Download Multiplication and Division of Rational Expressions

Chapter 6 Section 2
Multiplication and
Division of Rational
Expressions
1
Multiplying Rational Expressions
Multiplying Two Fractions
a c ac
 
, b  0, d  0
b d bd
Example:
3 4 (3)(4) 12
(2)( 6 )
2





2 15 (2)(15) 30
(5)( 6 )
5
1
or we can divide out common
factors first then multiply.
2
3 4
2


5
2 15
1
5
2
Multiplying Rational Expressions
 Factor the numerator and the denominator completely.
 Divide out common factors.
 Multiply numerators together and multiply denominators
together.
 Note: Factor -1 when numerator and denominator only
differ by their signs
 Rule:
a c a  c ac
 

, b  0 and d  0
b d b  d bd
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Multiplying Rational Expressions
Example:
1
5
1
1
2 x3 25 y 2
2 x x x 25 y y
5x2 y




 5x2 y
5 y 2x
1
5 y
2 x
1
1
1
1
Example:
2
2m 8n
2m m 8n
16
 4 2  

 
3
7n 3m
21n
7 n nnn 3 m m
4
Multiplying Rational Expressions
Example:
3
( x  4)
(3)
(3)( x  4 )
3


( x  4)   3

 2

 2
2 
2
1
( x )( x  4)
x
( x )( x  4 )
 x  4x 
Factor out common factor of x2
5
Multiplying Rational Expressions
Example:
Write as two factors
( x  3) 2 2 x
( x  3)( x  3 )
( 2 )( x )
x3
 2



2
8x
x 9
(4 x)( x  3)
( 8 )( x)( x ) ( x  3 )( x  3)
4
Difference in Two Squares: a2 – b2 = (a + b)(a – b)
a = x and b = 3
x2 – 32 = (x + 3)(x – 3)
We leave the numerator as a polynomial (in unfactored form)
and the denominator in factored form.
6
Multiplying Rational Expressions
Example:
4
m  6 20m
( m  6 ) ( 20 )( m )



 4
5m 6  m
( 5 )( m ) (1)( m  6 )
When only the signs differ in a numerator and denominator
factor out -1 (in either the numerator or denominator) then
divide out the common factor.
7
Multiplying Rational Expressions
Example:
Factor out -1 then GCF of 2
2x  5 2  6x
( 2 x  5 ) (-1)(6x-2)
(1)(2)( 3 x  1 )




 2
3x  1 2 x  5
(3x  1)
(2x+5)
( 3x  1 )
8
Multiplying Rational Expressions
Example:
(3 x 2  13 x  10) (2 x 2  x  1)

(3 x 2  x  2) (2 x 2  9 x  5)
(3 x 2  15 x  2 x  10) (2 x 2  2 x  x  1)

(3 x 2  3 x  2 x  2) (2 x 2  x  10 x  5)
3 x( x  5)  2( x  5) 2 x( x  1)  1( x  1)

3 x( x  1)  2( x  1) x(2 x  1)  5(2 x  1)
( 3 x  2 )( x  5 ) ( 2 x  1 )( x  1 )
1
( 3 x  2 )( x  1 ) ( x  5 )( 2 x  1 )
(a)(c) = (3)(-10)=-30
Factors of -30 that sum to 13
(15)(-2) = -30 and (15)+(-2) = 13
(a)(c) = (2)(-1)=-2
Factors of -2 that sum to 1
(2)(-1) = -2 and (2)+(-1) = 1
(a)(c) = (3)(-2)=-6
Factors of -6 that sum to 1
(3)(-2) = -6 and (3)+(-2) = 1
(a)(c) = (2)(-5)=-10
Factors of -10 that sum to 9
(-1)(10) = -10 and (-1)+(10) = 9
9
Multiplying Rational Expressions
Example:
4a
2a 2  9a  4

10a 2  5a
8a 2
(2a 2  8a  a  4) (1)(4)(a )

(5a )(2a  1)
(8a 2 )
2a (a  4)  1(a  4) (1)(4)(a )

(5)(a )(2a  1)
(8)(a )(a )
1
2a2 – 9a + 4
(a)(c) = (2)(4) = 8
Factors of 8 that sum to -9
(-8)(-1) = 8 and (-8)+(-1) = -9
Replace -9a with factors (-8) and
(-1) then factor by grouping.
10a2 – 5a
Factor out the common factor of 5a
(a  4)
( 2a  1 )(a  4) (1)( 4 )( a )


10a 2
(5)(a )( 2a  1 )
( 8 )(a )( a )
-4a
Factor out a -1
2
10
Multiplying Rational Expressions
Example:
x2  y 2
2x  5 y
 2
x  y 3 x  4 xy  y 2
x2 – y2
Difference in Two Squares: a2 – b2 = (a + b)(a – b)
a = x and b = y
x2 – y2 = (x + y)(x – y)
2 + 4xy + y2
( x  y( x  y)
(2 x  5 y )
3x

( x  y)
(3 x 2  3 xy  xy  y 2 ) (a)(c) = (3)(1) = 3
( x  y( x  y)
(2 x  5 y )

( x  y)
3 x( x  y )  y ( x  y )
( x  y )( x  y )
( x y)

Factors of 3 that sum to 4
(3)(1) = 3 and (3)+(1) = 4
Replace 4xy with the factors 3 and 1
Factor by grouping
(2 x  5 y )
2x  5 y

3x  y
(3 x  y )( x  y )
11
Divide Rational Expressions
Invert the devisor (the second fraction) and multiply
Rule: Divide Two Fractions
a c a d ad
   
, b  0, d  0, c  0
b d b c bc
Example:
1
2
3 9
3 10 2
  

5 10 5 9
3
1
3
12
Divide Rational Expressions
To Divide a Rational Expression we multiply the first
fraction by the reciprocal of the second fraction.
Example:
3
2a 3b

2
b
7
2
3
3
3
2a 7
(2)(7)(a )
14a
 2  2 

2
2
4
b 3b
(3)(b )(b )
3b
13
Divide Rational Expressions
To Divide a Rational Expression we multiply the first
fraction by the reciprocal of the second fraction.
Example:
Difference in Two Squares: a2 – b2 = (a + b)(a – b)
a = x and b = 4
x2 – 42 = (x + 4)(x – 4)
x 2  16 x  4
x 2  16 x  2
( x  4 )( x  4)( x  2 )




 x4
x2 x2
x2 x4
( x  2 )( x  4 )
14
Divide Rational Expressions
Example:
When only the signs differ in a numerator and denominator
factor out a -1 (in either the numerator or denominator) then
divide out the common factor.
1
4
1
(2  5 x)
(1) (1)( 5 x  2 )
1






5x  2 2  5x
(5 x  2)
4
(4)
4
( 5x  2 )
15
Divide Rational Expressions
Example:
x 2  3x  18
2

(
x

6)
x3
x  3x  18
1

3
x
( x  6) 2
2
X2 + 3x – 18
Factors of -18 that sum to 3
(6)(-3) = -18 and (6)+(-3) = 3
2
(
x

6)
( x  6) 2 Same as
1
( x  6)2 Same as ( x  6)( x  6)
( x  6 )( x  3)
(1)
x 3

 3
3
(x )
x ( x  6)
( x  6 )( x  6)
16
Divide Rational Expressions
Example:
Factor out any common factors
6 x 2  15 x  36 2 x 2  x  3
6 x 2  15 x  36 4 x 2  4 x


 2
10 x
4x2  4x
10 x
2x  x  3
(a)(c) = (2)(-12) = -24
Factors of -24 that sum to 5
(8)(-3) = -24 and (8)+(-3) = 5
(a)(c) = (2)(-3) = -6
Factors of -6 that sum to -1
(2)(-3) = -6 and (2)+(-3) = -1
(3)(2 x 2  5 x  12)
(4 x)( x  1)
(3)(2 x 2  8 x  3 x  12)
(4 x)( x  1)



(10)( x)
(2 x 2  2 x  3 x  3)
(10)( x)
2 x( x  1)  3( x  1)
2
(3)( 2 x  3 )( x  4) ( 4 )( x )( x  1 )
(3)(2)( x  4)
6( x  4)
6 x  24




(5)
5
5
(10 )( x )
( 2 x  3 )( x  1 )
5
17
Special Factoring
Difference of Two Squares
(a  b )  (a  b)(a  b)
2
2
Sum of Two Cubes
(a  b )  (a  b)(a  ab  b )
3
3
2
2
Difference of Two Cubes
(a  b )  (a  b)(a  ab  b )
3
3
2
2
18
Remember
 Factor completely before you simplify.
 Change a division problem to a
multiplication problem before factoring and
canceling.
 Factor out a -1 when the terms only differ
by the signs.
 Special factoring makes the problem easier.
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HOMEWORK 6.2
Page 366-367:
# 19, 23, 27, 33, 39, 45, 63, 87
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