Rewriting Fractions: Factoring, Rationalizing, Embedded

Simplifying Rational Expressions

2

3

2 3 20

4 64

x x

x x

2

2 5 4

4 16

x x

x x

2 5 4

4 4 4

x x

x x x

2

2 5or

4 16

x

x x

2 5

4 4

x

x x

Simplify:Can NOT cancel since

everything does not have a common factor and its not in

factored form

CAN cancel since the top and bottom have a

common factor

Factor Completely

This form is more convenient in order to

find the domain

Polynomial Division: Area Method

Simplify:

x4 +0x3 –10x2 +2x + 3

x

- 3

x3

x4

-3x3

3x3

3x2

-9x2

-x2

-x

3x

-x

-1

3

x3 + 3x2 – x – 1

Div

isor

Dividend (make sure to include

all powers

of x)

The sum of these boxes must be the dividend

Needed Needed Needed Needed Check

Quotient

4 210 2 33

x x xx

7 7x

x

Rationalizing Irrational and Complex Denominators

The denominator of a fraction typically can not contain an imaginary number or any other radical. To rationalize the

denominator (rewriting a fraction so the bottom is a rational number) multiply by the conjugate of the

denominator.

Ex: Rationalize the denominator of each fraction.

7 7

7 7

x

x

7 7

4 7 7 7 7 4

x x

x x x

7 7x x

x

6 7

6 7

24 4 7

36 6 7 6 7 7

24 4 7294

6 7

b.

a.

7 7x

Simplifying Complex Fractions1

1

1

1x

y

xy

xy

1

1

xyxxyy

xy

xy

Simplify:

It is not simplified since it has embedded

fractions

To eliminate the denominators of the embedded fractions,

multiply by a common denominator

Check to see if it can be simplified more:

xy y

xy x

1

1

xyxxyy

xy

xy

xy y

xy x

1

1

y x

x y

No Common Factor. Not everything can be

simplified!

Trigonometric Identitiestan cot

sin cos

x x

x x

tan cot

sin cos sin cos

x x

x x x x

Simplify:

Split the fraction

Write as simple as possible

sin 1 cos 1

cos sin cos sin sin cos

x x

x x x x x x

1 1tan cot

sin cos sin cosx x

x x x x

2 2

1 1

cos sinx x

2 2sec cscx x

Use Trigonometric

Identities