8-4: multiplying and dividing rational...
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Lesson 3-3 Multiplying and Dividing Rational Expressions
Math-3
Quiz 3-1:
Simplify the following:
2
64
x
x
x )2(
64*
)2(
)2(
x
x
xx
x
)2(
6
)2(
84
x
x
xx
x
x
x
x
x
xx
x*
)2(
6
)2(
84
)2(
6
)2(
84 2
xx
xx
xx
x
)2(
684 2
xx
xxx
)2)(1(
8102
xx
xx
Your Turn: Simplify
)7(
2832
x
xx1.
xx
xxx
3
122
23
2.
3.
72
32
x
x
BE CAREFUL!!!!!
)9(7
7
x
x
Addition and Subtraction mean:
Combine the terms into one term (if you can)
No, no, no, no, no!!!
If you can’t combine them (unlike terms)
they still are connected to each other.
Put binomials into a parentheses. )9(7
)7(
x
x
Multiplying Fractions
9
14*
7
6
Do you multiply first or do you simplify THEN multiply?
multiply 63
84 simplify
9*7
14*6
3*3*7
7*2*3*2
9*7
14*6
simplify 9
14*
7
6
3*3*7
7*2*3*2
3
4
3
4
Multiplying Rational Expressions
)3(2
)1(
x
xSimplify before you multiply.
)1)(3(2
)3)(3)(1(
xx
xxx
)1(2
)3)(1(
x
xx
DON’T multiply the simplified version of the product,
just leave it in factored form.
)1(
9*
2
x
x
multiplying allows you to put all the factors together
in one numerator and one denominator so you can simplify.
Your turn:
)2(6
)3(*
)4(
)2(*
)3(
)4(3
x
x
x
x
x
x
)4(
)16(*
)5(
)3( 2
x
x
x
x
Multiply the expressions
6
3
2
1
)4)(5(
)4)(4)(3(
xx
xxx
)5(
)4)(3(
x
xx
Your turn:
16
152*
9
122
2
2
2
x
xx
x
xx
Multiply the expressions
)4)(4(
)3)(5(*
)3)(3(
)4)(3(
xx
xx
xx
xx)4(
)5(
x
x
(Solutions on this slide)
Your turn:
242
127*
32
24822
2
2
2
xx
xx
xx
xx
Multiply the expressions
)4)(6(
)4)(3(*
)1)(3(
)124(2 2
xx
xx
xx
xx
(Solutions on this slide)
)4)(6(
)4)(3(*
)1)(3(
)124(2 2
xx
xx
xx
xx
)6)(1(
)124(2 2
xx
xx
)6)(1(
)2)(6(2
xx
xx
)1(
)2(2
x
x
Divide Rational Expressions
7
5
3
2 What do we do? Multiply by the reciprocal
5
7*
3
2
15
14
Divide Rational Expressions
mn
zx
mnp
xyz
5
4
5
12 2
Multiply by the reciprocal
zx
mn
mnp
xyz24
5*
5
1224
1*
12
xp
xy
xxp
xy
**4
1*
3*4
xp
y 1*
3
px
y3
bbb
pmm
ppm
ba
***2*2
***7*2*
***7
**3*2
Your turn:
pm
zb
mp
abz2
3
2 14
4
7
6
2
3
pb
am
zb
pm
mp
abz3
2
2 4
14*
7
6
)8)(2)(3(
)2)(3(
xxx
xx
Dividing Rational Expressions
?2
8
6
32
x
x
xx
x
8
2*
6
32
x
x
xx
x
simplify then multiply!
8
2*
)2)(3(
3
x
x
xx
x
)8(
1
xOR ? )8( x
Your turn:
6
8
6
322
xx
x
xx
x
)8(
)3(
x
x
8
6*
6
3 2
2
x
xx
xx
x
)8(
)6(*
)6(
)3( 2
2
x
xx
xx
x
Your turn:
149
152
124
3522
2
2
2
xx
xx
xx
xx
)3)(5(
)7)(2(*
)2)(6(
)5)(7(
xx
xx
xx
xx
)3)(6(
)7)(7(
xx
xx
152
149*
124
3522
2
2
2
xx
xx
xx
xx
Your Turn:
divide xx
xx
xx
xx
5
45
152
202
2
2
2
Divide
23
232
6
1428
yx
x
y
yx
Your Turn:
Multiply
Multiply
)5(2
)1(*
164
)5(4
x
xx
x
x
)2110(*152
103 2
2
2
xx
xx
xx
divide 255
42
54
22
2
x
x
xx
xx
Compound (or Complex) Fraction
52
31
5
2
3
1
2
5*
3
1
2*3
5*1
6
5
Simplifying a Compound Rational Function
1
2
x
x
1
4
x
x
Convert division
to multiplication
x
x
x
x
4
1*
1
2
2
1
4
2
remove common factors in
Numerator/denominator
Compound Rational Functions
4
3*
1
32
x
x
x
x
)4)(1(
)3)(32(
xx
xx
1
32
x
x
3
4
x
x
Convert Division into multiplication
=
Leave as is
Your turn:
x2
1
2
1
x
Simplify
Sum/difference in Numerator and Denominator
63
x
x
32
1. Combine the numerator fractions into one fraction.
x
32
= 1
6
3
x
x
32
= 3
18x
2. Combine the denominator fractions into one fraction.
x
x 32
3
18x
= 32
*3
18
x
xx
)32(3
)18(
x
xx
Your turn:
4
33
21
x
xx
Simplify the complex fraction.
Another Example: Electrical Circuits
1R
2R
321 RRRRtotal
3R
Voltage
Source
Parallel Resistors 321
1111
RRRR total
1R2R 3R
Voltage
Source
321
21
321
31
321
321
RRR
RR
RRR
RR
RRR
RR
RT
321
2131321
RRR
RRRRRR
RT
213132
321
RRRRRR
RRRRT
Parallel Resistors 321
1111
RRRR total
1R2R 3R
Voltage
Source
213132
321
RRRRRR
RRRRT
Electrical resistance is measured in “ohms.” Given the following resistances, find the total resistance of the circuit. 51R 102R 153R
72.2
10*515*515*10
15*10*5
People working together is similar to parallel resistance because they are working “in parallel”.
bat T
1
T
1
T
1The total time to complete a job is a function of the times
required to complete the job by each worker separately.
Alfonso takes 5 hours to paint a room. Bob takes 6 hours to paint the same room. How long does it take them to paint the room while working together?
6
1
5
1
T
1
t
30
5
30
6
T
1
t
30
11
T
1
t
hrs 7.211
30Tt
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