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

Slides:

Advertisements

Similar presentations

4.4 Rational Root Theorem.

Advertisements

Dr. Claude S. Moore Danville Community College

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.

3.3 Zeros of polynomial functions

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.

Descartes’s Rule of Signs & Bounds: Things that make your life easier TS: Making decisions after reflection and review.

2.5 Zeros of Polynomial Functions

Upper and Lower Bounds for Roots

Section 5.5 – The Real Zeros of a Rational Function

Zeros of Polynomial Functions

The Real Zeros of a Polynomial Function

Warm-up Find all the solutions over the complex numbers for this polynomial: f(x) = x4 – 2x3 + 5x2 – 8x + 4.

Problem of the day Can you get sum of 99 by using all numbers (0-9) and only one mathematical symbols ?

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

The Fundamental Theorem of Algebra And Zeros of Polynomials

Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Three Polynomial & Rational 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)

SECTION 3.5 REAL ZEROS OF A POLYNOMIAL FUNCTION REAL ZEROS OF A POLYNOMIAL FUNCTION.

Quick Crisp Review Zeros of a polynomial function are where the x-intercepts or solutions when you set the equation equal to zero. Synthetic and long division.

Key Concept 1. Example 1 Leading Coefficient Equal to 1 A. List all possible rational zeros of f (x) = x 3 – 3x 2 – 2x + 4. Then determine which, if any,

4.5: More on Zeros of Polynomial Functions The Upper and Lower Bound Theorem helps us rule out many of a polynomial equation's possible rational roots.

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

Real Zeros of Polynomial Functions

Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.4 Real Zeros of Polynomial Functions.

Section 3.3 Real Zeros of Polynomial Functions. Objectives: – Use synthetic and long division – Use the Remainder and Factor Theorem – Use the Rational.

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.

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

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

3.4 Zeros of Polynomial Functions. The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where n>0, then f has at least one zero in.

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

Zeros of Polynomials 2.5.

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

Sullivan PreCalculus Section 3.6 Real Zeros of a Polynomial Function Objectives Use the Remainder and Factor Theorems Use Descartes’ Rule of Signs Use.

Sullivan Algebra and Trigonometry: Section 5.2 Objectives Use the Remainder and Factor Theorems Use Descartes’ Rule of Signs Use the Rational Zeros Theorem.

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 Rational Zero Theorem The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all.

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.

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.

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.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

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.

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.

Slide Copyright © 2009 Pearson Education, Inc. Active Learning Lecture Slides For use with Classroom Response Systems © 2009 Pearson Education, Inc.

Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.

Jonathon Vanderhorst & Callum Gilchrist Period 1.

4-5 Exploring Polynomial Functions Locating Zeros.

3.3 Real Zeros of Polynomials. Rational Zero Theorem If the polynomial P(x) has integer coefficients then every rational zero of P is of the form p/q.

Dividing Polynomials Two options: Long Division Synthetic Division.

Descartes’ Rule of Signs

College Algebra Chapter 3 Polynomial and Rational Functions

4.5 Locating Zeros of a Polynomial Function

3.7 The Real Zeros of a Polynomial Function

3.3 Real Zeros of Polynomials

Locating Real Zeros of Polynomials

Sullivan Algebra and Trigonometry: Section 5

4.2 Real Zeros Finding the real zeros of a polynomial f(x) is the same as solving the related polynomial equation, f(x) = 0. Zero, solution, root.

Finding Zeros of Polynomials

3.7 The Real Zeros of a Polynomial Function

Zeros of a Polynomial Function Shortcuts?

College Algebra Chapter 3 Polynomial and Rational Functions

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

Section 2.4: Real Zeros of Polynomial Functions

Find all the real zeros of the functions.

Presentation transcript:

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

Descartes’ Rule of Signs The number of positive real zeros of a polynomial function P(x), with real coefficients, is equal to the number of variations in sign of the terms of P(x) or is less than this number by a multiple of 2. The number of negative real zeros is equal to the number of variations in sign of the terms of P(-x) or is less than this number by a multiple of 2.

Example 1 Find the possible number of positive and negative real zeros of The problem isn’t asking for the zeros themselves, but what are the possible number of them. This can help narrow down the possibilities when you do go on to find the zeros.

Example 2 Find the possible number of positive and negative real zeros of

Upper and Lower Bounds Let P(x), a polynomial function with positive leading coefficient, be divided by x – c. If c > 0 and all coefficients in the quotient and remainder are nonnegative, then c is an upper bound of the zeros. If c < 0 and the coefficients in the quotient and remainder alternate in sign, then c is a lower bound of the zeros.

Example 3 Show that all real roots of the equation Lie between –4 and 4

The Intermediate Value Theorem If f(x) is a polynomial function and f(a) and f(b) have different signs, then there is at least one value, c, between a and b such that f(c)=0.

This works because 0 separates the positives from the negatives. So to go from positive to negative or vice a versa you would have to hit a point in between that goes through 0.

Example 4 Show that has real zeros between 3 and 4 and 0 and -1.

Example 5 Find the upper and lower bounds. Find the possible number of positive and negative zeros. Find the location of the zeros.

Example 6 Find the upper and lower bounds. Find the possible number of positive and negative zeros. Find the location of the zeros.

Summary Descartes’ Rule of Signs is used to determine the number of possible positive and negative zeros. The Upper and Lower Bound Theorem is used to determine the bounds of the zeros. The Intermediate Value Theorem is used to determine between which real numbers a zero occurs.