Chapter 6 Final Touches. Pg 325 #42-47 Determine whether each binomial is a factor of Reminds me of a “bank” Limits the ones I have to check! Of course.

Slides:

Advertisements

Similar presentations

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

Advertisements

Notes 6.6 Fundamental Theorem of Algebra

Problem of the Day. Division and Rational Root Theorem TS: Making decisions after reflection and review Obj: Review polynomial division and how to find.

Rational Root Theorem.

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

Zeros of Polynomial Functions

Pre Calc Lesson 2.2 Synthetic Division ‘Remainder’ and ‘Factor’ Theorems Review Long Division: 5365 ÷ 27 Now review ‘long division’ of polynomials: (2x.

Chapter 4 – Polynomials and Rational Functions

Remainder and Factor Theorem Unit 11. Definitions Roots and Zeros: The real number, r, is a zero of f(x) iff: 1.) r is a solution, or root of f(x)=0 2.)

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.

1 Polynomial Functions Exploring Polynomial Functions Exploring Polynomial Functions –Examples Examples Modeling Data with Polynomial Functions Modeling.

Polynomials and Polynomial Functions

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

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

9.9 The Fundamental Theorem of Algebra

4-5, 4-6 Factor and Remainder Theorems r is an x intercept of the graph of the function If r is a real number that is a zero of a function then x = r.

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.

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

Using Technology to Approximate Roots of Polynomial Equations.

6.6 The Fundamental Theorem of Algebra

Lesson 2-6 Solving Polynomial Equations by Factoring – Part 2.

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

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

6-5 Theorems About Roots of Polynomial Equations

2.6 – Find Rational zeros Coach Bianco.

5.5 Theorems about Roots of Polynomial Equations P

Theorems About Roots of Polynomial Equations

 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 +

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’

Essential Question: How do I divide polynomials? 1.How do I perform polynomial long division? 2.How do I perform polynomial synthetic division?

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.

Section 3.3 Theorems about Zeros of Polynomial Functions.

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

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

Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the.

3.6 The Real Zeros of Polynomial Functions Goals: Finding zeros of polynomials Factoring polynomials completely.

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.

Solving polynomial equations

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

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

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

Zeros of Polynomial Functions A-APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is.

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

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.

Warm-ups Week 8 10/8/12 Find the zeros of f(x) = x3 + 2x2 – 13x + 10 algebraically (without a graphing calculator). What if I told.

HOMEWORK CHECK.

Essential Question: How do I divide polynomials?

Rational Root Theorem and Fundamental Theorem of Algebra

3.4 Zeros of Polynomial Functions

Objectives Use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation of least degree with given roots. Identify all of the.

Section 6.6 The Fundamental Theorem of Algebra

Real Zeros Intro - Chapter 4.2.

Rational Root Theorem and Fundamental Theorem of Algebra

7.5 Zeros of Polynomial Functions

Lesson 2.5 The Fundamental Theorem of Algebra

Zeros of a Polynomial Function

Rational Root Theorem.

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

Presentation transcript:

Chapter 6 Final Touches

Pg 325 #42-47 Determine whether each binomial is a factor of Reminds me of a “bank” Limits the ones I have to check! Of course a graph helps too!!!

If “it” goes in evenly, it’s a

FACTOR

If “it” goes in evenly, it’s a FACTOR We know the remainder is 0

Ultimate Goal To solve polynomials To graph polynomials If we can get the polynomial in factored form, then life is pretty easy!

Tool to help find factors Long Division – Always works Synthetic Division – Works with linear factors Especially when written in the form (x-a)

Tool to help find factors We need a BANK to limit the choices RATIONAL ROOT THEOREM If your polynomial is in standard form and If your polynomial has integer coefficients Then the bank consists of The possible rational roots are found in the bank

Example List the possible rational roots for

Before we extend this…

BOGO, new perspective Irrational Root Theorem – If the is a root, then is also a root – If is a root, then is also a root Imaginary Root Theorem – If is a root, then is also a root They are called conjugates of each other!

Examples Given the polynomial, find the other roots

Intermission Stand up and stretch for 1 minute Then use your calc to answer pg 318 #49

Back to finding all the roots Solve:

Back to finding all the roots Solve:

Back to finding all the roots Solve:

Fundamental Theorem of Algebra (Corollary) n th degree polynomials have exactly n roots (including real and imaginary) has 3 roots. Period. has 4 roots. Period.

All done! Solve

All done! Solve

Study Form a study group tonight! Do 6.5 pg 339 #4, 7, 12, 13, 17, 19, 21, 38 Do 6.6 pg 343 #10-12 Download Geist’s notes (from my website) Review Solving Quadratics Do Chap 5 Review pg. 300 #24, 27, 31, 38 Friday test: Chap 6 / part of Chap 5 Retest Monday: Review for Cumulative Tuesday: Cumulative 5-6 test Wed: Prepare for Final and take probability test Friday: 200 point Final