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

Slides:

Advertisements

Similar presentations

Notes 6.6 Fundamental Theorem of Algebra

Advertisements

SECTION 3.6 COMPLEX ZEROS; COMPLEX ZEROS; FUNDAMENTAL THEOREM OF ALGEBRA FUNDAMENTAL THEOREM OF ALGEBRA.

Zeros of Polynomial Functions

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.

5.7 Apply the Fundamental Theorem of Algebra

Academy Algebra II/Trig 5.5: The Real Zeros of a Polynomial Functions HW: p.387 (14, 27, 30, 31, 37, 38, 46, 51)

Complex Zeros; Fundamental Theorem of Algebra

9.9 The Fundamental Theorem of Algebra

Zeros of Polynomial Functions Section 2.5 Page 312.

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

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

7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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.

COMPLEX ZEROS: FUNDAMENTAL THEOREM OF ALGEBRA Why do we have to know imaginary numbers?

3.4 Zeros of Polynomial Functions Obj: Find x-intercepts of a polynomial Review of zeros A ZERO is an X-INTERCEPT Multiple Zeros the zeros are x = 5 (mult.

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

Chapter 2 Polynomial and Rational Functions. Warm Up

2.5 The Fundamental Theorem of Algebra Students will use the fundamental theorem of algebra to determine the number of zeros of a polynomial. Students.

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.

3.6 Complex Zereos. The Fundamental Theorem of Algebra The Fundamental Theorem of Algebra says that every polynomial with complex coefficients must have.

7.5 Roots and Zeros Objectives: The student will be able to…

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

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall. Section 5.6 Complex Zeros; Fundamental Theorem of Algebra.

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

Chapter 3 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Zeros of Polynomial Functions.

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.

Solving Polynomials.

7.5 Roots and Zeros Objectives:

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

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

The Fundamental Theorem of Algebra Intro - Chapter 4.6.

Objective: SWBAT understand and apply the Fundamental Theorem of Algebra and the Linear Factorization Theorem to find a polynomial given real and complex.

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

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.

REMAINDER THEOREM HONORS ADVANCED ALGEBRA PRESENTATION 2-7.

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.

Conjugate Pairs Theorem Every complex polynomial function of degree n  1 has exactly n complex zeros, some of which may repeat. 1) A polynomial function.

3.5 Complex Zeros & the Fundamental Theorem of Algebra.

“Is it better to be feared or respected? And I say, is it too much to ask for both?”

Fundamental Theorem of Algebra

Section 3.4 Zeros of Polynomial Functions

Section 6.6 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra

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

Fundamental Theorem of Algebra

Warm - Up Perform the operation and write the result in standard form

3.8 Complex Zeros; Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions

Lesson 2.5 The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra (Section 2-5)

Lesson 2.5 The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Zeros of a Polynomial Function

The Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

Lesson: _____ Section 2.5 The Fundamental Theorem of Algebra

4.6 - Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

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.

(1) Find all of the zeros of f.

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

Pre-AP Pre-Calculus Chapter 2, Section 5

Fundamental Theorem of Algebra

Fundamental Theorem of Algebra Roots/Zeros/X-Intercepts

Presentation transcript:

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

HWQ Perform the operation and simplify. Work must be shown to receive credit.

2.5 The Fundamental Theorem of Algebra Students will find all zeros of polynomial functions, including complex zeros. Students will find conjugate pairs of complex zeros. Students will find zeros of polynomials by factoring.

The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, then f(x) has n zeros, real and (or) imaginary.

Example 1: Real Zeros of a Polynomial Function Counting multiplicity, justify that the second-degree polynomial function has exactly two factors and zeros.

Example 2: Real and Imaginary Complex Zeros of a Polynomial Function Justify that the third degree polynomial function has exactly three factors and zeros.

Example 3: Finding the zeros of a Polynomial Function Write as the product of linear factors, and list all the zeros of f.

Example 4: Finding a Polynomial with Given Zeros Find a fourth degree polynomial function with real coefficients that has - 1, - 1, and 3i as zeros.

Find all zeros without graphing the polynomial first given is a zero: Example 4: More practice

Example 5:Factoring a Polynomial For the following polynomial list all zeros.

You try: Solve by factoring:

Solve: Example 6:More practice

Homework 2.5 pg. 140 #9-35 odd, 45, 47, 57 **QUIZ TOMORROW OVER SECTIONS 2.4 & 2.5**