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

Slides:

Advertisements

Similar presentations

The Rational Zero Theorem

Advertisements

4.4 Rational Root Theorem.

Chapter 11 Polynomial Functions

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.

6.5 & 6.6 Theorems About Roots and the Fundamental Theorem of Algebra

Using Our Tools to Find the Zeros of Polynomials

2.5 Zeros of Polynomial Functions

LIAL HORNSBY SCHNEIDER

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

Chapter 4 – Polynomials and Rational Functions

The Rational Zero Theorem

The Fundamental Theorem of Algebra And Zeros of Polynomials

Sullivan Algebra and Trigonometry: Section 5.6 Complex Zeros; Fundamental Theorem of Algebra Objectives Utilize the Conjugate Pairs Theorem to Find the.

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

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.

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1 3 Polynomial and Rational Functions Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Roots & Zeros of Polynomials III

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

6.6 The Fundamental Theorem of Algebra

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,

5.5 Theorems about Roots of Polynomial Equations P

6.9 Rational Zero Theorem Parts of a polynomial function f(x) oFactors of the leading coefficient = q oFactors of the constant = p oPossible rational roots.

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

Lesson 3.4 – Zeros of Polynomial Functions Rational Zero Theorem

Ch 2.5: The Fundamental Theorem of Algebra

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

Copyright © 2009 Pearson Education, Inc. CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions.

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.

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’

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

Zeros of Polynomials 2.5.

Section 3.3 Theorems about Zeros of Polynomial Functions.

Copyright © 2009 Pearson Education, Inc. CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions.

Section 4.4 Theorems about Zeros of Polynomial Functions Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

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

Factor Theorem Using Long Division, Synthetic Division, & Factoring to Solve Polynomials.

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models.

Remainder Theorem If f(x) is divided by x – r, then the remainder is equal to f(r). We can find f(r) using Synthetic Division.

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 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.

Zero of Polynomial Functions Factor Theorem Rational Zeros Theorem Number of Zeros Conjugate Zeros Theorem Finding Zeros of a Polynomial Function.

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.

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

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

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.

Section 4.3 Polynomial Division; The Remainder and Factor Theorems Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

And the zeros are x =  3, x = , and x = 2 Since the remainder is –64, we know that x + 3 is not a factor. Quotient Divisor Dividend.

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

Real Zeros of Polynomial Functions

Pre-Calculus.  Step 1: How many complex zeros exist in the above function? 3 OR 6 36.

Remainder Theorem Let f(x) be an nth degree polynomial. If f(x) is divided by x – k, then the remainder is equal to f(k). We can find f(k) using Synthetic.

Real Zeros Intro - Chapter 4.2.

5-5 Theorems About Roots of Polynomial Equations

The Rational Zero Theorem

Section 3.4 Zeros of Polynomial Functions

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

5.7: Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra And Zeros of Polynomials

The Rational Zero Theorem

Roots & Zeros of Polynomials III

Remainder Theorem If P(x) is divided by x – r, then the remainder is equal to P(r). We can find P(r) using Synthetic Division.

Presentation transcript:

Rational Root Theorem

Finding Zeros of a Polynomial Function Use the Rational Zero Theorem to find all possible rational zeros. Use Synthetic Division to try to find one rational zero (the remainder will be zero). If “n” is a rational zero, factor the original polynomial as (x – n)q(x). Test remaining possible rational zeros in q(x). If one is found, then factor again as in the previous step. Continue in this way until all rational zeros have been found. See if additional irrational or non-real complex zeros can be found by solving a quadratic equation.

Finding Rational Zeros So which one do you pick? Pick any. Find one that is a zero using synthetic division... Possible zeros are + 1, + 2, + 4, + 8

Let’s try 1. Use synthetic division – –8 1 2 –8 0 1 is a zero of the function The depressed polynomial is x 2 + 2x – 8 Find the zeros of x 2 + 2x – 8 by factoring or (by using the quadratic formula)… (x + 4)(x – 2) = 0 x = –4, x = 2 The zeros of f(x) are 1, –4, and 2

Use synthetic division Find all possible rational zeros of:

Example Continued This new factor has the same possible rational zeros: Check to see if -1 is also a zero of this: Conclusion:

Example Continued This new factor has as possible rational zeros: Check to see if -1 is also a zero of this: Conclusion:

Example Continued Check to see if 1 is a zero: Conclusion:

Example Continued Check to see if 2 is a zero: Conclusion:

Example Continued Summary of work done:

Using The Linear Factorization Theorem We can now use the Linear Factorization Theorem for a fourth- degree polynomial.

Using The Linear Factorization Theorem

Descartes Rule of Signs is a method for determining the number of sign changes in a polynomial function.

Polynomial FunctionSign Changes Conclusion There is one positive real zero. Descarte’s Rule of Signs and Positive Real Zeros

Descarte’s Rule of Signs Example Determine the possible number of positive real zeros and negative real zeros of P(x) = x 4 – 6x 3 + 8x 2 + 2x – 1. We first consider the possible number of positive zeros by observing that P(x) has three variations in signs. + x 4 – 6x 3 + 8x 2 + 2x – 1 Thus, by Descartes’ rule of signs, f has either 3 or 3 – 2 = 1 positive real zeros. For negative zeros, consider the variations in signs for P(  x). P(  x) = (  x) 4 – 6(  x) 3 + 8(  x) 2 + 2(  x)  1 = x 4 + 6x 3 + 8x 2 – 2x – 1 Since there is only one variation in sign, P(x) has only one negative real root Total number of zeros 4 Positive:3 1 Negative:1 1 Nonreal:0 2