Chapter 6 FaCtoring and the distributive ProPerty 169

Adding/Subtracting Fractions

We now use what we learned about factoring algebraic expressions to add/subtract

fractions. Remember that we can only add/subtract fractions that have the same

denominator. If the denominators are different, we can factor them so that we can

identify the LCD (least common denominator). Once we have the LCD, we

rewrite the fractions so that they have the same denominator and then perform

the arithmetic on the numerators. For now, we will practice finding the LCD.

EXAMPLES

Factor each denominator completely and find the LCD.

4

34

2

1

4

41

2

xx^2211

x

xxx

x

−− xx

+

−

=

−+

+

()()()−+()

From the first fraction, we see that the LCD needs x – 4 and x + 1 as fac-

tors. From the second fraction, we see that the LCD includes x – 1 and

x + 1, but x + 1 has been accounted for by the first fraction. The LCD is

(x – 4)(x –1)(x + 1).

75

2636

10 1

6

75

236

10 1

22

x

xx

x

xx

x

xx

+ x

−−

− −

+−

= +

+−

− −

()()(()xx+− 32 ()

LCD = 2(x+ 3)(x− 6)(x− 2)

x

xx xx

x

xx xx

−

++

+

++

= −

++

+

++

2

65

1

31815

2

51

1

(^22) ()() 35 ()( 1 1)
LCD = 3(x+ 5)(x+ 1)
1
1
3
1
1
1
3
1
1
1
3
xx− xxxx 1

• 
  

  −

  
  

  −

  
  


• 
  

  −−

  
  

  −

  
  


• −
  () −
  LCD =x− 1
  4 29
  5
  4
  1
  29
  5
  − +
  −
  =−+
  −
  x
  x
  x
  x
  LCD =x− 5
  3
  816
  2
  68
  3
  44
  2
  (^2242)
  x
  xx xx
  x
  ++ xx xx


• 
  

  ++

  
  

  ++

  
  


• 
  

  ()()()++()
  LCD = (x+ 4)(x+ 4)(x+ 2) = (x+ 4)^2 (x+ 2)
  EXAMPLES
  Factor each denominator completely and find the LCD.