"
SECTION7
Rational Expressions
55
In Problems 29-34, factor the sum or difference of two cubes.
'\..
I.
29. x' - 27
30. x' + 125
33. 8x' + 27
34. 64 - 27x'
31. x' + 27
32. 27 - 8x'
41.
37. x' + 7x + 6
x' - lOx + 16
45. x' + 7x - 8
38. x' + 9x + 8
42. x' - 17x + 16
46.x'+2x-8
49. 2x' - 4x + x - 2
50. 3x'
55. 2z' + 5z + 3
59. 3x' - 2x - 8
63. 3x' + lOx - 8
56. 6z' + 5z + I
60. 3x' - lOx + 8
64. 3x' - lOx - 8
In Problems 35-46, factor eaeh polynomial.
35. x' + 5x + 6
39. x' + 7x + 10
43. x' - 7x - 8
'\..
In Problems
36. x' + 6x + 8
40. x' + llx + 10
44. x' - 2x - 8
47-52, factor by grouping.
47. 2x' + 4x + 3x + 6
51. 6x' + 9x + 4x + 6
'\..
48. 3x' - 3x + 2x - 2
52. 9x' - 6x + 2x - 3
+ 6x - x - 2
In Problems 53-64, factor each polynomial.
53. 3x' + 4x + 1
57. 3x' + 2x - 8
61. 3x' + 14x + 8
'\..
I.
54. 2x' + 3x + 1
58. 3x' + lOx + 8
62. 3x' - 14x + 8
In Problems 65-112, factor completely
'\..
'<o-
If the polynomial
cannot be factored, say it is prime.
65.
69.
73.
77.
81.
85.
x' - 36
66. x' - 9
67. 2 - 8x'
x' + 7x + 10
70. x' + 5x + 4
71. x' - lOx + 21
4x' - 8x + 32
74. 3x' - 12x + 15
75. x' + 4x + 16
15 + 2x - x'
78. 14 + 6x - x'
79. 3x' - 12x - 36
y' + Ill' + 30y'
82. 3y' - 18y' - 48y
83. 4x' + 12x + 9
6x' + 8x + 2 .
86. 8x' + 6x - 2
87. x' - 81
89. xl> - 2x3 + 1
90. x6 + 2x3 + 1
91. x 7 - x5
93. 16x' + 24x + 9
94. 9x' - 24x + 16
95. 5 + 16x - 16x'
97. 4y' - 16y + 15
98. 9y' + 9y - 4
99. 1 - 8x' - 9x'
101. x(x + 3) - 6(x + 3) 102. 5(3x - 7) + x(3x - 7) 103. (x + 2)' - 5(x + 2)
105. (3x - 2)3 - 27
106. (5x + I)' - 1
107. 3(x' + lOx + 25) - 4(x
108. 7(x' - 6x + 9) + 5(x - 3)
109. x3 + 2x' - x - 2
111. x4 - xJ + x-I
112. x4 + x3 + x + 1
",_113.
.•
each polynomial.
Show thatx2
+
114. Show that x2
4 is prime.
115. Make up a polynomial
+x +
68. 3 - 27x'
72. x' - 6x + 8
76. x' + 12x + 36
80. x' + 8x' - 20x
84. 9x' - 12x + 4
4
88. x - 1
R
5
92. x - x
96. 5 + lIx - 16x'
4
100. 4 - 14x' - 8x
104. (x - I)' - 2(x - 1)
+ 5)
110. x' - 3x' - x + 3
1 is prime .
that factors into a perfect square.
W 116. Explain to a fellow student
what you look for first when presented
with a factoring
problem. What do you do next?
m RATIONAL EXPRESSIONS
2
Reduce a Rational Expression to Lowest Terms
Multiply and Divide Rational Expressions
3
Add and Subtract Rational Expressions
~
Use the Least Common Multiple Method
5
Simplify Mixed Quotients
OBJECTIVES
If we form Ihe quotient of two polynomials, the result is called a rational
expression. Some examples of rational expressions are
3
(a) x + 1
x
(b) 3x' + x - 2
x' + 5
(c) --
x
x' -
1
(d)
xl
(x _ y)'
Expressions (a), (b), and (c) are rational expressions in one variable, x, where.
as (d) is a rational expression in two variables, x and y.
56
CHAPTER R
'-
Review
Rational expressions are described in the same manner as rational numbers. In expression (a), the polynomial xl + 1 is called the numerator, and x
is called the denominator. When the numerator and denominator of a rational expression contain no common factors (except 1and -1), we say that
the rational expression is reduced tu luwest terms, or simplified.
The polynomial in the denominator of a rational expression cannot be
equal to 0 because division by 0 is not defined. For example, for the expression
l
x +1
---,
x
1
x cannot take on the value O.The domam.
'{I x
e vana'b Ie x IS
X '"
0 }.
A rational expression is reduced to lowest terms by factoring completely the numerator and the denominator and canceling any common factors by
using the cancellation property:
a£
b£
___
0 fht
a
=
if b '" 0,
b
C '"
(1)
0
Reducing a Rational Expression to Lowest Terms
x2
x2
Reduce to lowest terms:
Sol uti 0 n
+ 4x + 4
+ 3x + 2
We begin by factoring the numerator
and the denominator.
x2 + 4x + 4
=
(x + 2)(x + 2)
x2 + 3x + 2
=
(x + 2)(x + 1)
Since a common factor, x + 2, appears, the original expression
est terms. To reduce it to lowest terms, we use the cancellation
x2
x2
+ 4x + 4
+ 3x + 2
~(x+2)
~(x+
x+2
1)
x
+ I
is not in lowproperty:
x ",-2,-1
-
WARNI~G: Apply the cancellation property only to rational expressions written in
factored form. Be sure to cancel only common factors!
•
~
Reducing Rational Expressions to Lowest Terms
Reduce each rational expression
8
Xl -
(a)
Solution
(a)
(b)
•
x3
3 -
x
8
-
2
2x
x2
=
-
X -
12
+ 2x + 4)
~(X2
X2~
2(4 - x)
8 - 2x
x2-x-12
8 - 2x
(b)
2x2
Xl -
to lowest terms.
(x-4)(x+3)
NOW
WORK
=
x2 + 2x + 4
x2
2(-1)¥-41
¥-41(x
PROBLEM
1.
+3)
x'" 0,2
-2
x+3
-
x '" -3,4
-'
"
MULTIPLYING AND DIVIDING
2
Rational Expressions
SECTION 7
57
RATIONAL EXPRESSIONS
The rules for multiplying and dividing rational expressions are the same as
the rules for multiplying and dividing rational numbers. If:!' and ~, b '" 0,
d '" 0, are two rational expressions, then
b
,-
a c
ac
-.-=b d
bd
if b '" 0, d '" 0
a
bad
-=-.-=c
b c
ad
(3)
if b '" 0, e '" 0, d '" 0
be
d
In using equations (2) and (3) with rational expressions, be sure first to factor each polynomial completely so that common factors can be canceled.
Leave your answer in factored form.
__
Multiplying and Dividing Rational Expressions
Perform the indicated operation and simplify the result. Leave your answer
in factored form.
(a)
Solution
(a)
x2-2x+l
x3 +
X
x2-2x+l
x3
+
X
x+3
x2 - 4
(b) 2
x - X - 12
x3 - 8
(x - 1)2
4(x2 + 1)
4x2 + 4
=
x2 + X - 2
x(x2 + 1) (x+2)(x-l)
4x2 + 4
x2 + X - 2
(x - 1)2(4)~
+ 2)\r--l}
x~(x
4(x - 1)
+ 2)'
x(x
x+3
x+3
x2 - 4
=--,
(b) 2
x - X - 12 x2 - 4
x3 - 8
x
-
=
I!!l!
~-
NOW
x3
x2 -
+
8
X
x'" -2,0,1
-
3
12
(x - 2)(x2
+ 2x +
(x-2)(x+2)
(x-4)(x+3)
~21(x2
+ 2x + 4)
~rcx
+ 2)(x - 4)~
x2+2x+4
x '" -3, -2, 2, 4
(x+2)(x-4)
WORK
PROBLEMS
13
AND
21.
4)
•
58
CHAPTER
R
Review
ADDING
3
AND SUBTRACfING
RATIONAL EXPRESSIONS
The rules for adding and subtracting rational expressions are the same as
the rules for adding and subtracting rational numbers. Thus, if the denominators of two rational expressions to be added (or subtracted) are equal, we
add (or subtract) the numerators and keep the common denominator.
If ~ and ~ are two rational expressions,
a
c
a+e
a
c
a-c
b
b
b
b
b
b
-+-=-___
then
if b
*0
Adding and Subtracting Rational Expressions
with Equal Denominators
Perform the indicated
in factored form.
Solution
operation
2x'-4
(a) 2x + 5
+ 2x +5
x+3
2x' - 4
(a) 2x + 5
x +3
(2x' - 4) + (x + 3)
+ 2x + 5 =
2x + 5
x
3x + 2
(b) x - 3 - x - 3
=
=
__
and simplify the result. Leave your answer
3x + 2
x-3
(b) _x __
x-3
x*3
2x' + x - I
(2x - l)(x + 1)
2x+5
2x+5
x - (3x + 2)
x - 3x - 2
x - 3
x - 3
-2(x + I)
x-3
-2x - 2
x-3
•
Adding Rational Expressions Whose Denominators Are
Additive Inverses of Each Other
Perform the indicated operation
in factored form.
2x
x-3
and simplify the result. Leave your answer
5
3-x
--+-Sol uti
0
n
x
*3
Notice that the denominators
of the two rational expressions are different.
However, the denominator of the second expression is just the additive inverse of the denominator of the first. That is,
3 - x
= -x + 3 = -I'
(x - 3)
= -(x - 3)
Then,
2x
x - 3
5
--+--=--+
3 - x
t
2x
x - 3
I
J - x ~ -(x
5
-(x - 3)
- J)
2x
x - 3
---+--
t
a I
-b =
-5
x - 3
-0
b
2x + (-5)
x-3
NOW
WORK
PROBL£MS
33
AND
39.
2x x-3.
5
SECftON 7
Rational
Expressions
59
If the denominators of two rational expressions to be added or subtracted
are not equal, we can use the general formulas for adding and subtracting
quotients.
ad +bc
bd
ad - b,
bd
a c
a.d
b.c
-+-=--+--=
b d b.d
b.d
b.c
a.d
a c
---=-----=
b.d
b d b.d
EXAMPLE
Adding and Subtracting
Unequal Denominators
6
Solution
x
x-2
--+--
(a)
--+--=--.--+--.-x+4
x-3
x-21x+4
=
(b) --
x'
x' - 4
1
- -
x
= --.
r
x'
1
x' - 4
x
x+4
x
x-2
x+4
x-2
(x - 3)(x - 2)
(x
x' - 4
+ 4)(x - 2)
x
x'-4
X
x' - 4
- = ----X
1
(5b)
NOW
WORK
2x'-x+6
(x+4)(x-2)
- - --.
x3-x'+4
(x - 2)(x
x'" -2,0,2
+ (x + 4)(x)
x' - 5x +6 +x' +4x
(x + 4)(x - 2)
x'
(511)
x-2
(Sa)
=
b '"n. d '"n
(b) ----
x '" -4,2
x
if
and simplify the result. Leave your answer
(a)
x-3
(Sa)
Rational Expressions with
Perform the indicated operation
in factored form.
x - 3
x+4
O,d '"0
ifb '"
(x' - 4)(x)
-
+ 2)(x)
PROBL£M
x'(x)-(x'-4)(I)
43.
LEAST COMMON MULTIPLE (LCM)
4
If the denominators of two rational expressions to be added (or subtracted)
have common factors, we usually do not use the general rules given by equations (Sa) and (5b). Just as with fractions, we apply the least common multi.
pie (LCM) method. The LCM method uses the polynomial of least degree
that has each denominator polynomial as a factor.
60
CHAPTER R
Review
THE
LCM
SUBTRACTING
METHOD
FOR
RATIONAL
The Least Common
ADDING
OR
EXPRESSIONS
Multiple (LCM) Method requires four steps:
STEP
1:
Factor completely the polynomial
rational expression.
STEP
2:
The LCM of the denominator is the product of each of these
factors raised to a power equal to the greatest number of times
that the factor occurs in the polynomials.
STEP
3:
Write each rational
expression
in the denominator
of each
using the LCM as the com-
mon denominator.
STEP
4:
Add or subtract the rational expressions
using equation
(4).
Let's work an example that only requires Steps I and 2.
Finding the Least Common Multiple
____
Find the least common multiple of the following pair of polynomials:
x(x - 1)'(x
Sol uti
0
n
STEP
1: The polynomials
and
4(x - 1)(x
+ 1)3
are already factored completely
x(x - I)'(x
STEP
+ 1)
+ 1)
and
4(x - l)(x
2: Start by writing the factors of the left-hand
could start with the one on the right.)
x(x - 1)'(x
- 1)'(x
+ 1)3
polynomial.
(Or you
+ 1)
Now look at the right-hand polynomial.
not appear in our list, so we insert it.
4x(x
as
Its first factor, 4, does
+ 1)
The next factor, x-I,
is already in our list, so no change is necessary. The final factor is (x + 1 )3. Since our list has x + 1 to the first
power only, we replace x + 1 in the list by (x + I )3. The LCM is
4x(x
- 1)'(x
+ I)'
•
Notice that the LCM is, in fact, the polynomial of least degree that containsx(x - 1)'(x + 1) and4(x - I)(x + 1)3asfactors.
NOW
WORK
PROBLIM
49.
......__
Using the Least Common Multiple to Add Rational
Expressions
Perform the indicated
in factored form.
operation
and simplify the result. Leave your answer
x
2x - 3
I
-----+--- 2
x' + 3x + 2
x
x'" -2,-1, 1
Rational Expressions
SECTION 7
Sol uti 0 n
STEP
1: Factor completely the polynomials in the denominators.
+
x2
+2=
3x
x2 STEP
61
1
+
(x
2)(x
+ 1)
l)(x
+ 1)
= (x -
2: The LCM is (x + 2)(x + 1)(x - 1). Do you see why?
STEP3: Write each rational expression using the LCM as the denominator.
x
X
x2
+ 3x + 2
(x
= ------ x
. --x-I
+ 1) (x + 2)(x + 1) x-I
i
+ 2)(x
x(x - 1)
=
(x
+ 2)(x +
1)(x - 1)
Multiply numerator and
x -
denominator by
1 to get the
LCM in the denominator.
2
x
-
1
r
2x - 3
2x - 3
+ 1)
(x - l)(x
2x - 3
+ 1) .
(x - 1)(x
x
+2
x
+2
(2x - 3)(x
= (x -
+
1)(x
+ 2)
l)(x
+ 2)
Multiply numerator and
+
denominator by x
2 to get
the lCM in the denominator.
STEP
x
x2
+
3x
4: Now we can add hy using equation (4).
+2x-3=
+2
x2
-
+ 2)(x +
(x
=
(x2
x)
-
(x
+
(x
1)(x - 1)
2
+ (2x +
+
2)(x
3x2
_
+
x(x-l)
1
X
(2x-3)(x+2)
(x
+
+
l)(x
-
2)
+ 2)(x +
l)(x
- 1)
6)
-
1)(x - 1)
3(x2
6
-
2)(x
+ 2)(x + l)(x - 1)
(x
- 1)
•
Using the Least Common Multiple to Subtract
Rational Expressions
Perform the indica led operations and simplify the result. Leave your answer
in factored form.
x + 4
3
---2
x
Sol uti 0 n
STEP
+
+
x2
X
x2
STEP
x
1
* -1,0
1: Factor completely the polynomials in the denominators.
x2
STEP
+
2x
+
2x
+x
+1
= x(x
= (x
+ 1)
+ 1)2
2: The LCM is x(x + If
3: Write each rational expression using the LCM as the denominator.
3
x2
+
3
X
x
+
2x
+
1)
x+4
x+4
2
x(x
+
1
(x
+
=
3
+
x(x
x + 1
.--=
1) x + 1
x
x+4
1)'
(x
+
.-
1)2
X
=
x(x
x(x
3(x
x(x
+ 4)
+ 1)2
+ 1)
+ 1)2
62
CHAPTER R
Review
STEP 4: ,Subtract, using equation
3
x
---------
x'+x
(4).
+4
+ I)
x(x + 1)'
x(x + 4)
x(x + 1)'
+
+ 4)
3(x
x'+2x+l
3( x
1) - x( x
x(x
3x
+ I)'
+ 3 - x' - 4x
x(x
+ I)'
-x' - x + 3
=----x(x + I)'
NOW
WORK
PROBLEM
59.
-
MIXED QUOTIENTS
5
When sums and/or differences of rational expressions appear as the numerator and/or denominator of a quotient, the quotient is called a mixed quotient,' For example,
x'
1
+x
1
and
x
---3
x' - 4
x-3
---I
x+2
are mixed quotients. To simplif)' a mixed quotient means to write it as a rational exprcssion reduced to lowest terms. This can be accomplished in either
of two ways.
SIMPLIFYING
A
MIXED
QUOTIENT
METHOD
1: Treat the numerator and denominator of the mixed quotient separately, performing whatevcr operations are indicated and simplifying
the results. Follow this by
simplifying the resulting rational expression.
METHOD
2: Find thc LCM of the denominators of all rational expressions that appear in the mixed quotient. Multiply the numerator and denominator
of the mixcd quotient by the
LCM and simplify the result.
We will use both methods in the next example. By carefully studying
each method, you can discover situations in which one method may be easier to use than the other.
• Some texts use the term complex rr~ction.
Rational
SECfION 7
__
Expressions
63
Simplifying a Mixed Quotient
1
3
2
x
x +3
-+Simplify:
x'"
-3,0
4
So I uti 0 n
METHOD
1: First, we perform the indicated operation in the numerator, and
then we divide.
1
3
x
-+2
i
x+3
4
l'x+2.3
2.x
x+6
2x
x;3i
X;3
Rule for adding quotients
r
2x
x+3
2' .2. (x + 6)
2' . x . (x + 3)
+ 3)
Rule for multiplying
METHOD
4
Rule for dividing quotients
(x+6).4
2. x . (x
x+6
=--=--.--
2(x
+ 6)
x(x + 3)
quotients
2: The rational expressions that appear in the mixed quotient are
1
2'
3
x
x+3
4
The LCM of their denominators is 4x. We multiply the
numerator and denominator of the mixed quotient by 4x and
then simplify.
1
1
3
4x' - + 4x'2
x
4x.(x+3)
3
x
-+2
4
Distributive
Multiply the numerator
and denominator by 4x
1
3
'I< . 2x . - + 4.x' . 'I<
.x'
= --------
'4x . ~
+ 3)
property in numerator
i i
-
+ 12
2(x + 6)
x(x + 3)
x(x + 3)
2x
Simplify
__
Simplifying a Mixed Quotient
x'
--+2
Simplify:
x-4
2x - 2
-
x
-\
Factor
_
64
CHAPTER R
Review
Sol uti 0 n
We will use Method
1.
x'
x'
2(x - 4)
x-4
x-4
2x - 2
x
x
x
x'+2x-8
--+----
--+2
x-4
=
2x - 2
----
1
x
x-4
2x - 2 - x
x
(x+4)(x-2)
x-4
x-2
x
(x
+ 4)¥--Zt
x-4
x
-<--2
.---
(x+4).x
x-4
W;I
-..
NOW
WORK
•
69.
PROBLEM
Solving an Application in Electricity
An electrical circuit contains two resistors connected in parallel, as shown in
Figure 18. If the resistance of each is RJ and R, ohms, respectively, their combined resistance R is given by the formula
Figure 18
1
R=--1
1
-+RJ
R,
Express R as a rational expression; that is, simplify the right-hand side of this
formula. Evaluate the rational expression if RJ = 6 ohms and R, = 10 ohms.
Solution
We will use Method 2. If we consider 1 as the fraction.!:., then the rational expressions in the mixed quotient are
1
1
l'
RJ'
R,
The LCM of the denominators is RJR,. We multiply the numerator
nominator of the mixed quotient by R J R, and simplify.
and de-
1
1
RJ
1
R,
-+Thus,
R=
If RJ
=
6 and R,
=
10, then
6. 10
R
=
10
+
6
60
=
15
16 = '4
ohms
•
SECfION 7
7
1-12, reduce each rational expression
3x + 9
I. -,-x -9
+
+
Xl
6.
12x' - 6x
4x - 5
x Xl
J 3-30, perform
3x + 6
5x1
4x - 8
17.--'---3x
Xl -
4
y' - 25
4
7.
+
2y
6x - 27
2
5x
4x - 18
16
20.
Xl
23.x'-1
lOx
-x + 1
xl
Xl
Xl -
4x
+
x' -
+ 7x + 12
-4 + x
25.
4
x2+7x+6
28.
YOllr
x'
x+l
x-3
2x-3
x-3
35.
3x+52x-4
37.----2x-12x-l
5x - 4
x + 1
38.----3x+43x+4
40. _6
x-I
41. -----
4
39,
4
x-2
x
2-x
--+--
4
2
x+2
42, ----x+5
x-5
x+3
x-3
45. ----x+2
x+4
x-2
x-I
x)
x
x2+1
2x-32x+l
46.----x-I
x+l
47.
x-4
x
--+Xl
4
-
3
- --2x-3
x-I
44.~+~
x-I
1
2x-5
x+4
36. --+--3x + 2
3x + 2
--+--
x
2x-3
43.--+--
2
2
answer in factored form.
33. -2x-3
3x'
9
34.----2x-l
2x-l
x+l
9x' + 3x 12x' + 5x 30.
,
9x - 6x +
8x' - lOx -
2x' - x - 28
3x'-x-2
29.
4x' + 16x + 7
3x' + llx + 6
+X - 6
x' + 5x - 6
Xl + 5x + 6
Xl
9
9x'
3
6
32. - - x
x
x_
I-x
x' - I
4x - 2
16
x -
4x
x' - 16
In Problems 31-48, perform the indicated operations and simplifY the result. Leave
t
5
31. '-2+ -2
X
3 + x
3 - x
26. -,--
4 - x
12x
x' - 7x + 12
27.
Xl + X - 12
Xl - X - 12
-
12x
x - 2
-4x
24.
+ 5x - 3
1 - 2x
5x + 20
22. --4x'
-
-2x + 4
8x
3y + 5y + 2
x' - 3x - 10 x' + 4x - 21
19. x' + 2x - 35 . x' + 9x + 14
4
21. 3x _ 9
Xl
15
3y' - Y - 2
,
16._12_.
6x
+X - 6
x' - 25
+ 4x - 5 x' + 2x -
Xl
8.
12. 2x'
x - 4
2x
4x'
x' _
15.
18. ---
12 - 6x
15x' + 24x
3x'
and simplify the result. Leave your answer in factored form.
6x + 10
12
- 8y - 10
It. -----
2
x'
3
2x
14. -.---
,
x' + 5x - 14
2 - x
Xl
X -
the indicated operation
x
13 --.--
+
4x
4.
2x
Xl -
3. 3x - 6
x' - 16
10.
x2-2x+l
In Problems
•
65
to lowest terms.
4x' + 8x
2. 12x+ 24
24x2
Xl
9.
Expressions
EXERCISES
In Problems
5.
Rational
2
1
48
x
.
5
--+--
In Problems 49-56.jind ,he LCM of'he given polynomials.
49. x' - 4, x' - x - 2
50. x' - x - 12. x' - 8x + 16
52. 3x' - 27,
53. 4x3
2x' - x - 15
-
4x2 +
X,
2x3
-
Xl,
56, x' + 4x + 4, x' + 2x',
x]
(x + 2)'
54. x - 3,
Xl
+ 3x, x3
-
9x
66
CHAPTER R
Review
In Problems 57-68, perform the indicated operations and simplify the result. Leave
x
57.
-----
x
x' - 7x + 6
x' - 2x - 24
4x
2
59. --- ---x2-4
x2+x-6
+
3
61.,
(x-I)(x+l)
x+4
x -x-2
65
----+--2
3
-,----
2
x +x
3
.!-)
x+h
2x - 3
64. x' + 8x + 7
x - 2
(x + 1)'
x
x+l
x' - x'
66.
X _X2
67. .!- (_1__
h
,
2x+3
x +2x-8
,
1
.x
2
(x-l)(x+l)
63.
answer in factored form.
YOllr
x+l
58.
- --,---x - 3
x + 5x - 24
3x
x - 4
60. -- ----x-I
xl-2x+l
2
6
62.,
,
(x + 2) (x - 1)
(x + 2)(x - I)
x
68.
x
2
(x - I)'
H(x ~
+ - - --X
h)'
:,]
In Problems 69-78, perform the indicated operations and simplifY the result. Leave your answer in factored form.
1
1
1
x
1+4+x-1---
x
69.--I
1-x
73.
Xl
x+4
x-3
----x-2
x+1
x +
77. 1 ---
Xl
X
74. ------
x +3
78.1----1
1- x
x
79. The Lensmaker's Equation The focal length
with index of refraction n is
1
- =
f
[1
(n - 1) R,
f of a lens
80. Electrical Circuits
An electrical circuit contains three
resistors connected in parallel. If the resistance of each is
RI, R2• and RJ ohms. respectively, their combined resistance R is given by the formula
1]
+-
R,
,~
81. The following
expressions
1
I
1
I
R
R,
R,
R,
-=-+-+-
where R1 and Rz are the radii of curvature of the front
and back surfaces of the lens. Express f as a rational expression. Evaluate the rational expression for n = 1.5,
Ii, = 0.1 meter. and Ii, = 0.2 meter.
'iii
X
x
2x + 5
-----x
3
x
76.
(x + I)'
x'
--x - 3
x +3
x-I
x-2
--+-x+2
x+l
75.
2x - 3
X
-----x +1
x
x - 2
x
----x+1
x-2
1
1 --
71.
1
x+-
I
X + 1
72.
1
x2---
X
70.-1
3--
Express R as a rational expression. Evaluate
Ii, = 5 ohms. R, = 4 ohms. and R, = 10 ohms.
R for
are called continued fractions:
1
1
+x. 1+-- 1 • 1 +----I
1++-1
x
+x
1
+-----1 +---1
+-1+.!x
Each simplifies to an expression of the form
ax + b
bx
+
c
Trace the successive values of a, b. and c as you "continue" the fraction. Can you discover the patterns that these values follow? Go to the library and research Fibonacci numbers. Write a report on your findings.
82. Explain to a fellow student when you would use the LCM
method to add two rational expressions. Give two examples of adding two rational expressions. one in which you
use the LCM and the other in which you do not.
83. Which of the two methods given in the text for simplifying mixed quotients do you prefer? Write a brief paragraph stating the reasons for your choice.