Long and Synthetic Division of Polynomials

Section 2-3

2

Objectives

• I can use long division to divide two polynomials

• I can use synthetic division to divide a polynomial by a binomial (x – r)

3

Dividing Numbers

4164

When you divide a number by another number and there is no remainder:

Then the divisor is a factor!!

Also the quotient becomes another factor!!!

Dividend

Divisor

Quotient

4

Dividing Polynomials

Long division of polynomials is similar to long division of whole numbers.

dividend = (quotient • divisor) + remainder

The result is written in the form:

quotient +divisor

remainder divisor dividend

When you divide two polynomials you can check the answer using the following:

5

+ 2 2 3 1 2 xxx

Example: Divide x2 + 3x – 2 by x + 1 and check the answer.

x

x2 + x2x – 22x + 2

– 4

remainder

xx

xxx

22 1.

xxxx 2)1(2.

xxxxx 2)()3( 22 3.

22

2 x

xxx4.

22)1(2 xx5.

4)22()22( xx6.

Answer: x + 2 +1x

– 4

Dividing Polynomials

6

Example: Divide 4x + 2x3 – 1 by 2x – 2 and check the answer.

1 4 0 2 2 2 23 xxxx Write the terms of the dividend in

descending order.

23

2

2x

x

x1.

x2

232 22)22( xxxx 2.

2x3 – 2x2

2233 2)22(2 xxxx 3.

2x2 + 4x

xx

x

2

2 2

4.

+ x

xxxx 22)22( 2 5.

2x2 – 2x

xxxxx 6)22()42( 22 6.

6x – 1

32

6

x

x7.

+ 3

66)22(3 xx8.

6x – 6

remainder5)66()16( xx9.

5

Answer: x2 + x + 322

x5

Since there is no x2 term in the

dividend, add 0x2 as a placeholder.

7

6 5 2 2 xxxx

x2 – 2x

– 3x + 6

– 3

– 3x + 60

Answer: x – 3 with no remainder.

Example: Divide x2 – 5x + 6 by x – 2.

8

Dividing by Synthetic Division

• Synthetic Division is a method to divide any polynomial by a binomial.

• The steps must be followed exactly in order or you will not get the correct end result

• The following slide shows the steps for one complete problem.

9

Find: (6x3- 19x2 + x + 6) (x-3)

• Step 1: Rewrite the dividend with all terms. If a term is missing, insert a zero for that term.

• Bring down the coefficients from the dividend and make a row.

• Next identify the divisor. It must be in the format (x-r). Bring down r and put in a box on the left. Draw a line.

• Bring down 1st coefficient under the line. Multiply it by “r” and add to next column. Then repeat.

• New row of numbers are the coefficients of the quotient starting with one power less.

• 6x3 – 19x2 + 1x + 6

6 -19 1 6

3

6

18

-1

-3

-2

-6

0

6x2 – 1x – 2 (No remainder)

10

Find: (4x4- 5x2 + 2x + 4) (x+1)

• Step 1: Rewrite the dividend with all terms. If a term is missing, insert a zero for that term.

• Bring down the coefficients from the dividend and make a row.

• Next identify the divisor. It must be in the format (x-r). Bring down r and put in a box on the left. Draw a line.

• Bring down 1st coefficient under the line. Multiply it by “r” and add to next column. Then repeat.

• New row of numbers are the coefficients of the quotient starting with one power less.

• 4x4 + 0x3 – 5x2 + 2x + 4

4 0 -5 2 4

-1

4

-4

-4

4

-1

1

3

-3

1

1

13144 23

xxxx

11

16

Synthetic division is a shorter method of dividing polynomials.

This method can be used only when the divisor is of the form

x – a. It uses the coefficients of each term in the dividend.

Example: Divide 3x2 + 2x – 1 by x – 2 using synthetic division.

3 2 – 12

Since the divisor is x – 2, a = 2.

3

1. Bring down 3

2. (2 • 3) = 6

6

8 15

3. (2 + 6) = 8

4. (2 • 8) = 16

5. (–1 + 16) = 15coefficients of quotient remainder

value of a coefficients of the dividend

3x + 8Answer: 2

x15

12

Homework

• WS 4-1