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Copyright © 2011 Pearson, Inc. P.6 Complex Numbers
Copyright © 2011 Pearson, Inc. Slide P What youll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations … and why The zeros of polynomials are complex numbers.
Copyright © 2011 Pearson, Inc. Slide P Complex Number A complex number is any number that can be written in the form a + bi, where a and b are real numbers. The real number a is the real part, the real number b is the imaginary part, and a + bi is the standard form.
Copyright © 2011 Pearson, Inc. Slide P Addition and Subtraction of Complex Numbers If a + bi and c + di are two complex numbers, then Sum: (a + bi ) + (c + di ) = (a + c) + (b + d)i, Difference: (a + bi ) – (c + di ) = (a – c) + (b –d)i.
Copyright © 2011 Pearson, Inc. Slide P Example Multiplying Complex Numbers
Copyright © 2011 Pearson, Inc. Slide P Solution
Copyright © 2011 Pearson, Inc. Slide P Complex Conjugate
Copyright © 2011 Pearson, Inc. Slide P Complex Numbers The multiplicative identity for the complex numbers is 1 = 1 + 0i. The multiplicative inverse, or reciprocal, of z = a + bi is
Copyright © 2011 Pearson, Inc. Slide P Example Dividing Complex Numbers
Copyright © 2011 Pearson, Inc. Slide P Solution the complex number in standard form.
Copyright © 2011 Pearson, Inc. Slide P Discriminant of a Quadratic Equation
Copyright © 2011 Pearson, Inc. Slide P Example Solving a Quadratic Equation
Copyright © 2011 Pearson, Inc. Slide P Example Solving a Quadratic Equation
Copyright © 2011 Pearson, Inc. Slide P Quick Review
Copyright © 2011 Pearson, Inc. Slide P Quick Review Solutions
You will learn about: Complex Numbers Operations with complex numbers Complex conjugates and division Complex solutions of quadratic equations Why: The.
Chapter 2 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Complex Numbers.
Copyright © 2010 Pearson Education, Inc. Complex Numbers Perform arithmetic operations on complex numbersPerform arithmetic operations on complex numbers.
OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 1 Complex Numbers Define complex numbers. Add and subtract complex numbers. Multiply complex.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Fundamental Theorem of Algebra ♦ Perform arithmetic operations on complex.
Chapter Complex Numbers What you should learn 1.Use the imaginary unit i to write complex numbers 2.Add, subtract, and multiply complex numbers 3.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 8 Rational Exponents, Radicals, and Complex Numbers.
Copyright © 2011 Pearson, Inc. 2.5 Complex Zeros and the Fundamental Theorem of Algebra.
7-9: MORE About Complex Numbers Goal: Be able to solve equations with complex numbers, multiply complex numbers, find conjugates of complex numbers, and.
Solve the quadratic equation x = 0. Solving for x, gives x 2 = – 1 We make the following definition: Bell Work #1.
Complex Numbers Write imaginary numbers using i. 2.Perform arithmetic operations with complex numbers. 3.Raise i to powers.
Complex Numbers Section 3.2 Beginning on page 104.
Complex Numbers OBJECTIVES Use the imaginary unit i to write complex numbers Add, subtract, and multiply complex numbers Use quadratic formula to find.
Notes Over 5.4 Imaginary Numbers Notes Over 5.4Solving a Quadratic Equation Solve the equation.
5.3 Complex Numbers; Quadratic Equations with a Negative Discriminant.
Chapter 1 Equations and Inequalities Copyright © 2014, 2010, 2007 Pearson Education, Inc Complex Numbers.
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.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 7.7 Complex Numbers.
5-7: COMPLEX NUMBERS Goal: Understand and use complex numbers.
Solve the equation. 1.) 3x = 23 2.) 2(x + 7) 2 = 16 Warm Up.
Finding Complex Roots of Quadratics. Complex Number A number consisting of a real and imaginary part. Usually written in the following form (where a and.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1.
1.3 Complex Number System. Complex Numbers Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit.
CHAPTER 2 Introduction to Integers and Algebraic Expressions Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 2.1Integers and the Number.
2.4 Complex Numbers What is an imaginary number What is a complex number How to add complex numbers How to subtract complex numbers How to multiply complex.
Sec 3.4 & Sec 3.5 Complex Numbers & Complex Zeros Objectives: To understand complex numbers. To add, subtract & multiply complex numbers. To solve equations.
1.5 COMPLEX NUMBERS Copyright © Cengage Learning. All rights reserved.
Copyright © 2014, 2010, 2006 Pearson Education, Inc. 1 Chapter 3 Quadratic Functions and Equations.
Sullivan Algebra and Trigonometry: Section 1.3 Quadratic Equations in the Complex Number System Objectives Add, Subtract, Multiply, and Divide Complex.
Section 1.3 Complex Numbers; Quadratic Equations in the Complex Number System.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.
Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational.
Complex Numbers Properties & Powers of i Operations with Complex Numbers Practice Problems.
Chapter 5 Section 4: Complex Numbers. VOCABULARY Not all quadratics have real- number solutions. For instance, x 2 = -1 has no real-number solutions because.
Copyright © 2009 Pearson Addison-Wesley Complex Numbers, Polar Equations, and Parametric Equations.
Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 8.4 Quadratic Formula.
SAT Problem of the Day. 5.6 Quadratic Equations and Complex Numbers 5.6 Quadratic Equations and Complex Numbers Objectives: Classify and find all roots.
Copyright © 2011 Pearson Education, Inc. Complex Numbers Section P.7 Prerequisites.
Complex Numbers. The imaginary unit i is defined as.
5.6 Quadratic Equations and Complex Numbers. The Discriminant When using the Quadratic Formula you will find that the value of b 2 - 4ac is either positive,
Copyright © 2011 Pearson, Inc. 2.7 Solving Equations in One Variable.
Imaginary & Complex Numbers. Once upon a time… -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented.
Section 7.8 Complex Numbers The imaginary number i Simplifying square roots of negative numbers Complex Numbers, and their Form The Arithmetic.
SECTION 2.7 COMPLEX ZEROS OF A QUADRATIC FUNCTION COMPLEX ZEROS OF A QUADRATIC FUNCTION.
4-8 Complex Numbers Today’s Objective: I can compute with complex numbers.
4.8 Quadratic Formula and Discriminant. Quadratic Formula Formula: x = -b ± √(b 2 – 4ac) 2a This formula can be used to solve any quadratic equation.
Objective SWBAT simplify rational expressions, add, subtract, multiply, and divide rational expressions and solve rational equations SWBAT simplify rational.
1 Warm-up Divide the following using Long Division: (6x x x - 6) (3x –2 ) Divide the following with Synthetic Division (5x 3 – 6x 2 + 8) (x.
Copyright © 2011 Pearson, Inc. 7.1 Solving Systems of Two Equations.
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