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22

MULTIPLYING AND DIVIDING
ALGEBRAIC FRACTIONS

The rule

Section 2

Complex fractions -- Division

TO MULTIPLY FRACTIONS, multiply the numerators and multiply the denominators, as in arithmetic.

multiply fractions

Problem 1.   Multiply.

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Do the problem yourself first!

  a)    2
x
·   5
x
  =   10
x2
  b)    3ab
 4c
·   4a2b
 5d
  =   3a³b2
 5cd
   The 4's cancel.
  c)      3x  
x + 1
·    6x2 
x − 1
  =    18x³ 
x2 − 1
   The Difference of Two Squares
  d)    x − 3
x + 1
·   x − 2
x + 1
  =   x2 − 5x + 6
x2 + 2x + 1
If a multiplication looks like this:  a·   b
c
  or    b
c
·  a,  multiply only

the numerator.

a·   b
c
 =   ab
 c

Problem 2.   Multiply.

  a)    x
·   2x
 3
  =   2x2
 3
  b)    3x2
 4
·  7x3   =   21x5
  4
  c)   (x + 3)·   x − 3
x + 6
  =   x2 − 9
 x + 6
  d)     x2 − 2x + 5 
6x2 − 4x + 1
·  2x3   =   multiply fractions
  6x2 − 4x + 1
   No canceling!

Reducing

If any numerator has a divisor in common with any denominator,
they may be canceled.

a
b
·    c
d
·   e
a
  =    ce
bd

The a's cancel.

For if we took the trouble to multiply, and write

 ace 
bda

then it's obvious that we could divide both the numerator and denominator by a. It is more skillful, then, to reduce before multiplying.

Problem 3.   Multiply.  Reduce first.

  a)   ab
cd
·    ed
 fg
·    hcf
ake
  =   bh
gk
  b)    (x − 2)(x + 2)
        8x
·        __2x__     
(x + 2)(x − 1)
  =      x − 2 
4(x − 1)
  c)         __x³__     
(x + 2)(x + 3)
·   x + 3
  x7
  =     __1_   
(x + 2)x4
  d)    x(x + 1)
     6
·       2    
x2 − 1
  =   x(x + 1)
     6
·        __2__     
(x + 1)(x − 1)
  =      _x_   
3(x − 1)
  e)   aq·   b
cq
  =   ab
 c
  f)   10·  x + 2
   2
  =   5(x + 2)   =  5x + 10
  g)  3x·  5x
 6
  =   5x2
 2
  h)    a
 b
·   1
a
  =   1
b
  The a's cancel as −1, which on multiplication with 1 makes the fraction itself negative (Lesson 4).
  Example 1.   Multiply     x2 − 4x − 5
x2x − 6
·   x2 − 5x + 6
x2 − 6x + 5

Solution.   Although the problem says "Multiply," that is the last thing to do in algebra.  First factor.  Then reduce.  Finally, multiply.

And remember:  Only factors can be divided.

x2 − 4x − 5
x2x − 6
·   x2 − 5x + 6
x2 − 6x + 5
  =   (x + 1)(x − 5)
(x + 2)(x − 3)
·   (x − 3)(x − 2)
(x − 1)(x − 5)
 
    =   x + 1
x + 2
·   x − 2
x − 1
 
 
    =   x2x − 2
x2 + x − 2

Problem 4.   Multiply.

  a)       __x2__   
x2 + x − 12
·   x2 − 9
  2x6
  =        __x2__     
(x + 4)(x − 3)
·   (x − 3)(x + 3)
       2x6
 
    =      1   
x + 4
·   x + 3
  2x4
 
 
    =   _ x + 3 _
2x5 + 8x4
  b)    x2 − 2x + 1
x2x − 12
·   x2 + x − 6
x2 − 6x + 5
  =   __(x − 1)2__
(x − 4)(x + 3)
·   (x + 3)(x − 2)
(x − 1)(x − 5)
 
    =   x − 1
x − 4
·   x − 2
x − 5
 
 
    =   x2 − 3x + 2
x2 − 9x + 20
  c)    x2 + 3x − 10
x2 + 4x − 12
·   x2 + 5x − 6
x2 + 4x − 5
  =   (x + 5)(x − 2)
(x + 6)(x − 2)
·   (x − 1)(x + 6)
(x − 1)(x + 5)
 
    =   1  
  d)     _x³_ 
x2 − 1
·   x2 + x − 2
      x4
·    __x2__ 
x2 + 4x + 4
  =      ___x³___   
(x + 1)(x − 1)
·   (x − 1)(x + 2)
       x4
·    __x2__
(x + 2)2
  =   multiply fractions
 x + 1
·      1   
x + 2
 
    =   _    _x_   _
x2 + 3x + 2
end

Section 2:  Complex fractions -- Division

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