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

Slides:

Advertisements

Similar presentations

7-5 Roots and Zeros 7-6 Rational Zero Theorem

Advertisements

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Section 6.6 Finding Rational Zeros. Rational Zero Theorem Synthetic & Long Division Using Technology to Approximate Zeros Today you will look at finding.

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

Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.

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.

6.7 – Using the Fundamental Theorem of Algebra

Zeros of Polynomials PolynomialType of Coefficient 5x 3 + 3x 2 + (2 + 4i) + icomplex 5x 3 + 3x 2 + √2x – πreal 5x 3 + 3x 2 + ½ x – ⅜rational 5x 3 + 3x.

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

Complex Zeros; Fundamental Theorem of Algebra

Zeros of Polynomial Functions Section 2.5 Page 312.

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.

Section 5.5 Theorems about Roots of Polynomial Equations.

Real Zeros of a Polynomial Function Objectives: Solve Polynomial Equations. Apply Descartes Rule Find a polynomial Equation given the zeros.

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

7.5.1 Zeros of Polynomial Functions

Finding Real Roots of Polynomial Equations

6.6 The Fundamental Theorem of Algebra

6-5 Theorems About Roots of Polynomial Equations

2.6 – Find Rational zeros Coach Bianco.

5.5 Theorems about Roots of Polynomial Equations P

Solving Polynomial Equations Section 4.5 beginning on page 190.

 Evaluate a polynomial  Direct Substitution  Synthetic Substitution  Polynomial Division  Long Division  Synthetic Division  Remainder Theorem 

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 +

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Ch 2.5: The Fundamental Theorem of Algebra

Do Now: Find all real zeros of the function.

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’

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

Section 3.3 Theorems about Zeros of Polynomial Functions.

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

Topic: U4L5 Remainder and Factor Theorems EQ: Can I correctly apply the Remainder and Factor Theorems to help me factor higher order polynomials?

Chapter 2 Polynomial and Rational Functions. Warm Up

Sec. 6-5: Theorems About Roots of Polynomial Equations.

Solving Polynomial Equations by Factoring Factoring by grouping Ex. 1. Solve:

6.5 Theorems About Roots of Polynomial Equations

Remainder and Factor Theorems

6-5 & 6-6 Finding All Roots of Polynomial Equations Warm Up: Factor each expression completely. 1. 2y 3 + 4y 2 – x 4 – 6x 2 – : Use factoring.

Fundamental Theorem of Algebra Every polynomial function of positive degree with complex coefficients has at least one complex zero.

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

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

Warm-Up: November 16, Homework Questions? Zeros of Polynomial Functions Section 2.5.

Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem.

Algebra 2 List all the integer factors for the number below: 36.

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

Algebra Finding Real Roots of Polynomial Equations.

Algebra 2. Solve for x Algebra 2 (KEEP IN MIND THAT A COMPLEX NUMBER CAN BE REAL IF THE IMAGINARY PART OF THE COMPLEX ROOT IS ZERO!) Lesson 6-6 The Fundamental.

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

Section 3.4 Zeros of Polynomial Functions

Roots and Zeros 5.7.

Section 6.6 The Fundamental Theorem of Algebra

Real Zeros Intro - Chapter 4.2.

Rational Root and Complex Conjugates Theorem

Finding Real Roots of Polynomial Equations

5-5 Theorems About Roots of Polynomial Equations

7.5 Zeros of Polynomial Functions

Rational Root Theorem Math 3 MM3A1.

Lesson 2.5 The Fundamental Theorem of Algebra

Notes 5.6 (Day 1) Find Rational Zeros.

Zeros of a Polynomial Function

Warm Up – Look over your notes/examples from last week Any questions??

Zeros of polynomial functions

Writing Polynomial Functions

Divide using long division.

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

Fundamental Theorem of Algebra Roots/Zeros/X-Intercepts

Presentation transcript:

OUTLINE  Homework (and homework questions)  Ask any review questions you want  Review long division, solve by factoring and graphing calculators  BREAK  6.5 Notes (not on quiz)  Homework time / study time

Homework:

Quiz Friday! About pts  Key is online! Class notes are online!  Questions?

Review Long division

Review Solve by Factoring

Graphing calculators  Find all the real zeros of each  How many roots or zeros should each have?

Section 6.5 Rational Root Theorem!

Objectives  Use the rational, irrational, and imaginary root theorems to locate all the roots of a given polynomial equation

Rational Root Theorem  This only works if we have integer coefficients.  Possible rational roots =

Find the rational roots of  List the possible rational roots:  Test using substitution or synthetic division

Use the rational root theorem to find ALL roots

The Irrational Root Theorem  If is a root of a polynomial equation with rational coefficients, then the conjugate is also a root  In other words, irrational roots occur in conjugate pairs.  If is a root, then is a root

The Imaginary Root Theorem  If the imaginary number a + bi is a root of a polynomial equation with real coefficients, then the conjugate a – bi also is a root  Imaginary roots also occur in conjugate pairs

 If 3, and are roots of a polynomial with integer coefficients, find two additional roots.  What is the degree of this polynomial?

Find a third-degree polynomial equation with rational coefficients that has roots 3 and 1 + i

Homework  Page 333 (1-3), 7, 8, 11, (16-18), 23, 24 (32-36) even  Due Monday