Chapter 6 FaCtoring and the distributive ProPerty 171

5.^3
2448

7

624

1

42024

2

246

7

6

xx^22 xxx

xx x

+−

+

−

−

+−

=

−+

+

()()( −−

−

4 +−

1

)( 46 xx)( 1 )

LCD = 12(x − 4)(x + 6)(x − 1)

6.^24
71269

24

(^223433)
x
xx
x
xx
x
xx
x
xx
−
−+
−
−+
= −
−−
−
()()()−−( ))
LCD = (x − 3)(x − 3)(x − 4) = (x − 3)^2 (x − 4)

7.^67
5

3 67

5

3

(^221)
x
x
x
x
−
−
+= −
−

• LCD = x^2 − 5
  Once we find the LCD, we rewrite each fraction in terms of the LCD, that
  is, we multiply each fraction by the “missing” factors over themselves. Once we
  do this, each fraction has the same denominator, so we can add or subtract the
  numerators.
  EXAMPLES
  Find the sum or difference.
  1
  23 9
  1
  xx^223133
  x
  xxx
  x
  +− xx


• 
  

  −

  
  

  +−

  
  


• 
  

  ()()()−+()
  LCD = (x + 3)(x − 1)(x − 3)
  The factor x – 3 is “missing” in the first denominator so multiply the first
  fraction by x
  x
  −
  −
  3
  3

  
  

. An x – 1 is “missing” from the second denominator so
multiply the second fraction by x
x

−

−

1

1

.

1

31

3

333

1

1

3

3

()() ()()

()(

xx

x

x

x

xx

x

x

x

x

+−

⋅ −

−

+

−+

⋅ −

−

= −

+ xxx

xx

−−xxx

+ −

13 +−−

1

)( ) 313

()

()()()

xxx

xxx

xxx

xxx

−+ −

+−−

= −+ −

+−−

31

313

3

31

()^2

()()()()()( 33

3

313

2

)( )( )( )

= −

+−−

x

xxx

EXAMPLES

Find the sum or difference.