FINDING THE LEAST COMMON MULTIPLE (LCM)
What is the LCM of
4
12 and
6
8?
1. Find the prime factors.
x 2  4 x  12 :  x  6  x  2 
x2  6x  8 :
 x  4  x  2 
2. Write the product of primes (use the highest power for
each factor)
LCM:  x  6  x  4  x  2 
Complete Got It? #1b p.535 1b).  x  1 x  2   x  4 
2
1
ADDING RATIONAL EXPRESSIONS
• Find LCM of denominators and use as the Least
Common Denominator (LCD).
• Use LCD just like you did to add fractions.
1 1
1
1
5 1
1 22
5
4
9
  3
  3
 2 


8 10 2 25 5 2 25 2
40 40 40
• Make sure answer is in simplest form
ADDING RATIONAL EXPRESSIONS
What is the sum
in simplest form? State
any restriction on the variables.
4
x2
 2
x  3x x  6 x  9
2
LCM: x  x  3
2

4
x2

x  x  3  x  3 2

 x  3  4  x  2  x
 x  3 x  x  3  x  32 x

Complete Got
It? #2b p.535
.
2
2

2
2
;
4 x  12
x  x  3
2

x 2  2 x  12
x  x  3
2
x2  2x
x  x  3
2
x  0 or  3
2
2
SUBTRACTING RATIONAL EXPRESSIONS
What is the difference
in simplest form?
State any restrictions on the variable.
See next slide for solution
x 1
x
x 1
x



x  2x  8 4x  8
 x  4  x  2  4  x  2 
2
4
x 1
x
x4
 


4  x  4  x  2  4  x  2  x  4
LCM:
4  x  2  x  4 


Complete Got
It? #3b p.536
3 .
4
;
6
5
5or 1
4x  4
x2  4x

4  x  4  x  2  4  x  4  x  2 
4x  4   x2  4x 
4  x  4  x  2 
  x2  4
 x2  4
=
=
4  x  4  x  2  4  x  4  x  2 
=
  x  2  x  2    x  2 
=
4  x  4  x  2  4  x  4 
x  -4 or 2
3
Homework: p.539 #7, 10, 11-21 odd, 31-35 odd
SIMPLIFYING A COMPLEX FRACTION
Complex Fractions:
• A rational expression with at least one fraction in the
numerator, denominator or both.
• Two methods to simplify complex fractions:
1. Multiply both numerator and denominator by the LCD of all
the rational expressions.
2. Combine the fractions in the numerator and the fractions in
the denominator. Multiply the new numerator by the
reciprocal of the new denominator.
4
SIMPLIFYING A COMPLEX FRACTION
3x 
What is a simpler form of
y2
x
x
• Method 1:
3x 
1
y
y2
x
x
LCD of all rational
expressions:

1
 3 x  y  xy

 2
y

  x  xy
 x


xy
1
y
1
 xy
 y
Multiply by the LCD
 3x  xy  
 y2 
  xy   x  xy
 x 
3x 2 y  x
 3
y  x2 y
Distribute and Simplify
SIMPLIFYING A COMPLEX FRACTION
• Method 2:
3x 
LCD of
numerator:
1
y
y2
x
x
y
LCD of
denominator:
x
y
1
 3x  
y
y
Combine rational expressions in
 2
numerator and denominator
y
x
  x
x
x
3 xy  1
y
 2
Simplify
y  x2
x
3 xy  1  x 
Multiply by the

 2
2 
reciprocal
y  y x 
3x 2 y  x
 3
y  x2 y
5
Homework: p.539 #22-29, 41-44
6