In multiplying rational expressions, we use the following rule: Dividing by a rational expression is...
18
In multiplying rational expressions, we use the following rule: Dividing by a rational expression is the same as multiplying by its reciprocal. 5.2 Multiplying and Dividing Rational Expressions. P R PR Q S QS P R P S Q S Q R
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Transcript of In multiplying rational expressions, we use the following rule: Dividing by a rational expression is...
In multiplying rational expressions, we use the following rule:
Dividing by a rational expression is the same as multiplying by its reciprocal.
5.2 Multiplying and Dividing Rational Expressions.
P R P R
Q S Q S
P R P S
Q S Q R
To multiply rational expressions we use the following steps:
1. Multiply the numerator and multiply the denominator.
2. Factor completely the numerator and the denominator.
3. Cancel common factors and simplify
Multiply:2 3
4 2
16 5
4
x x
y y
2 3
4 2
16 5
4
x x
y y
2 3
4 2
6 5
4
1 x x
y y
5
6
20x
y
Factor 16 and use product rule to multiply.
2 3
4 2
4 4 5
4
x
y
Cancel common
factors and simplify
Multiply:4 12 5
5 7 5
80 14
49 25
m x y
x y m
12 516 2x
7
2
32
35
x
y m
4 12 5
57580 14
2549
m x y
mx y
4 12 5
5 7 5
80 14
49 25
m x y
x y m
12
5
5
5 7
45 16 2 7
7 7 5 5
ym
m
x
x y
Factor numbers on numerator and denominator.
Cancel common factors and use quotient rule.
7 5 5 47 5y m
Multiply:2
2
( 2) 2
4 2
t t
t t
2
2
( 2) 2
4 2
t t
t t
2 ( 2)( 2)
2 2 ( 2)
t t t
t t t
( 2)
2
t
t
Multiply:2( ) 30
5 2( )
a b
a b
2( ) 30
5 2( )
a b
a b
2( )
5
3
2 )
0
(
a b
a b
2
1
( )
5 2 ( )
2 3 5 a b
a b
2 13( )
1
a b
= 3(a + b)
Factor numerator
Cancel common factors and use quotient rule
Divide:8 6
5 5
35 25
9 10
q q
q q
Dividing by a rational expression is the same as multiplying by its reciprocal.
Change the division to a multiplication and invert the divisor.
8 6
5 5
35 25
9 10
q q
q q
8 5
5 6
35 10
9 25
q q
q q
8 5
5 69
35 10
25
q q
q q
8 5
5 6
5 5
5
7 2
9 5
q
q
Factor numbers and use product rule
13
11
14
9
q
q
13 1114
9
q
214
9q
Cancel common factors
Use quotient rule
8 5
5 6
5 5
5
7 2
9 5
q
q
Divide:2
2
2
4 ( 4)
a a
a a
Change the division to a multiplication and invert the divisor.
2
2
2
4 ( 4)
a a
a a
2
2
2 ( 4)
4
a a
a a
2
2( 4)2
(
4)
aa
a a
2 1
2 1
( 4)2
a
a
2( 4)a
a
Use quotient rule
Divide: 2 2
( 1) 3 3
17 30 7 18
x x
x x x x
Change the division to a multiplication and invert the divisor.
2 2
( 1) 3 3
17 30 7 18
x x
x x x x
2
2
( 1) 7 18
17 30 3 3
x x x
x x x
Factor each expression
2
2
( 1) 7 18
17 30 3 3
x x x
x x x
( 1) ( 9)( 2)
( 15)( 2) 3( 1)
x x x
x x x
Cancel common factors
( 9)
3( 15)
x
x
x2 + 7x – 18
x x
+9 -2
= (x + 9)(x – 2)
x2 – 17x + 30 x x
-15 -2
= (x – 15)(x – 2)
3x + 3 = 3(x + 1)
Divide:3 2 3 2
2 2
6 2
2 1 1
x x x x x x
x x x
Change the division to a multiplication and invert the divisor.
Factor common factors.
Cancel common factors
3 1
1
x
x
3 2 2
2 3 2
6 1
2 1 2
x x x x
x x x x x
2 2
2 2
(6 1) 1
2 1 ( 2 1)
x x x x
x x x x x
Factor all expressions
(3 1)(2 1) ( 1)( 1)
(2 1)( 1) ( 1)( 1)
x x x x x
x x x x x
Divide:2 2 2
2 2 2
( )
2 ( )
x y x y
x xy y x y
Change the division to a multiplication and invert the divisor.
Factor all expressions.
x2 – 2xy + y2 =
2 2 2
2 2 2
( )
2 ( )
x y x y
x xy y x y
x2 – y2 = (x + y) (x – y)x x
- y- y
2
2
( )( ) ( )
( ( )) ( )
x y x y
x y xy y
x y
x
(x – y) (x – y) = (x – y)2
2
2
( )( ) ( )
( ( )) ( )
x y x y
x y xy y
x y
x
x y
x y
Cancel common factors
Multiply:3 2 2
2 2
2 7 3 3
2 3 ( 3)
x x x x x
x x x
Factor common factors.
Factor all expressions2
2 2
(2 7 3) ( 3)
2 3 ( 3)
x x x x x
x x x
2x2 – 7x + 3 =
2x x
- 1 - 3
(2x – 1) (x – 3)
x2 + 2x – 3 =
x x
+ 3 - 1 (x + 3) (x – 1)
2
2 2
(2 7 3) ( 3)
2 3 ( 3)
x x x x x
x x x
(2 1)( 3) ( 3)
( 3)( 1) ( 3)( 3)
x x x x x
x x x x
Cancel common factors
(2 1)( 3) ( 3)
( 3)( 1) ( 3)( 3)
x x x x x
x x x x
2 (2 1)
( 1)( 3)
x x
x x
Divide:2 2 2 2
2 2 2 2
3 17 10 6 2
6 13 5 6 5
r rs s r rs s
r rs s r rs s
Change division to multiplication.
Factor all expressions
3r r
+2s +5s
3r 2r
- s +5s
2 2 2 2
2 2 2 2
3 17 10 6 5
6 13 5 6 2
r rs s r rs s
r rs s r rs s
3r 2r
-s-s3r 2r
+2s -s
(3 2 )( 5 )r s r s
5
2 5
r s
r s
Cancel common factors
(3 )(2 5 )r s r s (3 )(2 )r s r s
(3 2 )(2 )r s r s
Divide:
Change the division to a multiplication.
Factor all expressions.
Cancel common factors
r2 + 4r – 12 =
r2 + r – 6 = (r + 3) (r – 2)r r
- 2+6(r + 6) (r – 2)
2
2
6 3
4 12 1
r r r
r r r
2
2
6 1
4 12 3
r r r
r r r
2
2
6 1
4 12 3
r r r
r r r
( 3)( 2)( 1)
( 6)( 2)( 3)
r r r
r r r
1
6
r
r
