4.4 Finding Rational Zeros. The rational root theorem If f(x)=a n x + +a 1 x+a 0 has integer coefficients, then every rational zero of f has the following.

Slides:

Advertisements

Similar presentations

The Rational Zero Theorem

Advertisements

4.4 Rational Root Theorem.

9.5 Addition, Subtraction, and Complex Fractions By: L. Kealii Alicea.

$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.

6.6 – Finding Rational Zeros. The Rational Zero Theorem If f(x) = a n x n + … + a 1 x + a 0 has integer coefficients, then every rational zero of f has.

4.3 – Location Zeros of Polynomials. At times, finding zeros for certain polynomials may be difficult There are a few rules/properties we can use to help.

Example 1 ±1 ± 2 Disregard the negative sign Factors of the constant:

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

3.3 Zeros of polynomial functions

2.5 Zeros of Polynomial Functions

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

5.5 Completing the Square p. 282 What is completing the square used for? ► Completing the square is used for all those not factorable problems!! ► It.

2.5 Real Zeros of Polynomial Functions Descartes Rule of Signs

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.

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

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.

5.6 Quadratic Formula & Discriminant p. 291 Discriminant: b 2 -4ac The discriminant tells you how many solutions and what type you will have. If the.

Section 2.5 Part Two Other Tests for Zeros Descartes’s Rule of Signs Upper and Lower Bounds.

EXAMPLE 2 Find all real zeros of f (x) = x 3 – 8x 2 +11x SOLUTION List the possible rational zeros. The leading coefficient is 1 and the constant.

The Rational Zero Theorem

The Fundamental Theorem of Algebra And Zeros of Polynomials

Finding Rational Zeros.

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

Finding Rational Zeros 6.6 pg. 359!. The rational zero theorem If f(x)=a n x + +a 1 x+a 0 has integer coefficients, then every rational zero of f has.

6.6 Finding Rational Zeros pg. 359! What is the rational zero theorem? What information does it give you?

2.1 Using Properties of Exponents

6.1 Using Properties of Exponents p Properties of Exponents a&b are real numbers, m&n are integers Follow along on page 323. Product Property :

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

Complex and Rational Zeros of Polynomials (3.4)(2) Continuing what you started on the worksheet.

Finding Real Roots of Polynomial Equations

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,

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

Section 4.3 Zeros of Polynomials. Approximate the Zeros.

Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm.

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’

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

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.

EXAMPLE 5 Use the result to write f (x) as a product of two factors. Then factor completely. f (x) = x 3 – 2x 2 – 23x + 60 The zeros are 3, – 5, and 4.

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.

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.

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.

7.2 Properties of Rational Exponents By: L. Keali’i Alicea.

1.8 Quadratic Formula & Discriminant p. 58 How do you solve a quadratic equation if you cannot solve it by factoring, square roots or completing the square?

Determine the number and type of roots for a polynomial equation

8.5 Add & Subtract Rational Expressions Algebra II.

Real Zeros of Polynomial Functions

PreCalculus 4-R Unit 4 Polynomial and Rational Functions Review Problems.

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

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.

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

3.3 Real Zeros of Polynomials

Real Zeros Intro - Chapter 4.2.

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

The Rational Zero Theorem

2.6 Find Rational Zeros pg. 128 What is the rational zero theorem?

Rational Root Theorem Math 3 MM3A1.

5.6 Find Rational Zeros.

The Rational Zero Theorem

The Rational Zero Theorem

Roots & Zeros of Polynomials III

5.6 Finding Rational Zeros Algebra II

Notes Over 6.6 Possible Zeros Factors of the constant

Presentation transcript:

4.4 Finding Rational Zeros

The rational root theorem If f(x)=a n x + +a 1 x+a 0 has integer coefficients, then every rational zero of f has the following form: p factors of constant term a 0 q factors of leading coefficient a n n … =

Example 1: Find rational roots of f(x)=x 3 +2x 2 -11x-12 1.List possible q=1(1) p=-12 (1,2,3,4,6,12) X= ±1/1,± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1 2.Test: X= x= Since -1 is a zero: (x+1)(x 2 +x-12)=f(x) Factor: (x+1)(x-3)(x+4)=0 x=-1 x=3 x=-4 You can use a graphing calculator to narrow your choices before you test!

Extra Example 1: Find rational zeros of: f(x)=x 3 -4x 2 -11x q=1 (1) p=30 (1,2,3,5,6,10,30) x= ±1/1, ± 2/1, ±3/1, ±5/1, ±6/1, ±10/1, ±15/1, ±30/1 2.Test: x= x= X= (x-2)(x 2 -2x-15)= (x-2)(x+3)(x-5)= x=2 x=-3 x=5

Example 2: f(x)=10x 4 -3x 3 -29x 2 +5x+12 1.List: q=10 (1,2,5,10) p=12 (1,2,3,4,6,12) x= ± 1/1, ± 2/1, ± 3/1, ± 4/1, ± 6/1, ±12/1, ± ½, ± 3/2, ± 1/5, ± 2/5, ± 3/5, ± 4/5, ± 6/5, ± 12/5, ± 1/10, ± 3/10, ± 12/10 2.w/ so many – sketch graph on calculator and find reasonable solutions: x= -3/2, -3/5, 4/5, 3/2 Check: x= -3/ Yes it works * (x+3/2)(10x 3 -18x 2 -2x+8)* (x+3/2)(2)(5x 3 -9x 2 -x+4) -factor out GCF (2x+3)(5x 3 -9x 2 -x+4) -multiply 1 st factor by 2 __ ____

Descartes’ Rule of Signs The number of positive real zeros = the number of sign changes in f(x) or less than by an even #. NO YES There are four changes. So there may be 4, 2 or 0 positive real zeros.

Descartes’ Rule of Signs cont. The number of negative real zeros = the number of sign changes in f(-x) or less than by an even #. YES NO There is one change. So there is 1 negative real zero.

Find the number of positive and negative real zeros. Then determine the rational zeros for g(x)= 5x 3 - 9x 2 – x + 4 There are 2 positive and 1 negative real zero 1. LC=5 CT=4 x:±1, ±2, ±4, ±1/5, ±2/5, ±4/5 *The graph of original shows 4/5 may be: x=4/ (x-4/5)(5x 2 -5x-5)= (x-4/5)(5)(x 2 -x-1)= mult.2 nd factor by 5 (5x-4)(x 2 -x-1)= -now use quad for last- * 4/5, 1+, __ ____

AAAA ssss ssss iiii gggg nnnn mmmm eeee nnnn tttt: page ,12,20,22