Bell Ringer. Multiplying and Dividing with Complex Numbers Monday, February 29, 2016.

Slides:

Advertisements

Similar presentations

Operations with Complex Numbers

Advertisements

Complex Numbers.

Warm-upAnswers Compute (in terms of i) _i, -1, -i, 1.

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.

5.4 Complex Numbers Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can!

Complex Zeros; Fundamental Theorem of Algebra

9.9 The Fundamental Theorem of Algebra

How do we divide complex numbers?

I am greater than my square. The sum of numerator and denominator is 5 What fraction am I?

OUTLINE  Homework (and homework questions)  Ask any review questions you want  Review long division, solve by factoring and graphing calculators  BREAK.

6-5 Theorems About Roots of Polynomial Equations

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

5.5 Theorems about Roots of Polynomial Equations P

Ch 2.5: The Fundamental Theorem of Algebra

Imaginary & Complex Numbers EQ : Why do imaginary numbers exists and how do you add/subtract/multiply/divide imaginary numbers?

5.4 Complex Numbers. Let’s see… Can you find the square root of a number? A. E.D. C.B.

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

1.3 Multiplying and Divide Complex Numbers Quiz: Thursday.

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

4.6 Perform Operations With Complex Numbers. Vocabulary: Imaginary unit “i”: defined as i = √-1 : i 2 = -1 Imaginary unit is used to solve problems that.

Do Now What is the exact square root of 50? What is the exact square root of -50?

How do we divide complex numbers? Do Now: What is the conjugate? Explain why do we multiply a complex number and its conjugate Do Now: What is the conjugate?

6.5 Theorems About Roots of Polynomial Equations

Objective: SWBAT establish bounds for the real zeros of a function & identify potential rational zeros. Bell Ringer: pg 218 #50 HW Requests: Go over problems.

Warm-Up Solve Using Square Roots: 1.6x 2 = x 2 = 64.

5-7: COMPLEX NUMBERS Goal: Understand and use complex numbers.

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.

Complex Number System Reals Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Whole (0, 1, 2, …) Natural (1, 2, …) Irrationals.

Unit 2 – Quadratic, Polynomial, and Radical Equations and Inequalities Chapter 5 – Quadratic Functions and Inequalities 5.4 – Complex Numbers.

Chapter 4 Section 8 Complex Numbers Objective: I will be able to identify, graph, and perform operations with complex numbers I will be able to find complex.

6.6 – Complex Numbers Complex Number System: This system of numbers consists of the set of real numbers and the set of imaginary numbers. Imaginary Unit:

Bell Ringer. Complex Numbers Friday, February 26, 2016.

Introduction to Complex Numbers Adding, Subtracting, Multiplying And Dividing Complex Numbers SPI Describe any number in the complex number system.

Zeros (Solutions) Real Zeros Rational or Irrational Zeros Complex Zeros Complex Number and its Conjugate.

Complex Numbers We haven’t been allowed to take the square root of a negative number, but there is a way: Define the imaginary number For example,

Algebra Operations with Complex Numbers. Vocabulary Imaginary Number i -

DIVIDING COMPLEX NUMBERS. DO NOW Today’s Objective IWBAT multiply and divide complex numbers... And Why To work with the square roots of negative numbers.

Section 6.5 Theorems about Roots of Polynomial Equations Objective: Students will be able to solve equations using Theorems involving roots. Rational Root.

Bell Ringer When solving polynomial equations, what should you do first? 2. How many zeros (roots) will a polynomial have? 3. To get a polynomial.

Bell Ringer In elementary school, you learned long division. Describe, in detail, how to divide 232 by 4 without a calculator.

Daily Check!!! (Irrational Roots)

The Fundamental Theorem of Algebra

4.4: Complex Numbers -Students will be able to identify the real and imaginary parts of complex numbers and perform basic operations.

Complex Numbers.

Complex Numbers Imaginary Numbers Vectors Complex Numbers

Bell Ringer 1. What is the Rational Root Theorem

Rational Root and Complex Conjugates Theorem

5.6 Complex Numbers.

Operations with Complex Numbers

Bell Ringer What is a factor tree?

Explain why you don’t have to subtract when using Synthetic Division

Bell Ringer 1. What are the “special cases” for binomials and how do they factor? 2. What is the “special case” for a trinomial and how does it factor?

Lesson: _____ Section 2.5 The Fundamental Theorem of Algebra

Complex numbers Math 3 Honors.

Section 4.6 Complex Numbers

College Algebra Chapter 1 Equations and Inequalities

Unit 11 Math-2 (Honors) 11.1: Dividing Square root numbers

1.2 Adding And Subtracting Complex Numbers

1.2 Adding And Subtracting Complex Numbers

Lesson 2.4 Complex Numbers

Homework Check.

Homework Check.

Explain why you don’t have to subtract when using Synthetic Division

Bell Ringer When solving polynomial equations, what should you do first? 2. How many zeros (roots) will a polynomial have? 3. To get a polynomial.

4.6 Complex Numbers Algebra II.

1.3 Multiply & Divide Complex Numbers

Bell Ringer In elementary school, you learned long division. Describe, in detail, how to divide 230 by 4 without a calculator.

1.3 Multiply & Divide Complex Numbers

7.7 Complex Numbers.

Fundamental Theorem of Algebra Roots/Zeros/X-Intercepts

Presentation transcript:

Bell Ringer

Multiplying and Dividing with Complex Numbers Monday, February 29, 2016

Multiplying Complex Numbers  When we multiply complex numbers, sometimes the imaginary part “disappears”.  Ex: (2 + 3i) (2 – 3i) =  (4 – 6i + 6i – 9i 2 ) =  (4 + 9) = 15

Theorems say…  Complex roots come in pairs…  Ex: You can’t have a root of 3i without a root of -3i

Multiplying Complex Numbers  If a polynomial has complex roots, when you multiply the factors to find the original polynomial, the imaginary part “disappears”.

What about dividing complex numbers?  You can’t divide by an imaginary number.  To make the denominator real, we must multiply by a special 1 called the CONJUGATE.  We make the middle term “disappear”.  6 – 3i ÷ 2 + 4i  6 – 3i (2 – 4i) 2 + 4i (2 – 4i)  12 – 30i + 12i 2 4 – 16i 2  12 – 30i + (-12) 4 – (-16)  - 30i 20  -3 i 2

Complex numbers and Conjugates  Complex numbers are in the form a + bi  Complex conjugates are in the form a - bi  Complex: 2 + 4i  Conjugate: 2 – 4i  Complex: i  Conjugate: -3 – 5i  Complex: 1 – 7i  Conjugate: 1 + 7i

Time to Practice!  Classwork: Find Polynomial Functions given the roots (multiplying)  Homework: Dividing Complex Numbers (multiplying by i and complex conjugates)

Exit Ticket  1. What happens when you multiply complex conjugates?  2. Both complex and irrational roots must come in _____.  3. How do you make the denominator real if it is a complex number?