4.4 Rational Root Theorem.

Slides:

Advertisements

Similar presentations

The Rational Zero Theorem

Advertisements

Notes 6.6 Fundamental Theorem of Algebra

$100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300 $400 $500 $100 $200 $300.

Rational Root Theorem.

6.5 Theorems About Roots of Polynomial Equations Imaginary Roots.

Zeros of Polynomial Functions Section 2.5. Objectives Use the Factor Theorem to show that x-c is a factor a polynomial. Find all real zeros of a polynomial.

4.4 Notes The Rational Root Theorem. 4.4 Notes To solve a polynomial equation, begin by getting the equation in standard form set equal to zero. Then.

2.5 Zeros of Polynomial Functions

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

2.5 Descartes’ Rule of Signs To apply theorems about the zeros of polynomial functions To approximate zeros of polynomial functions.

2.5 Real Zeros of Polynomial Functions Descartes Rule of Signs

2.7 Apply the Fundamental Theorem of Algebra day 2

EXAMPLE 4 Use Descartes’ rule of signs Determine the possible numbers of positive real zeros, negative real zeros, and imaginary zeros for f (x) = x 6.

Rational Root Theorem. Finding Zeros of a Polynomial Function Use the Rational Zero Theorem to find all possible rational zeros. Use Synthetic Division.

Chapter 4 – Polynomials and Rational Functions

The Rational Zero Theorem

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)

Zeros of Polynomial Functions Section 2.5 Page 312.

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

Using Technology to Approximate Roots of Polynomial Equations.

Splash Screen. Example 1 Identify Possible Zeros A. List all of the possible rational zeros of f(x) = 3x 4 – x Answer:

Rational Root and Complex Conjugates Theorem. Rational Root Theorem Used to find possible rational roots (solutions) to a polynomial Possible Roots :

5.5 Theorems about Roots of Polynomial Equations P

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

Real Zeros of Polynomial Functions 2-3. Descarte’s Rule of Signs Suppose that f(x) is a polynomial and the constant term is not zero ◦The number of positive.

Do Now: Find all real zeros of the function.

Warm up Write the quadratic f(x) in vertex form..

Chapter 3 – Polynomial and Rational Functions Real Zeros of Polynomials.

Copyright © 2011 Pearson Education, Inc. The Theory of Equations Section 3.3 Polynomial and Rational Functions.

Section 5.5 The Real Zeros of a Polynomial Function.

The Real Zeros of a Polynomial Function Obj: Apply Factor Theorem, Use Rational Zero Theorem to list roots, Apply Descartes’ Rule of Signs to determine.

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.

The Original f(x)=x 3 -9x 2 +6x+16 State the leading coefficient and the last coefficient Record all factors of both coefficients According to the Fundamental.

Chapter 4: Polynomial and Rational Functions. Warm Up: List the possible rational roots of the equation. g(x) = 3x x 3 – 7x 2 – 64x – The.

UNIT 2, LESSON 5 DESCARTES LAW OF SIGNS. FINDING POLYNOMIALS WITH GIVEN ZEROS If we are given the zeros of a polynomial, we can generate the polynomial.

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

7.6 Rational Zero Theorem Objectives: 1. Identify the possible rational zeros of a polynomial function. 2. Find all the rational zeros of a polynomial.

5.5 – Apply the Remainder and Factor Theorems The Remainder Theorem provides a quick way to find the remainder of a polynomial long division problem.

4.4 The Rational Root Theorem

Chapter 4: Polynomial and Rational Functions. Determine the roots of the polynomial 4-4 The Rational Root Theorem x 2 + 2x – 8 = 0.

POLYNOMIALS 9/20/2015. STANDARD FORM OF A POLYNOMIAL ax + bx + cx … mx + k nn-1n-2 The degree is ‘n’

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

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

PreCalculus Section 2.6 Solve polynomial equations by factoring and the Rational Roots Theorem. Solve by factoring: x 3 + 5x 2 – 4x – 20 = 0 x 6 – x 3.

Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.

Determine the number and type of roots for a polynomial equation

TRASHKETBALL PRECALCULUS CHAPTER 2 QUIZ. WHAT IS THE VERTEX AND WHAT ARE THE INTERCEPTS?

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

Section 3.4 Zeros of Polynomial Functions

Descartes Rule of Signs Positive real zeros = Negative real zeros =

Roots and Zeros 5.7.

Notes Over 3.4 The Rational Zero Test

4.4 The Rational Root Theorem

Real Zeros Intro - Chapter 4.2.

Rational Zero Theorem Rational Zero Th’m: If the polynomial

Rational Root and Complex Conjugates Theorem

Finding Real Roots of Polynomial Equations

Lesson 7.2: Finding Complex Solutions of Polynomial Equations

5-5 Theorems About Roots of Polynomial Equations

Section 3.4 Zeros of Polynomial Functions

4.4 The Rational Root Theorem

Section 3 – The Remainder and Factor Theorems

The Rational Zero Theorem

Rational Root Theorem.

2.5 The Real Zeros of a Polynomial Function

Notes Over 6.6 Possible Zeros Factors of the constant

Warm-up: CW: Cumulative Review 5 F(x) = 2x3 + 3x2 – 11x – 6

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.

Presentation transcript:

4.4 Rational Root Theorem

Rational Root Theorem give direction in testing possible zeros. Let a0xn + a1xn-1 + …an-1x + an = 0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of an and q is a factor of a0.

Ex 1 List all the possible rational roots then determine the rational roots. 3x3 – 13x2 + 2x + 8 = 0

Ex 2 Find ALL of the roots. x3 + 6x2 – 13x – 6 = 0

Descartes’ Rule of Signs Used to determine the possible number of positive real zeros a polynomial has. P(x) is a polynomial in descending order. The # of positive real zeros is the same as the number of sign changes of the coefficients or is less than this by an even number. The # of negative real zeros is the same as the number of sign changes of the coefficients of P(-x), or less than by an even number. (Ignore zero coefficients.)

Ex 3 find the number of possible positive and negative real zeros for then determine the rational zeros: P(x) = 2x4 – x3 – 2x2 + 5x + 1