9.4 Polynomial Division, Factors, and Remainders

©2001 by R. Villar

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Polynomial Division, Factors, and Remainders

In this section, we will look at two methods to divide polynomials:

long division (similar to arithmetic long division)

synthetic division (a quicker, short-hand method)

Let’s take a look at long division of polynomials...

Example: Divide (2x2 + 3x – 4) ÷ (x – 2)

(x – 2) 2x2 + 3x – 4

Rewrite in longdivision form...

divisor

dividendThink, how many timesdoes x go into 2x2 ?

2x

Multiply by the divisor. 2x2 – 4xSubtract. 7x – 4Think, how many timesdoes x go into 7x ?

+ 7

7x – 14

10 remainder

2x + 7 + 10 x – 2 divisor

Write the result like this...

Example: Divide (p3 – 6) ÷ (p – 1)

(p – 1) p3 + 0p2 + 0p – 6

Be sure to add “place-holders”for missing terms...

p2

p3 – p2

p2 + 0p

+ p

p2 – p

p – 6

p2 + p + 1 – 5 p – 1

+ 1

p – 1–5

Let’s look at an abbreviated form of long division, called synthetic division...

Synthetic division can be used when the divisor is in the form (x – k).

Example: Use synthetic division for the following (2x3– 7x2– 8x + 16) ÷ (x – 4)

First, write down the coefficients in descending order, and k of the divisor in the form x – k :

k

2

Bring downthe firstcoefficient.

8

Multiply thisby k

1Add thecolumn.

4

–4–16

0These are the coefficients of thequotient (and the remainder)

2x2 + x – 4Repeat the process.

2 -7 -8 164

Example: Divide (5x3 + x2 – 7) ÷ (x + 1)

–1 5 1 0 –7Notice thatk is –1 sincesynthetic divisionworks for divisorsin the form (x – k).

place-holder

5x2 – 4x + 4 – 11 x + 1

5

–5

–4

4

4

–4

–11