Mrs. McConaughy Honors Algebra 2 1 Lesson 6.5: Finding Rational Zeros Objective: To find the rational zeros of a polynomial function.

Slides:

Advertisements

Similar presentations

5.5: Polynomial Long Division and Synthetic Division

Advertisements

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

Digital Lesson Polynomial Functions.

2.5 Zeros of Polynomial Functions

Section 5.5 – The Real Zeros of a Rational Function

OBJECTIVE: I will be able to calculate the real zeros of a polynomial function using synthetic division and the Rational Zero Theorem through use of in-class.

Objective Video Example by Mrs. G Give It a Try Lesson 6.6  Find the rational and real zeros of a polynomial function.

1 5.6 Complex Zeros; Fundamental Theorem of Algebra In this section, we will study the following topics: Conjugate Pairs Theorem Finding a polynomial function.

The Fundamental Theorem of Algebra And Zeros of Polynomials

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.

Finding Rational Zeros.

Copyright © Cengage Learning. All rights reserved.

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.

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.

Daily Warm-UP Quiz Factor completely: 1. x 4 – 17x x x x 4 – 7x x 8 – x 6 – x 6 – 36.

Obj: To solve equations using the Rational Root Theorem.

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

2.6 – Find Rational zeros Coach Bianco.

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

7.6 Rational Zero Theorem Algebra II w/ trig. RATIONAL ZERO THEOREM: If a polynomial has integer coefficients, then the possible rational zeros must be.

Polynomials Integrated Math 4 Mrs. Tyrpak. Definition.

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.

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.

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

Chapter 2 Polynomial and Rational Functions. Warm Up

Objectives: 1. Use the factor theorem. 2. Factor a polynomial completely.

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

Remainder and Factor Theorems

1/27/2016 Math 2 Honors - Santowski 1 Lesson 21 – Roots of Polynomial Functions Math 2 Honors - Santowski.

January 19, 2012 At the end of today, you will be able to find ALL zeros of a polynomial. Warm-up: Check HW 2.4 Pg. 167 #20-21, 27-29, , 55, 65,

9.8 Day 2 – Finding Rational Zeros. The Rational Zero Theorem: If has integer coefficients, then every rational zero of f have the following form:

Solving Polynomials.

Solving polynomial equations

Long and Synthetic Division. Long Division Polynomial long division can be used to divide a polynomial d(x), producing a quotient polynomial q(x) and.

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.

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.

Warm Up Compute the following by using long division.

Welcome to Precalculus!

Theorems about Roots of Polynomial Equations and

Copyright © Cengage Learning. All rights reserved.

3.4 Zeros of Polynomial Functions

Section 6.6 The Fundamental Theorem of Algebra

Copyright © Cengage Learning. All rights reserved.

Lesson 23 – Roots of Polynomial Functions

5.6 – Find the Rational Zeros

Real Zeros Intro - Chapter 4.2.

Finding Real Roots of Polynomial Equations

Rational Root Theorem Math 3 MM3A1.

Notes 5.6 (Day 1) Find Rational Zeros.

Section 3.4 Zeros of Polynomial Functions

5.7 Apply the Fundamental Theorem of Algebra

Finding Zeros of Polynomials

Zeros of a Polynomial Function

The Rational Zero Theorem

Apply the Fundamental Theorem of Algebra

6.7 Using the Fundamental Theorem of Algebra

Section 2.4: Real Zeros of Polynomial Functions

Presentation transcript:

Mrs. McConaughy Honors Algebra 2 1 Lesson 6.5: Finding Rational Zeros Objective: To find the rational zeros of a polynomial function

Mrs. McConaughy Honors Algebra 2 2 Before we start, recall: A rational number is one that can be written ________________________ ______________________________. The real-number zeros of a polynomial function are either ________ or __________. as the quotient (ratio) of two integers. rationalirrational

Mrs. McConaughy Honors Algebra 2 3 EXAMPLES Find the zeros of the following functions: f(x) = x 2 - 3x – 4 f(x) = x 2 - 2

Mrs. McConaughy Honors Algebra 2 4 The Rational Zero Test Consider the polynomial f(x) = a n x n + a n-1 x n a 1 x + a 0 with integer coefficients. Every rational zero of f has the form p = ______________________ q factors of constant term factors of leading coefficient

Mrs. McConaughy Honors Algebra 2 5 EXAMPLE 1 Find the rational zeros of the following function: f(x) = x 4 - x 3 – 5x 2 + 3x + 6 STEP 1: Find all possible rational zeros: STEP 2: Use the Remainder Theorem and Synthetic Division to determine actual zeros: f(x) = ________________ ________________________________ (x-1) (x-2)(x 2 -3)

Mrs. McConaughy Honors Algebra 2 6 From the previous factorization, you can see that f has only ______ rational zeros: ______________ (The other two zeros, ______, are irrational.) two f(x) = x 4 -x 3 –5x 2 +3x+6 NOTE: Graphing can speed up the process of finding the first zero. x = – 1 and x = 2.

Mrs. McConaughy Honors Algebra 2 7 EXAMPLE 2 Find the real zeros of the following function: f(x) = 10x 3 – 15x x + 12 STEP 1: Find all possible rational zeros: STEP 2: Use the Remainder Theorem and Synthetic Division to determine actual zeros. Shorten your search for the initial rational zero by graphing on a graphics calculator, first. graphing f(x) = ___________________________

Mrs. McConaughy Honors Algebra 2 8 The Fundamental Theorem of Algebra Counting complex and repeated solutions, an nth-degree polynomial equation has exactly n solutions. Carl Friedrich Gauss

Mrs. McConaughy Honors Algebra 2 9 FINAL CHECKS FOR UNDERSTANDING 1.List the possible rational zeros given by the Rational-Zero Test for: f(x) = 3x 3 – 6x 2 + 7x Factor x 3 – 3x 2 - 6x + 8 completely. 3. Factor x 3 +4x 2 - x - 4 completely. 4.Which is the more efficient method, factoring or rational-zero test, for finding all real zeros of the function, g(x) = x 3 + x 2 -2x – 2? Explain.

Mrs. McConaughy Honors Algebra 2 10 In 4-6, find all real zeros of the polynomial function. y = 4x 3 – 12x 2 - x + 15y = -4x x 2 - 8x - 3y = -3x x x + 16 HOMEWORK ASSIGNMENT: ______________________________

Mrs. McConaughy Honors Algebra 2 11 Click to return to lesson. Three reasonable choices for x would be: _____________ -6/5, 1/2, 2 f(x) = _________________________