Real Zeros of Polynomial Functions. Solve x 3 – 2x + 1 = 0. How? Can you factor this? Can you use the quadratic formula? Now what if I tell you that one.

Slides:

Advertisements

Similar presentations

$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.

Advertisements

3.3 Zeros of polynomial functions

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

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

Zeros of Polynomial Functions

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 ?

Chapter 4 – Polynomials and Rational Functions

The Rational Zero Theorem

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.

Polynomial Functions Chapter 2 Part 1. Standard Form f(x)=ax 2 +bx+c Vertex Form f(x)=a(x-h) 2 +k Intercept Form f(x)=a(x-d)(x-e) y-int (0, c) let x =

Zeros of Polynomial Functions

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

Zeros of Polynomial Functions Section 2.5 Page 312.

Notes 2.4 –Real Zeros of Polynomial Functions

7.5.1 Zeros of Polynomial Functions

Finding Real Roots of Polynomial Equations

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,

Copyright © 2011 Pearson, Inc. 2.4 Real Zeros of Polynomial Functions.

Today in Pre-Calculus Go over homework Notes: –Real Zeros of polynomial functions –Rational Zeros Theorem Homework.

2.4 – Real Zeros of Polynomial Functions

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.

3.3 (2) Zeros of polynomials Descarte’s Rule of Signs Two theorems on bounds for zeros.

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’

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.

Zeros of Polynomials 2.5.

Using the Fundamental Theorem of Algebra 6.7. Learning Targets Students should be able to… -Use fundamental theorem of algebra to determine the number.

Plowing Through Sec. 2.4b with Two New Topics: Synthetic Division Rational Zeros Theorem.

Date: 2.4 Real Zeros of Polynomial Functions

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

Theorems About Roots of Polynomial Equations. Find all zeros: f(x)= x +x –x Synthetic Division one zero…need 2 more use (x – k), where.

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.

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.

6.5 Theorems About Roots of Polynomial Equations

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

Warm Up 9/2. Homework Questions Announcements Quiz Friday HW 2.4 pg. 206 #’s even, even.

LESSON 5.6 Rational Zeros of Polynomial Functions.

Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.

Warm Up: 1.  2 Finding Rational Zeros (Day 1) 3.

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

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

2.5 Zeros of Polynomial Functions Question to be answered: How do we determine zeros of functions that we cannot factor easily? Copyright © by Houghton.

Zeros (Solutions) Real Zeros Rational or Irrational Zeros Complex Zeros Complex Number and its Conjugate.

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.

Polynomial Long Division Review

Descartes’ Rule of Signs

Polynomial Long Division Review

3.7 The Real Zeros of a Polynomial Function

3.3 Real Zeros of Polynomials

2.5 Zeros of Polynomial Functions

5.8 Rational Zero Theorem.

Real Zeros Intro - Chapter 4.2.

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.

Zeros of Polynomial Functions

Polynomial Long Division Review

Warm Up #2 Factor completely. 2. 2x2 – 5x – 3 1. x2 – x – 12

Notes 5.6 (Day 1) Find Rational Zeros.

3.7 The Real Zeros of a Polynomial Function

Real Zeros of Polynomial Functions

Warm-up: Find f(2) for HW: pg. 248(2 – 14even, 18, 20, 26, 31, 46, 48)

Section 2.4: Real Zeros of Polynomial Functions

Find all the real zeros of the functions.

Divide using long division.

Presentation transcript:

Real Zeros of Polynomial Functions

Solve x 3 – 2x + 1 = 0. How? Can you factor this? Can you use the quadratic formula? Now what if I tell you that one root is x =1? Then x 3 – 2x + 1 = (x – 1)( ). Use synthetic division to find the missing factor. The missing factor is x 2 + x - 1, which cannot be factored. So use the quadratic formula to get (-1 ±√5)/2.

and this one… Find all of the roots of f(x) = x 4 + 2x 3 – 7x 2 – 20x – 12 given that x = 3 and x = -2 are roots. Answer: x = 3, -2 (double root) and -1. Verify this using your graphing calculator.

But what if you don’t know any zeros and cannot factor a polynomial? Rational Zero Theorem If f(x) = a n x n + … + a 0 and has integer coefficients, then every rational zero of f(x) has the form p/q where p is a factor of a 0 and q is a factor of a n. List the possible rational zeros for f(x) = 2x 3 + 3x 2 – 4x + 6. Answer: ±1,±2,±3,±6,±1/2,±3/2

Now that you know the possible zeros, you can use trial and error until you get down to a quadratic. Then you can factor or use the quadratic formula. Find the x-intercepts of f(x) = x 3 + 4x 2 + x – 6 Answer: 1, -2, -3 Find the x-intercepts of f(x) = x 4 - 6x 2 + 3x + 2 Answer: 1, 2, (-3 ±√5)/2

Upper and Lower Bounds Let f(x) be a polynomial with real coefficients and a positive leading coefficient. Suppose f(x) is divided by x – c using synthetic division. 1.If c > 0 and every number in the last row is non- negative, then c is an upper bound for the real zeros of f. 2.If c < 0 and every number in the last row alternates sign, then c is a lower bound. (Zero can be either positive or negative.)