2.3 polynomial and synthetic division why teach long division in grade school?

2.3 Polynomial and 2.3 Polynomial and Synthetic DivisionSynthetic Division

Why teach long division in Why teach long division in grade school?grade school?

Long Division

Find 2359 ÷ 51 by hand

Which one goes inside

√

Long Division

Find 2359 ÷ 51 by hand

51 √ 2359

Long Division

Find 2359 ÷ 51 by hand

4

51 √ 2359

204

What do you do now ?

Long Division

Find 2359 ÷ 51 by hand

4

51 √ 2359

- 204

319

Long Division

Find 2359 ÷ 51 by hand

46

51 √ 2359

- 204

319

- 306

13

Long Division

Find 2359 ÷ 51 by hand46 13/51

51 √ 2359

- 204

319

- 306

13

Lets do the same with a Polynomial

Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1

2x2 + 1 √ 6x3 + 4x2 – 10x - 5

Lets do the same with a Polynomial

Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1

3x

2x2 + 1 √ 6x3 + 4x2 – 10x – 5

6x3 + 3x

Lets do the same with a Polynomial

Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1

3x

2x2 + 1 √ 6x3 + 4x2 – 10x – 5

6x3 + 3x

4x2 – 13x - 5

Lets do the same with a Polynomial

Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1

3x + 2

2x2 + 1 √ 6x3 + 4x2 – 10x – 5

6x3 + 3x

4x2 – 13x - 5

4x2 + 2

- 13x - 7

Lets do the same with a Polynomial

Divide 6x3 + 4x2 – 10x – 5 by 2x2 + 1

3x + 2 +

2x2 + 1 √ 6x3 + 4x2 – 10x – 5

6x3 + 3x

4x2 – 13x - 5

4x2 + 2

- 13x - 7

12x

7 -13x -2

The Division Algorithm

f(x) = d(x)g(x) + r(x)

(6x3 + 4x2 – 10x – 5) = (3x + 2)(2x2 + 1) +(-13x – 7)

= 6x3 + 4x2 + 3x + 2 – 13x – 7

= 6x3 + 4x2 - 10x – 5

WE can use the division Algorithm to find G.C.D.(greatest common divisors )

What is the G.C.D. of 3461, 4879

4879 = 3461(1) + 14183461 = 1418(2) + 6251418 = 625(2) + 168625 = 168(3) + 121168 = 121(1) + 47121 = 47(2) + 2747 = 27(1) + 2027 = 20(1) + 720 = 7(2) + 67 = 6(1) + 1 ← G.C.D.6 = 1(6) + 0

Ruffini’s rule3 or Synthetic Division

• Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician and 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.

Synthetic Division

Can be used when dividing by x – r term, where r is a number.

(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0

x = 2

This will go in the little box in the

first line.

Synthetic Division

Can be used when dividing by x – r term, where r is a number.

(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8

The coefficients are written out in descending exponential order.

(even leaving a zero for the 1st degree term)

Synthetic Division

Can be used when dividing by x – r term, where r is a number.

(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8

4 The first number is dropped,

then multiply by 2 and add to 5

Synthetic Division

Can be used when dividing by x – r term, where r is a number.

(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0

2 | 4 5 0 8

8

4 13

Then the steps are repeated added and multiply by 2.

Synthetic Division

Can be used when dividing by x – r term, where r is a number.

(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0

2 | 4 5 0 8

8 26 52

4 13 26 60

Then the steps are repeated added and multiply by 2.

Synthetic Division

Can be used when dividing by x – r term, where r is a number.

(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0

2 | 4 5 0 8

8 26 52

4 13 26 60

60 is the reminder; 26 is the constant, 13 the 1st degree term, 4 the 2nd degree term

Synthetic Division

Can be used when dividing by x – r term, where r is a number.

(4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8

8 26 524 13 26 60

4x2 + 13x + 26 + 2

60

x

The Remainder Theorem

The remainder is the answer!

So in f(x) = 4x3 + 5x2 + 8

f(2) = 60

The Remainder Theorem

The remainder is the answer!

So in f(x) = 4x3 + 5x2 + 8

f(2) = 60

Check it out: 4(2)3 + 5(2)2 + 8

4(8) + 5(4) + 8

32 + 20 + 8 = 60

(x2 + 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?

HomeworkHomework

Page 140 – 142 Page 140 – 142

##2, 8, 14, 17, 2, 8, 14, 17,

21, 24, 28, 36,21, 24, 28, 36,

42, 44, 51, 55,42, 44, 51, 55,

63, 74, 82, 9263, 74, 82, 92

HomeworkHomework

Page 140-142 Page 140-142

# # 7, 13, 15, 22,7, 13, 15, 22,

27, 35, 40, 43,27, 35, 40, 43,

50, 53, 62, 70,50, 53, 62, 70,

81, 8681, 86