Dividing Polynomials

Simple Division -dividing a polynomial by a monomial

2 2 2 26 3 9

3

r s rs r s1.

rs

6r 2s2

3rs

3rs2

3rs

9r 2s

3rs

2rs s 3r

Simplify

3a2b

3ab

6a3b2

3ab

18ab

3ab

a 2a2b 6

2 3 23 6 18

3

a b a b ab2.

ab

Simplify

12x2 y

3x

3x

3x

4xy 1

212 3

3

x y x3.

x

Long Division -divide a polynomial by a polynomial

•Think back to long division from 3rd grade.

•How many times does the divisor go into the dividend? Put that number on top.

•Multiply that number by the divisor and put the result under the dividend.

•Subtract and bring down the next number in the dividend. Repeat until you have used all the numbers in the dividend.

-( )x 3 x 2 5x 24

x

x2 + 3x- 8x

- 8

- 24- 8x - 24

0-( )

x2/x = x

2 5 24

3

x x4.

x

-8x/x = -8

-( )

-( )

-( )h 4 h3 0h2 11h 28

h2

h3 - 4h2

4h2

+ 4h

- 11h4h2 - 16h

5h

h3/h = h2

13 11 28 45. h h h

+ 284h2/h = 4h

+ 5

5h - 2048

48

h 4

5h/h = 5

Synthetic Division -

4 26 : 5 4 6 ( 3)Ex x x x x

To use synthetic division:

•There must be a coefficient for every possible power of the variable.

•The divisor must have a leading coefficient of 1.

divide a polynomial by a polynomial

Step #1: Write the terms of the polynomial so the degrees are in descending order.

5x4 0x3 4x2 x 6

Since the numerator does not contain all the powers of x, you must include a 0 for the x3.

4 2

5 4 6 ( 3)x x x x

Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients.

Since the divisor is x-3, r=3

5x4 0x3 4x2 x 6

5 0 -4 1 63

4 25 4 6 ( 3)x x x x

5

Step #3: Bring down the first coefficient, 5.

3 5 0 - 4 1 6

4 25 4 6 ( 3) x x x x

5

3 5 0 - 4 1 6

Step #4: Multiply the first coefficient by r, so 3 5 15

and place under the second coefficient then add.

15

15

4 2

5 4 6 ( 3)x x x x

5

3 5 0 - 4 1 6

15

15

Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add.

15 3 45

45

41

4 2

5 4 6 ( 3)x x x x

5

3 5 0 - 4 1 6

15

15 45

41

Step #5 cont.: Repeat the same procedure.

123

124

372

378

Where did 123 and 372 come from?

4 2

5 4 6 ( 3)x x x x

Step #6: Write the quotient.The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

5

3 5 0 - 4 1 6

15

15 45

41

123

124

372

378

4 2

5 4 6 ( 3)x x x x

The quotient is:

5x3 15x2 41x 124 378

x 3

Remember to place the remainder over the divisor.

4 2

5 4 6 ( 3)x x x x

5x 5 21x4 3x3 4x2 2x 2 x 4 Ex 7:

Step#1: Powers are all accounted for and in descending order.

Step#2: Identify r in the divisor.

Since the divisor is x+4, r=-4 .

4 5 21 3 4 2 2

Step#3: Bring down the 1st coefficient. Step#4: Multiply and add.

4 5 21 3 4 2 2

-5

Step#5: Repeat.

20 4 -4 0 8-1 1 0 -2 10

4 3 2 105 2

4x x x

x

5 4 3 25 21 3 4 2 2 4x x x x x x

6x2 2x 4 2x 3 Ex 8:

6x2

2

2x

2

4

2

2x

2

3

2

Notice the leading coefficient of the divisor is 2 not 1.

We must divide everything by 2 to change the coefficient to a 1.

3x2 x 2 x 3

2

3

2 3 1 2

3

9

2

2

2

7

2

21

4

8

4

29

4

26 2 4 2 3x x x

297 43

322

xx

*Remember we cannot have complex fractions - we must simplify.

7 29 13

32 42

xx

7 293

324

2

x

x

7 29

32 4 6

xx

26 2 4 2 3x x x

x3 x 2 2x 7 2x 1 Ex 9:

3 2 2 7 2 1

2 2 2 2 2 2

x x x x

11

2

1

2

7

2 Coefficients

1 1 1 7 1

2 2 2 2

1

2

1

41

4

2

4

1

8

8

8

7

8

7

16

56

16

49

16

3 21 1 7 1

2 2 2 2x x x x

3 2 2 7 2 1x x x x