Sect. 2-2 Synthetic Division; The remainder and Factor theorems

Objective: SWBAT use the synthetic division and to apply the remainder

and factor theorems.

Using Long DivisionIn earlier Algebra we use Long Division to find the remainder

4 3 2

4 3

3) 2x 0 15 10 5

2x 6

x x x x

x

3 2

3 2

6x 15

6x 18

x

x

Example 5: 4 2 (x 9 - 5) by (x - 2)Divide x

4 3 2 x - 2 ) x + 0x - 9x + 0x - 54 3x - 2x

3 22x - 9x3 22x - 4x

2 - 5x2 - 5x + 10x - 10 x - 5

- 10x + 20 - 25

3 x 2+ 2x - 5x - 10

Exercise #18 2 (a - 81) by (a - 9 )Divide

2a - 9) a + 0a - 81 a

2 a - 9a 9a - 81

+ 9

9a - 81

C.) Synthetic Division

2 4 0 1 7

4 2 4 0 1 7

4 88

2 4 0 1 78 16

4 8 17

2 4 0 1 78 16

4 8 17

34

41Quotient Remainder

Writing the 2 of x -2 and the coefficients of the dividend

Bringing down the first coefficient

Multiplying 4 by 2 to get 8 and Adding 0 and 8

Multiplying 8 by 2 to get 16 and Adding 1 and 16

Multiplying 17 by 2 to get 34 and Adding 7 and 34

Example 2: Use synthetic Division to find the quotient and remainder

Writing the 2 of x -2 and the coefficients of the dividend

2

3 2( x + 6x - x - 30 ) (x - 2)

1 6 - 1 -30 2

1 81615

30

0 Using Synthetic Division

2The quotient is x + 8x + 15, the remainder is 0.

Example 3: Use synthetic Division

-2 8 0 -6 0 1 - 8

8-16-16

3226

-52-52

104105

-210-218

4 3 2The quotient is 8x - 16x + 26x - 52x + 105, the remainder is -218.

The Remainder Theorem When a polynomial P(x) is divided by x – a the remainder is P(a)

The Factor Theorem For a polynomial P(x) x – a is a factor If and only If P(a) = 0

Example of the remainder theorem3 2x 2 5 1x x by 1x

1 1 -2 5 1

1

1

-1

-1

4

4

5

2 - x + 4 remainder = 5x

Example of the Factor Theorem

• Any time you can divide and get a 0 for a remainder then you have an illustration of the factor theorem.

-1 1 -2 0 5 2

1

-1

-3

3

3

-3

2

-2

0

3 2 - 3 3 + 2 remainder = 0x x x

Finding the remaining roots3 22x 5 4 3 0; 3x x root

3 2 -5 -4 3

2

6

1

3

-1

-3

02( 3)(2x 1) 0x x

( 3)(2 1)( 1) 0x x x 1 x = 3 x= x= -12

1Sol= 3, , 12

Homework 1,3,4,5,11,15,19,20pg. 61