Today in Pre-Calculus Notes: –Fundamental Theorem of Algebra –Complex Zeros Homework Go over quiz.

Slides:

Advertisements

Similar presentations

2.5 complex 0’s & the fundamental theorem of algebra

Advertisements

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.

Zeros of Polynomial Functions

Solving Polynomial Equations. Fundamental Theorem of Algebra Every polynomial equation of degree n has n roots!

OBJECTIVES: 1. USE THE FUNDAMENTAL THEOREM OF ALGEBRA 2. FIND COMPLEX CONJUGATE ZEROS. 3. FIND THE NUMBER OF ZEROS OF A POLYNOMIAL. 4. GIVE THE COMPLETE.

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.

Simplify each expression.

The Fundamental Theorem of Algebra And Zeros of Polynomials

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Fundamental Theorem of Algebra ♦ Perform arithmetic operations on complex.

5.7 Apply the Fundamental Theorem of Algebra

Bell Ringer 1. What is the Rational Root Theorem (search your notebook…Unit 2). 2. What is the Fundamental Theorem of Algebra (search your notebook…Unit.

A3 3.4 Zeros of Polynomial Functions Homework: p eoo, odd.

9.9 The Fundamental Theorem of Algebra

PRE-AP PRE- CALCULUS CHAPTER 2, SECTION 5 Complex Zeros and the fundamental Theorem of Algebra.

7.5.1 Zeros of Polynomial Functions

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 +

Today in Pre-Calculus Go over homework Notes: Remainder and Factor Theorems Homework.

Lesson 2.5, page 312 Zeros of Polynomial Functions Objective: To find a polynomial with specified zeros, rational zeros, and other zeros, and to use Descartes’

Warm Up. Find all zeros. Graph.. TouchesThrough More on Rational Root Theorem.

Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number.

4.5 Quadratic Equations Zero of the Function- a value where f(x) = 0 and the graph of the function intersects the x-axis Zero Product Property- for all.

Chapter 2 Polynomial and Rational Functions. Warm Up

THE FUNDAMENTAL THEOREM OF ALGEBRA. Descartes’ Rule of Signs If f(x) is a polynomial function with real coefficients, then *The number of positive real.

Objective: To learn & apply the fundamental theorem of algebra & determine the roots of polynomail equations.

Complex Zeros and the Fundamental Theorem of Algebra.

Polynomials of Higher Degree 2-2. Polynomials and Their Graphs  Polynomials will always be continuous  Polynomials will always have smooth turns.

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:

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

1/27/ Fundamental Theorem of Algebra. Intro Find all zeros for each of the following: Multiplicity – When more than one zero occurs at the.

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

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.

1/27/2016 Math 2 Honors - Santowski 1 Lesson 21 – Roots of Polynomial Functions Math 2 Honors - Santowski.

Section 4.6 Complex Zeros; Fundamental Theorem of Algebra.

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

7.5 Roots and Zeros Objectives:

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

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

1 What you will learn today…  How to use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function  How to use your.

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.

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

Every polynomial P(x) of degree n>0 has at least one zero in the complex number system. N Zeros Theorem Every polynomial P(x) of degree n>0 can be expressed.

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

Copyright © Cengage Learning. All rights reserved. 4 Complex Numbers.

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

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

3.5 Complex Zeros & the Fundamental Theorem of Algebra.

1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Start Up Day 13.

Fundamental Theorem of Algebra

Complex Zeros and the Fundamental Theorem of Algebra

Complex Zeros Objective: Students will be able to calculate and describe what complex zeros are.

Review & 6.6 The Fundamental Theorem of Algebra

Rational Root and Complex Conjugates Theorem

3.8 Complex Zeros; Fundamental Theorem of Algebra

Lesson 2.5 The Fundamental Theorem of Algebra

4.5 The Fundamental Theorem of Algebra (1 of 2)

The Fundamental Theorem of Algebra (Section 2-5)

Warm Up Identify all the real roots of each equation.

Today in Precalculus Go over homework Notes: Remainder

Apply the Fundamental Theorem of Algebra

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

Fundamental Theorem of Algebra

4.5 The Fundamental Theorem of Algebra (1 of 2)

Fundamental Theorem of Algebra

6-8 Roots and Zeros Given a polynomial function f(x), the following are all equivalent: c is a zero of the polynomial function f(x). x – c is a factor.

Pre-AP Pre-Calculus Chapter 2, Section 5

5.8 Analyzing Graphs of Polynomials

Presentation transcript:

Today in Pre-Calculus Notes: –Fundamental Theorem of Algebra –Complex Zeros Homework Go over quiz

Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of the zeros may be repeated. The following statements about a polynomial function f are equivalent if k is a complex number: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. x – k is a factor of f(x) NOTE: If k is a nonreal zero, then it is NOT an x-intercept of the graph of f.

Example Write the polynomial function in standard form, and identify the zeros of the function and the x-intercepts of its graph. f(x) = (x – 3i)(x + 3i)(x + 5) f(x) = (x 2 + 3ix – 3ix – 9i 2 )(x + 5) f(x) = (x 2 + 9)(x + 5) f(x) = x 3 + 5x 2 + 9x + 45 Zeros: 3i, -3i, -5 x-intercepts: -5

Example Use the quadratic formula to find the zeros for: f(x) = 2x 2 + 5x + 6 These are called complex conjugates: a-bi and a+bi

Complex Conjugates For any polynomial, if a + bi is a zero, then a – bi is also a zero. Example: Write a standard form polynomial function of degree 4 whose zeros include: 3 + 2i and 4 – i So 3 – 2i and 4 + i are also zeros. f(x)= (x – 3 – 2i)(x – 3 + 2i)(x – 4 + i)(x – 4 – i) SHORTCUT: When [x – (a + bi)] and [x – (a – bi)] are factors their product always simplifies to: x 2 – 2ax + (a 2 + b 2 )

Complex Conjugates f(x)= (x – 3 – 2i)(x – 3 + 2i)(x – 4 + i)(x – 4 – i) SHORTCUT: x 2 – 2ax + (a 2 + b 2 ) f(x)= [x 2 – 2(3)x + ( )][x 2 – 2(4)x + (4 2 + (-1) 2 )] f(x)= (x 2 – 6x + 13)(x 2 – 8x + 17) x 4 – 8x x 2 –6x x 2 – 102x 13x 2 – 104x f(x) = x 4 – 14x x 2 – 206x + 221

Practice Write a polynomial function in standard form with real coefficients whose zeros are -1 – 2i and i. f(x)= (x i)(x +1 – 2i) f(x)= x 2 – 2(-1)x + ((-1) 2 +(-2) 2 ) f(x)= x 2 + 2x + 5

Practice Write a polynomial function in standard form with real coefficients whose zeros are -1, 2 and 1 – i. f(x)= (x + 1)(x – 2)(x – 1 + i)(x – 1 – i) f(x)= (x 2 – x – 2)(x 2 – 2x + 2) x 4 – 2x 3 + 2x 2 –x 3 + 2x 2 – 2x –2x 2 + 4x – 4 f(x) = x 4 – 3x 3 + 2x 2 + 2x – 4

Practice Write a polynomial function in standard form with real coefficients whose zeros and multiplicities are 1 (multiplicty 2); –2(multiplicity 3) f(x)= (x – 1)(x – 1)(x + 2)(x + 2)(x + 2) f(x)= (x 2 – 2x + 1)(x 2 + 4x + 4)(x + 2) x 4 + 4x 3 + 4x 2 –2x 3 – 8x 2 – 8x x 2 + 4x + 4 f(x) = (x 4 + 2x 3 –3x 2 – 4x + 4)(x + 2) f(x) = x 5 + 4x 4 + x 3 – 10x 2 – 4x + 8

Homework Pg. 234: 1-11odd, all