Presentation is loading. Please wait.

Presentation is loading. Please wait.

2.3 Polynomial and Synthetic Division Why teach long division in grade school?

Published byEmerald Robertson Modified over 8 years ago

Similar presentations

Presentation on theme: "2.3 Polynomial and Synthetic Division Why teach long division in grade school?"— Presentation transcript:

1 2.3 Polynomial and Synthetic Division Why teach long division in grade school?

2 Long Division Find 2359 ÷ 51 by hand Which one goes inside √

3 Long Division Find 2359 ÷ 51 by hand 51√ 2359

4 Long Division Find 2359 ÷ 51 by hand 4 51√ 2359 204 What do you do now ?

5 Long Division Find 2359 ÷ 51 by hand 4 51√ 2359 - 204 319

6 Long Division Find 2359 ÷ 51 by hand 46 51√ 2359 - 204 319 - 306 13

7 Long Division Find 2359 ÷ 51 by hand 46 13/51 51√ 2359 - 204 319 - 306 13

8 Lets do the same with a Polynomial Divide6x 3 + 4x 2 – 10x – 5 by 2x 2 + 1 2x 2 + 1 √ 6x 3 + 4x 2 – 10x - 5

9 Lets do the same with a Polynomial Divide6x 3 + 4x 2 – 10x – 5 by 2x 2 + 1 3x 2x 2 + 1 √ 6x 3 + 4x 2 – 10x – 5 6x 3 + 3x

10 Lets do the same with a Polynomial Divide6x 3 + 4x 2 – 10x – 5 by 2x 2 + 1 3x 2x 2 + 1 √ 6x 3 + 4x 2 – 10x – 5 6x 3 + 3x 4x 2 – 13x - 5

11 Lets do the same with a Polynomial Divide6x 3 + 4x 2 – 10x – 5 by 2x 2 + 1 3x + 2 2x 2 + 1 √ 6x 3 + 4x 2 – 10x – 5 6x 3 + 3x 4x 2 – 13x - 5 4x 2 + 2 - 13x - 7

12 Lets do the same with a Polynomial Divide6x 3 + 4x 2 – 10x – 5 by 2x 2 + 1 3x + 2 + 2x 2 + 1 √ 6x 3 + 4x 2 – 10x – 5 6x 3 + 3x 4x 2 – 13x - 5 4x 2 + 2 - 13x - 7

13 The Division Algorithm f(x) = d(x)g(x) + r(x) (6x 3 + 4x 2 – 10x – 5) = (3x + 2)(2x 2 + 1) +(-13x – 7) = 6x 3 + 4x 2 + 3x + 2 – 13x – 7 = 6x 3 + 4x 2 - 10x – 5 WE can use the division Algorithm to find G.C.D. (greatest common divisors )

14 What is the G.C.D. of 3461, 4879 4879 = 3461(1) + 1418 3461 = 1418(2) + 625 1418 = 625(2) + 168 625 = 168(3) + 121 168 = 121(1) + 47 121 = 47(2) + 27 47 = 27(1) + 20 27 = 20(1) + 7 20 = 7(2) + 6 7 = 6(1) + 1← G.C.D. 6 = 1(6) + 0

15 Ruffini’s rule3 or Synthetic Division Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and philosopher. Italianmathematician philosopher By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics. Among his work was an incomplete proof (Abel–Ruffini theorem1) that quintic (and higher-order) equations cannot be solved by radicals (1799), and Ruffini's rule3 which is a quick method for polynomial division.Abel–Ruffini theorem1 quintic (and higher-order) equations radicalsRuffini's rule3polynomial division

16 Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x 3 + 5x 2 + 8)÷(x – 2)What is x; x – 2 = 0 x = 2 This will go in the little box in the first line.

17 Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x 3 + 5x 2 + 8)÷(x – 2)What is x; x – 2 = 0 2 |4508 The coefficients are written out in descending exponential order. (even leaving a zero for the 1 st degree term)

18 Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x 3 + 5x 2 + 8)÷(x – 2)What is x; x – 2 = 0 2 |4508 4The first number is dropped, then multiply by 2 and add to 5

19 Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x 3 + 5x 2 + 8)÷(x – 2)What is x; x – 2 = 0 2 |4508 8 413 Then the steps are repeated added and multiply by 2.

20 Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x 3 + 5x 2 + 8)÷(x – 2)What is x; x – 2 = 0 2 |4508 82652 4132660 Then the steps are repeated added and multiply by 2.

21 Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x 3 + 5x 2 + 8)÷(x – 2)What is x; x – 2 = 0 2 |4508 82652 4132660 60 is the reminder; 26 is the constant, 13 the 1 st degree term, 4 the 2 nd degree term

22 Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x 3 + 5x 2 + 8)÷(x – 2)What is x; x – 2 = 0 2 |4508 82652 4132660 4x 2 + 13x + 26 +

23 The Remainder Theorem The remainder is the answer! So in f(x) = 4x 3 + 5x 2 + 8 f(2) = 60

24 The Remainder Theorem The remainder is the answer! So in f(x) = 4x 3 + 5x 2 + 8 f(2) = 60 Check it out: 4(2) 3 + 5(2) 2 + 8 4(8) + 5(4) + 8 32 + 20 + 8 = 60

25 (x 2 + 3x – 40) ÷ (x - 5) 5| 1 3 - 40 5 40 1 8 0 Since the reminder is 0, 5 is a root or zero of the equation. What is the other root?

26 Homework Page 140 – 142 #2, 8, 14, 17, 21, 24, 28, 36, 42, 44, 51, 55, 42, 44, 51, 55, 63, 74, 82, 92 63, 74, 82, 92

27 Homework Page 140-142 # 7, 13, 15, 22, 27, 35, 40, 43, 50, 53, 62, 70, 50, 53, 62, 70, 81, 86 81, 86

Download ppt "2.3 Polynomial and Synthetic Division Why teach long division in grade school?"

Similar presentations

Dividing Polynomials.

Dividing Polynomials.
Long and Synthetic Division of Polynomials Section 2-3.

Long and Synthetic Division of Polynomials Section 2-3.
Chapter 2 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 2.4 Dividing Polynomials: Remainder and Factor Theorems.

Chapter 2 Polynomial and Rational Functions Copyright © 2014, 2010, 2007 Pearson Education, Inc Dividing Polynomials: Remainder and Factor Theorems.
Dividing Polynomials Objectives

Dividing Polynomials Objectives
Example 1 divisor dividend quotient remainder Remainder Theorem: The remainder is the value of the function evaluated for a given value.

Example 1 divisor dividend quotient remainder Remainder Theorem: The remainder is the value of the function evaluated for a given value.
Warm up!! Pg. 130 #’s 16, 22, 36, 42, 52, 96.

Warm up!! Pg. 130 #’s 16, 22, 36, 42, 52, 96.
Section 3.3 Dividing Polynomials; Remainder and Factor Theorems

Section 3.3 Dividing Polynomials; Remainder and Factor Theorems
Dividing Polynomials; Remainder and Factor Theorems.

Dividing Polynomials; Remainder and Factor Theorems.
5.5 Apply the Remainder and Factor Theorem

5.5 Apply the Remainder and Factor Theorem
Synthetic Division. This method is used to divide polynomials, one of which is a binomial of degree one.

Synthetic Division. This method is used to divide polynomials, one of which is a binomial of degree one.
1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 3 Polynomial and Rational Functions.
Section 7.3 Products and Factors of Polynomials.

Section 7.3 Products and Factors of Polynomials.
3.3: Dividing Polynomials: Remainder and Factor Theorems Long Division of Polynomials 1.Arrange the terms of both the dividend and the divisor in descending.

3.3: Dividing Polynomials: Remainder and Factor Theorems Long Division of Polynomials 1.Arrange the terms of both the dividend and the divisor in descending.
Lesson 2.4, page 301 Dividing Polynomials Objective: To divide polynomials using long and synthetic division, and to use the remainder and factor theorems.

Lesson 2.4, page 301 Dividing Polynomials Objective: To divide polynomials using long and synthetic division, and to use the remainder and factor theorems.
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.

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.

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 / 3 8829 / 8.

5.4 – Apply the Remainder and Factor Theorems Divide 247 / / 8.
Polynomial Division and the Remainder Theorem Section 9.4.

Polynomial Division and the Remainder Theorem Section 9.4.
Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm.

Lesson 2.3 Real Zeros of Polynomials. The Division Algorithm.
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.

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.

Similar presentations

Ads by Google