8.4 Dividing Polynomials

CORD Math

Mrs. Spitz

Fall 2006

Objective

• Divide polynomials by binomials

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Assignment

• Pg. 320 #3-31 all

Introduction

• To divide a polynomial by a polynomial, you can use a long division process similar to that used in arithmetic. For example, you can divide x2 + 8x +15 by x + 5 as shown on the next couple of slides.

Step 1

• To find the first term of the quotient, divide the first term of the dividend (x2) by the first term of the divisor (x).

xx

xxx

5

15852

2

x

3x + 15

Step 2

• To find the next term of the quotient, divide the first term of the partial dividend (3x) by the first term of the divisor (x).

xx

xxx

5

15852

2

x + 3

3x + 153x - 15

0Therefore, x2 + 8x + 15 divided by x + 5 is x + 3. Since the remainder is 0, the divisor is a factor of the divident. This means that (x + 5)(x + 3) = x2 + 8x + 15.

What happens if it doesn’t go evenly?

• If the divisor is NOT a factor of the dividend, there will be a non-zero remainder. The quotient can be expressed as follows:

Quotient = partial quotient + divisor

remainder

Ex. 1: Find (2x2 -11x – 20) (2x + 3).

xx

xxx

32

20112322

2

- 14x - 20+ 14x+ 21

x - 7

+ 1

← Multiply by x(2x+3)

← Subtract, then bring down - 20

← Multiply -7(2x+3)

← Subtract. The remainder is 1

The quotient is x – 7 with a remainder of 1. Thus, (2x2 -11x – 20) (2x + 3) = x – 7 + .

32

1

x

Other note . . .

• In an expression like s3 +9, there is no s2 term and no s term. In such situations, rename the expression using 0 as a coefficient of these terms as follows:

s3 + 9 = s3 + 0s2 + 0s + 9

Ex. 2: Find

23

23

3

9003

ss

ssss

3

93

s

s

3s2 + 0s

s2+3s + 9

+ 36

← Insert 0s2 and 0s. Why?

← Subtract, then bring down 0s

← Multiply 3s(s - 3)

← Subtract. The remainder is 36

The quotient is s2+3s+9 with a remainder of 36. Thus,

(s3 + 9) (s - 3) = s2 + 3s + 9 + .3

36

s

3s2 - 9s

9s + 9-9s + 27

← Multiply by s2