Download Review: Rewriting and Simplifying Fractions

Review: Rewriting and
Simplifying Fractions
Simplifying Rational Expressions
Simplify:
2x 3x2
0
4x36
4x
2
2x5x4
4xx216
This form is more
convenient in order to
find the domain
2x5x4
4
xx
 4x4
Can NOT cancel since
everything does not have a
common factor and its not in
factored form
Factor Completely
CAN cancel since the
top and bottom have a
common factor
2x5
2
x5
o
r 2
4xx4
4
x
1
6
x
Polynomial Division: Area Method
Simplify:
4
2
x

1
0
x

2
x

3
x

3
Divisor
Quotient
x
x3
3x2
-x
-1
x4
3x3
-x2
-x
The sum of these boxes must be
the dividend
-3
x
4
Needed
3
-3x
2
-9x
3x
3
–10x +2x
+3
Check
Needed 3 Needed2 Needed
x + 3x – x – 1
3
+0x
2
Dividend
(make
sure to
include
all
powers
of 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.
a.
4
6 7

6 7
6 7

x
b.
x7 7


2
4

47
3
6

67

67

7
x
7 7
x
7 7

244 7
29

xx

7

7
x
4

7x

7

7x

7

4


 
x7

7
x x

7 7
x
Simplifying Complex Fractions
xy
1

x
y

1
xy
x
Simplify:
 
xy
1
x y 1 xy y
xy
1

x
y

x
It is not simplified since

xy
it has embedded
1 xy y
fractions
xy  y

xy  x
1
x
1
y
Check to see if it can
be simplified more:
xy  y y  x 1

xy  x x  y 1
To eliminate the
denominators of the
embedded fractions,
multiply by a common
denominator
No Common Factor.
Not everything can be
simplified!
Trigonometric Identities
tan x
cot x
tan x  cot x


Simplify:
sin x  cos x
sin x  cos x sin x  cos x
1
1
 tan x 
 cot x 
sin x  cos x
sin x  cos x
sin x
1
cos x
1




cos x sin x  cos x sin x sin x  cos x
1
1

 2
2
cos x sin x
Split the fraction
Use
Trigonometric
Identities
 sec x  csc x
2
2
Write as simple as
possible