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.

Slides:

Advertisements

Similar presentations

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

Advertisements

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

5.5 Apply the Remainder and Factor Theorem

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

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 5–5) CCSS Then/Now New Vocabulary Key Concept: Remainder Theorem Example 1:Synthetic Substitution.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 6–5) Then/Now New Vocabulary Key Concept: Remainder Theorem Example 1:Synthetic Substitution.

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

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.

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.

The Remainder and Factor Theorems

7.4 THE REMAINDER & FACTOR THEOREMS Objectives: The student will be able to… 1)evaluate functions using synthetic substitution 2)determine whether a binomial.

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.

7.4 The Remainder and Factor Theorems Use Synthetic Substitution to find Remainders.

1 Use the Remainder Theorem and the Factor Theorem. 2.3 Day 2 What You Should Learn.

6.5 The Remainder and Factor Theorems

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

6-7 The Division Algorithm & The Remainder Theorem dividend=quotient. divisor + remainder If a polynomial f(x) is divided by x - c, the remainder is the.

Synthetic Evaluation of Polynomials Be able to use synthetic division to evaluate polynomials.

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.

Section 2-2 Synthetic Division; The Remainder and Factor Theorems.

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.

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

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

WARM UP. Homework Q’s Dividing Polynomials using Synthetic Division EQ: How is Long Division utilized to divide a polynomial functions? Assessment:

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.

Polynomial and Synthetic Division Objective: To solve polynomial equations by long division and synthetic division.

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

Pre Calculus – Synthetic Division Unit 3. First, you have to write the coefficients of the polynomial to be divided at the top (remember to use 0’s.

Holt Algebra Dividing Polynomials Synthetic division is a shorthand method of dividing a polynomial by a linear binomial by using only the coefficients.

Then/Now You factored quadratic expressions to solve equations. (Lesson 0–3) Divide polynomials using long division and synthetic division. Use the Remainder.

Then/Now You used the Distributive Property and factoring to simplify algebraic expressions. Evaluate functions by using synthetic substitution. Determine.

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

Objective Use long division and synthetic division to divide polynomials.

Dividing Polynomials Section 4.3.

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.

5.2 Dividing Polynomials.

Section 5.4 – Dividing Polynomials

Division of a Polynomial

Name:__________ warm-up 5-6

Pre-Calculus Section 2.3 Synthetic Division

The Remainder and Factor Theorems

DIVIDING POLYNOMIALS Synthetically!

Polynomial Division; The Remainder Theorem and Factor Theorem

6.5 The Remainder and Factor Theorems

5.5 - Long and Synthetic Division

Remainder and Factor Theorem

5.5 Apply the Remainder and Factor Theorems

6.5 The Remainder and Factor Theorems

The Factor Theorem A polynomial f(x) has a factor (x − k) if and only if f(k) = 0.

The Remainder and Factor Theorems

The Remainder and Factor Theorems

5.5 Apply the Remainder and Factor Theorems

Dividing Polynomials (SYNTHETIC Division)

Presentation transcript:

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 synthetic substitution when x = 6]

Example 1: Synthetic Division Divide x 3 - 3x 2 - 7x + 6 by x + 2

Example 2: Synthetic Substitution If f(x) = 2x 4 – 5x 2 + 8x – 7, find f(6). Answer: The remainder is Thus, by using synthetic substitution, f(6) = 2453.

You try If f(x) = 2x 3 – 3x 2 + 7, find f(3). A.20 B.34 C.88 D.142

Example 4: Find Function Values The number of college students from the United States who study abroad can be modeled by the function S(x) = 0.02x 4 – 0.52x x x , where x is the number of years since 1993 and S(x) is the number of students in thousands. How many U.S. college students will study abroad in 2011? Answer:In 2011, there will be about 451,760 U.S. college students studying abroad.

Example 5: The number of high school students in the United States who hosted foreign exchange students can be modeled by the function F(x) = 0.02x 4 – 0.05x x 2 – 0.02x, where x is the number of years since 1999 and F(x) is the number of students in thousands. How many U.S. students will host foreign exchange students in 2013? A.616,230 students B.638,680 students C.646,720 students D.659,910 students

Same format as synthetic substitution. Use FILLERS! used when divisor is in the form x - k. Create a polynomial out of coefficients that are left under the line. Factor further if possible. If the remainder is 0, then (x - k) is a factor of the polynomial k is called a zero because f(k) = 0

Example 6: Use the Factor Theorem Determine whether x – 3 is a factor of x 3 + 4x 2 – 15x – 18. Then find the remaining factors of the polynomial. Answer: So, x 3 + 4x 2 – 15x – 18 = (x – 3)(x + 6)(x + 1).

Check example 6

Example 7: Factor f(x) = 3x x 2 + 2x - 8 given that f(-4) = 0.

You Try Determine whether x + 2 is a factor of x 3 + 8x x If so, find the remaining factors of the polynomial. A.yes; (x + 5)(x + 1) B.yes; (x + 5) C.yes; (x + 2)(x + 3) D.x + 2 is not a factor.