6.5 The Remainder and Factor Theorems

Slides:

Advertisements

Similar presentations

Long and Synthetic Division of Polynomials Section 2-3.

Advertisements

Rational Zeros Theorem Upper & Lower Bounds Long Division

6.3 Dividing Polynomials. Warm Up Without a calculator, divide the following Solution:

5.5 Apply the Remainder and Factor Theorem

Bell Problem Find the real number solutions of the equation: 18x 3 = 50x.

Warm Up #1 1. Use synthetic substitution to evaluate f (x) = x3 + x2 – 3x – 10 when x = – ANSWER –4.

HW: Pg #13-61 eoo.

Do Now: Factor the following polynomial:. By the end of this chapter, you will be able to: - Identify all possible rational zeroes - Identify all actual.

Division and Factors When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor.

2.5 Apply the Remainder and Factor Theorems p. 120 How do you divide polynomials? What is the remainder theorem? What is the difference between synthetic.

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

6.5 The Remainder and Factor Theorems p. 352 How do you divide polynomials? What is the remainder theorem? What is the difference between synthetic substitution.

Section 2.3 Polynomial and Synthetic Division Long Division of polynomials Ex. (6x 3 -19x 2 +16x-4) divided by (x-2) Ex. (x 3 -1) divided by (x-1) Ex (2x.

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

1 What we will learn today…  How to divide polynomials and relate the result to the remainder and factor theorems  How to use polynomial division.

 PERFORM LONG DIVISION WITH POLYNOMIALS AND DETERMINE WHETHER ONE POLYNOMIAL IS A FACTOR OF ANOTHER.  USE SYNTHETIC DIVISION TO DIVIDE A POLYNOMIAL BY.

Dividing Polynomials & The Remainder Theorem. Dividing Polynomials When dividing a polynomial by a monomial, divide each term in the polynomial by the.

3.2 Dividing Polynomials 11/28/2012. Review: Quotient of Powers Ex. In general:

5.3 Part 2 Polynomial Division

Algebraic long division Divide 2x³ + 3x² - x + 1 by x + 2 x + 2 is the divisor The quotient will be here. 2x³ + 3x² - x + 1 is the dividend.

6.4 Multiplying/Dividing Polynomials 1/10/2014. How do you multiply 1256 by 13?

1 Warm-up Determine if the following are polynomial functions in one variable. If yes, find the LC and degree Given the following polynomial function,

The Remainder and Factor Theorems 6.5 p When you divide a Polynomial f(x) by a divisor d(x), you get a quotient polynomial q(x) with a remainder.

2.3 Polynomial Division and Synthetic Division Ex. Long Division What times x equals 6x 3 ? 6x 2 6x x 2 Change the signs and add x x.

Warm Up 9-1 Use long division to find the quotient and remainder for the problems below ÷ ÷ 4.

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 4.3 Polynomial Division; The Remainder and Factor Theorems  Perform long division.

1. 2 Polynomial Function A polynomial function is a function of the form where n is a nonnegative integer and each a i (i = 0,1,…, n) is a real number.

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.

7.3 Products and Factors of Polynomials Objectives: Multiply polynomials, and divide one polynomial by another by using long division and synthetic division.

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

The Remainder Theorem A-APR 2 Explain how to solve a polynomial by factoring.

I CAN USE LONG DIVISION AND SYNTHETIC DIVISION. I CAN APPLY THE FACTOR AND REMAINDER THEOREMS. Lesson 2-3 The Remainder and Factor Theorems.

Using theorems to factor polynomials.  If a polynomial f(x) is divided by x-k, then the remainder r = f(k)  This is saying, when you divide (using synthetic.

Synthetic Division and Zeros. Synthetic division Only applies when the divisor is x-c and when every descending power of x has a place in the dividend.

7.4 Solving Polynomial Equations

Warm-Up Exercises 1. Use the quadratic formula to solve 2x 2 – 3x – 1 = 0. Round the nearest hundredth. 2. Use synthetic substitution to evaluate f (x)

Division of Polynomials Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Dividing Polynomials Long division of polynomials.

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.

3.3 Polynomial and Synthetic Division. Long Division: Let’s Recall.

Get out your homework, but keep it at your desk..

5.6 The Remainder and Factor Theorems. [If you are dividing by (x - 6), the remainder will be the same as if you were evaluating the polynomial using.

Quotient Dividend Remainder Divisor Long Division.

Division of Polynomials Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Dividing Polynomials Long division of polynomials.

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

Polynomial & Synthetic Division Algebra III, Sec. 2.3 Objective Use long division and synthetic division to divide polynomials by other polynomials.

3.2 Division of Polynomials. Remember this? Synthetic Division 1. The divisor must be a binomial. 2. The divisor must be linear (degree = 1) 3. The.

Objective Use long division and synthetic division to divide polynomials.

Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

Division of a Polynomial

Divide by x - 1 Synthetic Division: a much faster way!

Aim #3. 3 How do we divide Polynomials

DIVIDING POLYNOMIALS Synthetically!

Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

Apply the Remainder and Factor Theorems

Polynomial Division; The Remainder Theorem and Factor Theorem

6.5 The Remainder and Factor Theorems

Real Zeros of Polynomial Functions

Warm-up: Divide using Long Division

Remainder and Factor Theorem

5.5 Apply the Remainder and Factor Theorems

Today in Precalculus Go over homework Notes: Remainder

6.5 The Remainder and Factor Theorems

Dividing Polynomials WOW! I want to learn how to do that?

Dividing Polynomials.

Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

5.5 Apply the Remainder and Factor Theorems

Dividing Polynomials (SYNTHETIC Division)

Presentation transcript:

6.5 The Remainder and Factor Theorems OBJ: use long division to divide polynomials Do Now: Use long division to divide. Round to the nearest hundredth. 1) 4759 divided by 36 2) 12 divided by 5

Polynomial Long Division If a term is missing in standard form, fill it in with a 0 coefficient. Ex 1: 2x4 + 3x3 + 5x – 1 x2 – 2x + 2 2x2 2x4 = 2x2 x2

-( ) 2x2 +7x +10 2x4 -4x3 +4x2 7x3 - 4x2 +5x -( ) 7x3 - 14x2 +14x -( ) 2x4 -4x3 +4x2 7x3 - 4x2 +5x -( ) 7x3 - 14x2 +14x 10x2 - 9x -1 7x3 = 7x x2 -( ) 10x2 - 20x +20 11x - 21 remainder

The answer is written: Quotient + Remainder over Divisor 2x2 + 7x + 10 + 11x – 21 x2 – 2x + 2 Ex 2: y4 + 2y2 – y + 5 y2 – y + 1 y2 + y + 2 + 3 y2 – y + 1

Remainder Theorem: If a polynomial is divisible by (x – k), then the remainder is f(k) ONLY when you divide by x-k, check if the remainder is f(k) by directly substituting k back into the eqn!!! On your homework, this is possible on problems #6 and # 7 HW:

Divide f(x)= 3x3 – 2x2 + 2x – 5 by x - 2 continued… Do Now: Use long division to divide f(x)= 3x3 – 2x2 + 2x – 5 by x - 2 3x2 + 4x + 10 + 15 x – 2 Synthetic Division (like synthetic substitution) Ex 3: Divide f(x)= 3x3 – 2x2 + 2x – 5 by x - 2

f(x)= 3x3 – 2x2 + 2x – 5 Divide by x - 2 3 -2 2 -5 2 6 8 20 3 4 10 R: 15 + 15 x-2 3x2 + 4x + 10

Divide x3 + 2x2 – 6x -9 by x-2 1 2 -6 -9 2 Ex 4: 2 8 4 1 4 2 R:-5 1 2 -6 -9 2 2 8 4 1 4 2 R:-5 x2 + 4x + 2 + -5 x-2

Divide x3 + 2x2 – 6x -9 by x+3 1 2 -6 -9 -3 Ex 5: -3 3 9 1 -1 -3 R: 0 1 2 -6 -9 -3 -3 3 9 1 -1 -3 R: 0 x2 – x - 3

Factor Theorem: A polynomial has the factor x-k if f(k)=0 Factor Theorem: A polynomial has the factor x-k if f(k)=0 ** if the remainder is zero, x-k is a factor since it divides evenly ** it also means that k is a “zero” (a solution) of the function! HW:

Factoring a polynomial continued… Factoring a polynomial Do Now: Use synthetic division to divide f(x)= x4 -6x3 -40x +33 by x - 7 x3+ x2 + 7x +9 + 96 x – 7 Factor f(x) = 2x3 + 11x2 + 18x + 9 Given f(-3)=0 *Since the remainder is zero, x – k is a factor! x-(-3) or x+3 *So use synthetic division to find the others!! Ex 6:

2 11 18 9 -3 -15 -9 -6 2 5 3 R: 0 (x + 3)(2x2 + 5x + 3) *keep factoring (bustin ‘da ‘b’) gives you: (x+3)(2x+3)(x+1)

Factor f(x)= 3x3 + 13x2 + 2x -8 Given f(-4)=0 (x + 1)(3x – 2)(x + 4) Ex 7: Factor f(x)= 3x3 + 13x2 + 2x -8 Given f(-4)=0 (x + 1)(3x – 2)(x + 4)

Finding the zeros of a polynomial function Ex 8: One zero of f(x) = x3 – 2x2 – 9x +18 is x=2. Find the others. *if x=2 is a zero, then x-2 is a factor *Use synthetic division to factor completely. *then solve (x-2)(x2-9) (x-2)(x+3)(x-3) x=2 x=-3 x=3

One zero of f(x) = x3 + 6x2 + 3x -10 is x=-5. Find the others. Ex 9: One zero of f(x) = x3 + 6x2 + 3x -10 is x=-5. Find the others. (x-2)(x+1)(x+5) x=2 x=-1 x=-5 HW: