Complex Roots of Polynomials

Slides:

Advertisements

Similar presentations

Complex Numbers.

Advertisements

Section 2.4 Complex Numbers

Complex Numbers; Quadratic Equations in the Complex Number System

Roots & Zeros of Polynomials I

Roots & Zeros of Polynomials

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.

Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this.

6.2 – Simplified Form for Radicals

Pre-Calculus For our Polynomial Function: The Factors are:(x + 5) & (x - 3) The Roots/Solutions are:x = -5 and 3 The Zeros are at:(-5, 0) and (3, 0)

Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.

Complex Numbers OBJECTIVES Use the imaginary unit i to write complex numbers Add, subtract, and multiply complex numbers Use quadratic formula to find.

Lesson 1-5 The Complex Numbers. Objective: Objective: To add, subtract, multiply, and divide complex numbers.

The Rational Zero Theorem

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.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

1.3 Complex Number System.

Solving Quadratic Equations (finding roots) Example f(x) = x By Graphing Identifying Solutions Solutions are -2 and 2.

Sec 3.4 & Sec 3.5 Complex Numbers & Complex Zeros

The Fundamental Theorem of Algebra And Zeros of Polynomials

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

Polynomial Division: Dividing one polynomial by another polynomial to find the zeros of the polynomial. Ex 1: Find the zeros of Solution:1 st way: At.

Objectives Define and use imaginary and complex numbers.

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

Section 4.8 – Complex Numbers Students will be able to: To identify, graph, and perform operations with complex numbers To find complex number solutions.

GUIDED PRACTICE for Example How many solutions does the equation

RATIONAL EXPRESSIONS. Definition of a Rational Expression A rational number is defined as the ratio of two integers, where q ≠ 0 Examples of rational.

6.6 The Fundamental Theorem of Algebra

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 +

Math is about to get imaginary!

Ch 2.5: The Fundamental Theorem of Algebra

Complex Numbers MATH Precalculus S. Rook. Overview Section 2.4 in the textbook: – Imaginary numbers & complex numbers – Adding & subtracting complex.

5.6 – Quadratic Equations and Complex Numbers Objectives: Classify and find all roots of a quadratic equation. Graph and perform operations on complex.

Lesson 2.1, page 266 Complex Numbers Objective: To add, subtract, multiply, or divide complex numbers.

Zeros of Polynomials 2.5.

Complex Number System Adding, Subtracting, Multiplying and Dividing Complex Numbers Simplify powers of i.

Introduction Synthetic division, along with your knowledge of end behavior and turning points, can be used to identify the x-intercepts of a polynomial.

The Rational Zero Theorem The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all.

4-8 Complex Numbers Today’s Objective: I can compute with complex numbers.

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

1.5 COMPLEX NUMBERS Copyright © Cengage Learning. All rights reserved.

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

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

5.6 – Quadratic Equations and Complex Numbers Objectives: Classify and find all roots of a quadratic equation. Graph and perform operations on complex.

Lesson 1.8 Complex Numbers Objective: To simplify equations that do not have real number solutions.

I am able to solve a quadratic equation using complex numbers. I am able to add, subtract, multiply, and divide complex numbers. Solve the equation.

January 17, 2012 At the end of the today, you will be able to work with complex numbers. Warm-up: Correct HW 2.3: Pg. 160 # (2x – 1)(x + 2)(x.

Topic VII: Polynomial Functions Solving Polynomial Equations Roots and Zeros.

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

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Objectives Define and use imaginary and complex numbers.

With a different method

Mod. 3 Day 4 Graphs of Functions.

The Rational Zero Theorem

5.7: Fundamental Theorem of Algebra

Complex Numbers Using Complex Conjugates in dividing complex numbers and factoring quadratics -- Week 15 11/19.

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Complex Numbers Objectives Students will learn:

The Fundamental Theorem of Algebra And Zeros of Polynomials

10.1 Radical Expressions and Graphs

The Rational Zero Theorem

Sec. 1.5 Complex Numbers.

Section 2.4 Complex Numbers

5.4 Complex Numbers.

Find (x3 + 2x2 – 5x – 6) ÷ (x – 2) using synthetic division.

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

Lesson 5–5/5–6 Objectives Be able to define and use imaginary and complex numbers Be able to solve quadratic equations with complex roots Be able to solve.

Presentation transcript:

Complex Roots of Polynomials Lesson 3.6 Complex Roots of Polynomials At the conclusion of this lesson you will be able to: Write complex numbers in the form of a + bi Find zeros of quadratic functions involving complex numbers Determine how many complex roots a polynomial may have before factoring it completely Write polynomial functions that have complex roots.

We have seen several times where a graph does not intersect the x-axis. This means that there is no real number that makes a function equal to 0. Ex: To examine the zeros of a function like this we must extend our number system once more to the set of Complex Numbers. Before we can define complex numbers, we must introduce the number . From this we get cannot be a real number because there is no real number that satisfies

Complex Numbers The set of complex numbers consists of all expressions of the form Where a and b are real numbers. This set is often denoted C. The number a is called the real part of the complex number z, and the number b is called the imaginary part of z. When b = 0, the complex number is a real number, therefore, all real numbers are complex numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. a + bi = c + di if and only if a = c and b = d

Addition, Subtraction and Multiplication of Complex Numbers Addition and subtraction is done just like other numbers: combine like terms. Multiplication is just like multiplying binomials: FOIL method. Ex 1: Determine the sum and product of the complex numbers: Sum: Difference: Product:

Recall: quadratic formula: We use the quadratic formula to find the zeros of a quadratic function. When there are two real solutions. When there is one real solution. When there are two complex solutions. How to recover the complex roots from the quadratic formula: Multiplied both sides by -1 This is now a real number

Ex 2: Determine all the solutions to the equation a = 1 b = -4 c = 13 Recall: turning a zero into a factor: If x = 3, the factor would be… (x – 3). Now, turn the complex roots into factors. NOTE: The two complex solutions have the same real part and the imaginary parts only differ by sign. These are called Complex Conjugates.

Complex Conjugates The complex conjugate of the complex number is the complex number For any complex number z, which are both real numbers.

Ex 3: Determine the complex conjugates of and and then find: Solution: Conjugate Results For any pair of complex numbers, z and w, and for any integer n, we have

Complex roots come in conjugate pairs. Ex: If x = 2 – 3i is a complex root of a polynomial, then so is x = 2 + 3i. Ex 4: One solution to the equation is the complex number . Determine all the solutions to this equation. Solution: If x = 1 + 2i is a solution then so is x = 1 – 2i. Now, turn them into factors and simplify. (x – (1 + 2i)) = (x – 1 – 2i) (x – (1 – 2i)) = (x – 1 + 2i) The product of these two new factors is the divisor (D(x)) for your division algorithm. (x – 1 – 2i)(x – 1 + 2i) = x2 – 2x + 5 = D(x) Notice that when complex conjugates are multiplied that the i’s cancel out.

Now, divide using the division algorithm. Factor Q(x) We can see that we need to use the quadratic formula a = 1 b = -4 c = 1 What if the directions asked to factor completely? turn solutions into factors.

Ex 5: Factor completely. Solution: We need to find at least one zero on our own before we can do the division algorithm. (RATIONAL ZERO TEST) x = 3, the factor is: (x - 3) which is your D(x) for your division algorithm. Now use the quad. formula to find other zeros.