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