section 8-4 multiplying and dividing rational expressions
TRANSCRIPT
Section 8-4
Multiplying and Dividing
Rational Expressions
Objectives
• I can simplify rational expressions with multiplication
• I can simplify rational expressions with division
Review Key Concepts
• Factoring Methods– GCF– Reverse FOIL– Swing & Divide– Difference of 2 Squares
GCF
• 3x + 9
• 3(x + 3)
• Or
• (2 – x)
• -1(x – 2)
Reverse FOIL
• x2 - x – 12
• (x – 4)(x + 3)
Swing & Divide
• 3x2 + 2x – 8
• x2 + 2x – 24
• (x + 6)(x – 4)
• (x + 6/3)(x – 4/3)
• (x + 2)(3x – 4)
Difference of 2 Squares
• 16x2 – 9
• (4x + 3)(4x – 3)
Multiplying rational Expressions
• Usually you DON’T multiply, you just reduce
• 1. You will factor all numerators and denominators, then
• 2. Reduce or cancel like terms
Simplifying Property for Rational Expressions
• If a, b, and c are expressions with b and c not equal to zero, then
ac
bca
b
Example: Reducing
3
2
2
2
8
3
5
2
a
bc
cb
a
cbbaaa
ccbaa
85
32
cbbaaa
ccbaa
85
32ab
c
40
6
ab
c
20
3
Example: Factoring
2
33
34
822
2
x
x
xx
xx
2
33
34
822
2
x
x
xx
xx)2)(4( xx
)1)(3( xx
)1(3 x
)2)(1)(3(
)1(3)2)(4(
xxx
xxx
)3(
)4(3
x
x
EXAMPLE 1 Simplify a rational expression
x2 – 2x – 15x2 – 9
Simplify :
x2 – 2x – 15x2 – 9
(x +3)(x –5)(x +3)(x –3)= Factor numerator and denominator.
(x +3)(x –5)(x +3)(x –3)= Divide out common factor.
Simplified form
SOLUTION
x – 5x – 3=
ANSWER x – 5x – 3
GUIDED PRACTICE for Examples 1 and 2
2x2 + 10x3x2 + 16x + 5
6.
2x2 + 10x3x2 + 16x + 5 (3x + 1)(x + 5)
2x(x + 5)= Factor numerator and
denominator.
Divide out common factor.
2x3x + 1= Simplified form
(3x + 1)(x + 5)2x(x + 5)=
ANSWER 2x3x + 1
SOLUTION
Dividing Rational Expressions
• We actually never want to divide rational expressions.
• Instead, turn them into multiplication problems to simplify by reducing
• To turn division into multiplication, simply change the sign and invert the 2nd fraction
Division to Multiplication
c
d
b
a
d
c
b
a
Example
)1(
)3(
)22(
)9( 2
x
x
x
x
)3(
)1(
)22(
)9( 2
x
x
x
x
)3(
)1(
)22(
)9( 2
x
x
x
x)3)(3( xx
)1(2 x)3)(1(2
)1)(3)(3(
xx
xxx
2
)3( x
Handling Negatives
)2(
)2(3
)2(
)2(
a
x
x
a
)2(
)2(3
)2(
)2(
a
x
x
a
)2(
3)2(
a
a
)2(
3)2(1
a
a
)2(
3)2(1
a
a 3
GUIDED PRACTICE for Examples 6 and 7
Divide the expressions. Simplify the result.
4x5x – 20
x2 – 2xx2 – 6x + 8
11.
4x5x – 20
x2 – 2xx2 – 6x + 8
Multiply by reciprocal.
Divide out common factors.
Factor.
Simplified form
4x5x – 20 x2 – 2x
x2 – 6x + 8=
4(x)(x – 4)(x – 2)5(x – 4)(x)(x – 2)
=
4(x)(x – 4)(x – 2)5(x – 4)(x)(x – 2)
=
45=
SOLUTION
Homework
• WS 12-4