Chabot Mathematics
§6.3 Complex
Rational Fcns
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Review § 6.2
MTH 55
 Any QUESTIONS About
• §6.2 → Add-n-Sub Rational
Expressions
 Any QUESTIONS About HomeWork
• §6.2 → HW-20
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Complex Rational Expression
 Complex Rational Expression is a
rational expression that contains
rational expressions within its numerator
and/or its denominator.
 Some examples:
3 3x  5 a
4a
2x 

x , 2x  7 , 3
5 .
x2
a6
7x
x 3
a2
2
Chabot College Mathematics
3
The rational
expressions within
each complex
rational expression
are red.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Simplify Complex Rational
Expressions by Dividing
1. Add or subtract, as needed, to get a
single rational expression in the
numerator.
2. Add or subtract, as needed, to get a
single rational expression in the
denominator.
3. Divide the numerator by the
denominator (invert and multiply).
4. If possible, simplify by removing any
factors equal to 1
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
 SOLUTION
x
x4  x  3
3
x  4 2x  8
2x  8
Rewriting with a division symbol
x 2x  8


x4
3
Multiplying by the reciprocal of the
divisor (inverting and multiplying)
x 2( x  4 )


3
x4
Factoring and removing a
factor equal to 1.

Chabot College Mathematics
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x
x4
3
2x  8
2x
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
 SOLUTION
x
3 x
5
5 
3 
3 3
4 1
4 2 1

 
x 2x x 2 2x
Multiplying by 1 to get the LCD,
3, for the numerator.
Multiplying by 1 to get the LCD,
2x, for the denominator.
15 x
15  x

 3 3  3
8
1
7

2x 2x
2x
Chabot College Mathematics
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x
3
4 1

x 2x
5
Adding
Subtracting
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Solution cont.
15 x
15  x

 3 3  3
8
1
7

2x 2x
2x
Add and Subtract Using
COMMON Denominators
15  x 7


3
2x
Rewriting with a division symbol.
This is often done mentally.
15  x 2 x


3
7
Multiplying by the reciprocal of the
divisor (inverting and multiplying)
30 x  2 x

21
Chabot College Mathematics
7
x
3
4 1

x 2x
5
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
1
2
y
2
3
y
 SOLUTION
Write the numerator and denominator
as equivalent fractions.
1
2 y 1 1
2
  
y
1 y y 1

2
3 y 2 1
3
  
y
1 y y 1
2y 1

y y

3y 2

y y
2 y 1
y

3y  2
y
2 y 1 3y  2 2 y 1
y




y
y
y 3y  2
Chabot College Mathematics
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2 y 1

3y  2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Simplify Complex Rational
Expressions by LCD Mult.
1. Find the LCD of ALL rational expressions
within the complex rational expression.
2. Multiply the complex rational expression by
a factor equal to 1. Write 1 as the LCD
over itself (LCD/LCD).
3. Simplify. No fractional expressions
should remain within the complex
rational expression.
4. Factor and, if possible, simplify.
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
 SOLUTION - In This Case look
for the LCD of all four Terms.
1 3 1 3


2 4  2 4  12
2 1 2 1 12


3 4 3 4
1 3
1 3
 12


2 4  12   2 4 
2 1 12  2 1 

 12

3 4
3 4
Chabot College Mathematics
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1 3

2 4
2 1

3 4
Multiplying by a factor
equal to 1, using the LCD:
12/12=1
Multiplying the numerator by 12
Don’t forget the parentheses!
Multiplying the denominator by 12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Solution cont.
1 3
1 3
 12


2 4  12   2 4 
2 1 12  2 1 

 12

3 4
3 4
1
3
(12)  (12)
4
2
2
1
(12)  (12)
3
4
Using the distributive law
69
15

, or
83
11
Simplifying
Chabot College Mathematics
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1 3

2 4
2 1

3 4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
 SOLUTION - The LCD for all is x
1
1
1
1( x)  ( x)
1
1
x
x
x
x 
1
1
1 x
1( x)  ( x)
1
1
x
x
x
Using the distributive law
 When we multiply by x, all
fractions in the numerator
and denominator of the
complex rational
expression are cleared:
Chabot College Mathematics
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1
1
1
1
x
x
1
1
1
1
x
x
1
1( x)  ( x)
x 1
x


1
1( x)  ( x) x  1
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
 SOLUTION
6 2
 2
3
x x
3 4
 2
x x
6 2
6 2
 2
 2 3
3
3
The LCD for all is x3 so we
x x  x x x
multiply by 1 using x3/x3.
3 4
3 4 x3
 2
 2
x x
x x
6 3 2 3
x  2 x
3
x
Using the distributive law
 x
3 3 4 3
x  2 x
x
x
6  2x
2(3  x)
 2

3x  4 x x(3x  4)
Chabot College Mathematics
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All fractions have been
cleared and simplified.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
x 3

2 x
x 1
1
x
 SOLUTION: Multiply the
numerator and denominator by the LCD
of all the rational expressions; 2x here
 x 3
x 3
x 2x 3 2x


2
x



  
2
x

2 x  
 2 1 x 1
x 1  x 1 
1 2x x 1 2x
1
1


2
x
 



x
x 
1 1
x
1

x2  6

2 x  2( x  1)
Chabot College Mathematics
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x2  6
x2  6
x2  6


or
2x  2x  2
4x  2
2(2 x  1)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
 SOLUTION:
5x 2
5x 2
 2
 2 x2 y3
xy y
xy y


1
5
1
5
2 3


x
y
3
2
3
2
y
x y y
x y
 5x 2  2 3
 xy  y 2  x y

 
 1
5  2 3
 y3  x2 y  x y


Chabot College Mathematics
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5x 2
 2
xy y
.
1
5
 2
3
y
x y
Multiplying by 1,
using the LCD.
Multiplying the
numerator and the
denominator.
Remember to use
parentheses.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
 SOLN
cont.
5x 2
 2
xy y
.
1
5
 2
3
y
x y
5x 2 3 2 2 3
x y  2 x y
xy
y

1 2 3
5
2 3

x
y


x
y
3
2
y
x y
xy
y2
2 2
 5 x y  2  2 x 2 y Removing factors that equal
xy
y

1. Study this carefully. Take
3
2
y
x y
2
2
CARE with CANCELLING

x


5
y
3
2
y
x y
5x2 y 2  2 x2 y

x2  5 y 2
x 2 y(5 y  2)
 2
x  5 y2
Chabot College Mathematics
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Using the distributive law to
carry out the multiplications
Simplifying
Factoring. This does
not simplify further
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Example  Simplify
x 2  y 2
x 3  y 3
 SOLUTION:
Rewrite using only positive exponents
x 2  y 2
x 3  y 3
 1
1  3 3
1
1
 2  2  2  x y
2
x
y 
x
y



1 1
 1 1  3 3


x y
x 3 y 3  x 3 y 3 
xy 3  x3 y
 3 3
y x
Chabot College Mathematics
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xy( y 2  x 2 )

( y  x)( y 2  xy  x 2 )
LCD of all
individual
Rational
Expressions is
x 3y 3
Simplified
Version is still a
bit “Complex”
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
WhiteBoard Work
 Problems From §6.3 Exercise Set
• 38, 42, 48, 53

Three Resistors
in Parallel
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
All Done for Today
More Info
on
LCDs
Chabot College Mathematics
19
 Liquid Crystal Display
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Chabot College Mathematics
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5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
4
5
6
y
5
2
y
1
x
4
0
-6
-5
-4
-3
-2
-1
3
0
1
2
3
4
-1
-2
2
-3
1
-4
x
-5
5
-6
0
-3
-2
-1
0
1
2
3
-1
4
-7
-8
-2
-9
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
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file =XY_Plot_0211.xls
-10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-31_sec_6-3_Complex_Rationals.ppt
5
6