Section 2.4: Real Zeros of Polynomial Functions

Slides:

Advertisements

Similar presentations

Rational Zeros Theorem Upper & Lower Bounds Long Division

Advertisements

2.5 Zeros of Polynomial Functions

Section 5.5 – The Real Zeros of a Rational Function

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

The Remainder and Factor Theorems Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division,

Chapter 2 Polynomial and Rational Functions. Warm Up 2.3  An object is launched at 19.6 meters per second from a meter tall platform. The equation.

Notes 2.4 –Real Zeros of Polynomial Functions

Rational Root Theorem By: Yu, Juan, Emily. What Is It? It is a theorem used to provide a complete list of all of the possible rational roots of the polynomial.

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.

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

2.4 – Real Zeros of Polynomial Functions

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

Homework Lesson 2.3 Read: Pages Page 127: #1-61 (EOO)

5.4 – Apply the Remainder and Factor Theorems Divide 247 / / 8.

Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm.

Real Zeros of Polynomial Functions. Quick Review.

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

The Remainder and Factor Theorems

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

ACTIVITY 31: Dividing Polynomials (Section 4.2, pp )

Section 5.5 The Real Zeros of a Polynomial Function.

Section 5.3(d) Synthetic Substitution. Long division Synthetic Division can be used to find the value of a function. This process is called Synthetic.

2.4/2.52.4/2.5 Real Zeros of Polynomial Functions.

Date: 2.4 Real Zeros of Polynomial Functions

Section 2-4 Real Zeros of Polynomial Functions. Section 2-4 long division and the division algorithm the remainder and factor theorems reviewing the fundamental.

a. b.  To simplify this process, we can use a process called division.  Synthetic division works when dividing a polynomial by.  To get started, make.

Solving Polynomials. What does it mean to solve an equation?

Real Zeros of Polynomials Section 2.4. Review – Long Division 1. What do I multiply by to get the first term? 2. Multiply through 3. Subtract 4. Bring.

6.5 Theorems About Roots of Polynomial Equations

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

Remainder and Factor Theorems

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.

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.

Precalculus Lesson 2.5 The Fundamental Theorem of Algebra.

4-5 Exploring Polynomial Functions Locating Zeros.

Solving Polynomials. Factoring Options 1.GCF Factoring (take-out a common term) 2.Sum or Difference of Cubes 3.Factor by Grouping 4.U Substitution 5.Polynomial.

Products and Factors of Polynomials (part 2 of 2) Section 440 beginning on page 442.

Polynomial Long Division

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 Section 4.3.

Dividing Polynomials Two options: Long Division Synthetic Division.

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.

Descartes’ Rule of Signs

Division of a Polynomial

College Algebra Chapter 3 Polynomial and Rational Functions

3.3 Real Zeros of Polynomials

Remainder and Factor Theorems

The Remainder and Factor Theorems

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.

Polynomial Division/Factoring

Finding Zeros of Polynomials

Remainder and Factor Theorem

Lesson 2.3 Real Zeros of Polynomial Functions

Real Zeros of Polynomial Functions

The Remainder and Factor Theorems

Finding Zeros of a Polynomial Function

Digital Lesson Polynomial Functions.

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

Zeros of polynomial functions

The Remainder and Factor Theorems

4-3: Remainder and Factor Theorems

Digital Lesson Polynomial Functions.

Find all the real zeros of the functions.

The Real Zeros of a Polynomial Function

Dividing Polynomials (SYNTHETIC Division)

Presentation transcript:

Section 2.4: Real Zeros of Polynomial Functions

Polynomial Long Division Divide by w – 3. Divide by .

Synthetic Division Divide by x – 1. Divide by x – 2.

Remainder Theorem If dividing f(x) by x – a, f(a) will determine the remainder. What is the remainder when dividing by x – 4? Use the Remainder Theorem to evaluate f (x) = 6x3 – 5x2 + 4x – 17 at x = 3.

Factor Theorem When dividing polynomials, if the remainder is zero, then the divisor is a factor. Use the Factor Theorem to determine whether x – 1 is a factor of f (x) = 2x4 + 3x2 – 5x + 7. Using the Factor Theorem, verify that x + 4 is a factor of f (x) = 5x4 + 16x3 – 15x2 + 8x + 16.

Finding Exact Irrational Zeros Find the exact zeros for the function. Identify each zero as rational or irrational.

Writing Functions w/ Given Conditions If the zeros of the function are -3, 7, and -1, and the leading coefficient is 4, write an equation for the function.

Rational Zeros Theorem Given a function with constant of p and leading coefficient of q, all possible rational zeros can be found by Find the possible rational zeros of . Find the possible rational zeros of

Upper and Lower Bounds Test When dividing polynomials, if the quotient polynomial has all non-negative coefficients, then the “k” value is an upper bound. If the quotient polynomial has alternating sign coefficients, then the “k” value is a lower bound. Show that all real roots of the equation lie between - 4 and 4.