i )/(3 - 4 i ) 4. 5.

C ONJUGATE The conjugate of a complex number a + bi is the number a - bi The notation commonly used for conjugation is abar: For example,

P ROPERTIES OF CONJUGATE Suppose z and w are complex numbers. for n = 1, 2, 3, …. z is a real number if and only if

E XAMPLE Suppose we wish to find the zeros of f(x) = x 2 - 2x + 5 Two things are important to note 1. The zeros 1+2 i and i are complex conjugates 2. (x - [1+2i]) (x - [1 - 2i]) = x 2 - 2x + 5