intermediate algebra exam 2 material rational expressions
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Intermediate Algebra
Exam 2 Material
Rational Expressions
Rational Expression
• A ratio of two polynomials where the denominator is not zero (an “ugly fraction” with a variable in a denominator)
• Example:
• Will the value of the denominator ever be zero?If x = - 3, then the denominator becomes 0, so we say that – 3 is a restricted value of x
• What is the “domain” of the rational expression (all acceptable values of the variable)?Domain is the set of all real numbers except - 3.Domain: {x | x ≠ -3}
3
22
x
xx
Finding Restricted Values and Domains of Rational Expressions
• Completely factor the denominator
• Make equations by setting each factor of the denominator equal to zero
• Solve the equations to find restricted values
• The domain will be all real numbers that are not restricted
Example
Find the domain:
• Factor the denominator:(x – 2)(x + 2)
• Set each factor equal to zero and solve the equations:x – 2 = 0 and x + 2 = 0x = 2 and x = -2 (Restricted Values)
• Domain:{x | x ≠ -2, x ≠ 2}
4
532
x
x
Evaluating Rational Expressions
• To “evaluate” a rational expression means to find its “value” when variables are replaced by specific “unrestricted” numbers inside parentheses
• Example: 2for 32
32 Evaluate
2 -x
xx
x
32
322 22
2
344
34
3
7
3
7
Fundamental Principle of Fractions
• If the numerator and denominator of a fraction contain a common factor, that factor may be divided out to reduce the fraction to lowest terms:
ac
ab
c
b
18
12
332
322
3
2
place.each in left is "1" out, divided are factorscommon When
x5
5
x
1
1
1
1
1
1
1
1
1
Reducing Rational Expressions to Lowest Terms
• Completely factor both numerator and denominator
• Apply the fundamental principle of fractions: divide out common factors that are found in both the numerator and the denominator
Example of Reducing Rational Expressions to Lowest Terms
• Reduce to lowest terms:
• Factor top and bottom:
• Divide out common factors to get:
63
243 3
x
x
23
83 3
x
x
23
4223 2
x
xxx
422 xx
1
11
1
Example of Reducing Rational Expressions to Lowest Terms
• Reduce to lowest terms:
• Factor top and bottom:
• Divide out common factors to get:
x
x
3
3
31
31
x
x
31
31
x
x
1
3
3
x
x
1
Equivalent Formsof Rational Expressions
• All of the following are equivalent:
• In words this would say that a negative factor in the numerator or denominator can be moved, respectively, to a negative factor in the denominator or numerator, or can be moved to the front of the fraction, or vice versa
q
p
q
p
q
p
Example of Using Equivalent Forms of Rational Expressions
• Write equivalent forms of:
2
5
x
x
2
5
x
x
2
5
x
x
2
5
x
x
2
5
x
x
2
5
x
x
Homework Problems
• Section: 6.1
• Page: 401
• Problems: Odd: 3 – 9, 13 – 23, 27 – 63, 67 – 73
• MyMathLab Homework Assignment 6.1 for practice
• MyMathLab Quiz 6.1 for grade
Multiplying Rational Expressions (Same as Multiplying Fractions)
• Factor each numerator and denominator• Divide out common factors • Write answer (leave polynomials in factored
form)• Example:
28
15
9
41 1
1
1
1121
5
722
53
33
22
Example of MultiplyingRational Expressions
Completely factor each top and bottom:
Divide out common factors:
43
23
8143
8232
2
x
x
xx
xx
43
23
423
243
x
x
xx
xx
1
1
1
1
4
2
x
x
Dividing Rational Expressions(Same as Dividing Fractions)
• Invert the divisor and change problem to multiplication
• Example:
d
c
b
a
c
d
b
a
4
3
3
2
bc
ad
3
4
3
2
9
8
Example of DividingRational Expressions
27
48
9
2 352 yyy
35
2
48
27
9
2
yy
y
124
27
9
223
2
yy
y
122
32 yy
3
21
1 1
y
Homework Problems
• Section: 6.2
• Page: 408
• Problems: Odd: 3 – 25, 29 – 61
• MyMathLab Homework Assignment 6.2 for practice
• MyMathLab Quiz 6.2 for grade
Finding the Least Common Denominator, LCD, of Rational
Expressions
• Completely factor each denominator
• Construct the LCD by writing down each factor the maximum number of times it is found in any denominator
Example of Finding the LCD
• Given three denominators, find the LCD:, ,
• Factor each denominator:
• Construct LCD by writing each factor the maximum number of times it’s found in any denominator:
123 2 x
123 2x
16164 2 xx
16164 2 xx
126 x
126x
43 2x 223 xx
26 x
444 2 xx 2222 xx
232 x
LCD
LCD
222322 xxx
2212 2 xx
Equivalent Fractions
• The fundamental principle of fractions, mentioned earlier, says:
• In words, this says that when numerator and denominator of a fraction are multiplied by the same factor, the result is equivalent to the original fraction
.
ac
ab
c
b
18
12
36
263
2
Writing Equivalent Fractions With Specified Denominator
• Given a fraction and a desired denominator for an equivalent fraction that is a multiple of the original denominator, write an equivalent fraction by multiplying both the numerator and denominator of the original fraction by all factors of the desired denominator not found in the original denominator
• To accomplish this goal, it is usually best to completely factor both the original denominator and the desired denominator
24
20
Example
Write an equivalent fraction to the given fraction that has a denominator of 24:
622
522
6
5
6
524
? 326 322224
:rdenominatoeach Factor
Example
Write an equivalent rational expression to the given one that has a denominator of :
yyyyy
y
242
?2232
12 yyyy
112242 23 yyyyyy
yyy 242 23
1
222 yy
y
yy
y
121
122
yyy
yy
121
22 2
yyy
yy
yyy
yy
242
42223
2
:rdenominatoeach Factor
Homework Problems
• Section: 6.3
• Page: 414
• Problems: Odd: 5 – 43, 51 – 69
• MyMathLab Homework Assignment 6.3 for practice
• MyMathLab Quiz 6.3 for grade
Adding and Subtracting Rational Expressions (Same as Fractions)• Find a least common denominator, LCD,
for the rational expressions• Write each fraction as an equivalent
fraction having the LCD• Write the answer by adding or
subtracting numerators as indicated, and keeping the LCD
• If possible, reduce the answer to lowest terms
Example
• Find a least common denominator, LCD, for the rational expressions:
• Write each fraction as an equivalent fraction having the LCD:
• Write the answer by adding or subtracting numerators as indicated, and keeping the LCD:
• If possible, reduce the answer to lowest terms
yyy
y
yy
y 1
242
3222
y
1yy 112 yy 112 yyy
LCD
112
1112
112
3
112
122
yyy
yy
yyy
yy
yyy
yy
yyy
y
yy
y 1
112
3
1
2
112
122322 222
yyy
yyyyy
112
2423422 222
yyy
yyyyy
112
222
yyy
yy reduce!t on'fraction w factor,t won' topSince
Homework Problems
• Section: 6.4
• Page: 422
• Problems: Odd: 9 – 21, 25 – 47, 51 – 71
• MyMathLab Homework Assignment 6.4 for practice
• MyMathLab Quiz 6.4 for grade
Complex Fraction
• A “fraction” that contains a rational expression in its numerator, or in its denominator, or both
• Example:
• Think of it as “fractions inside of a fraction”• Every complex fraction can be simplified to a
rational expression (ratio of two polynomials)
y
x
65
231
Two Methods for Simplifying Complex Fractions
• Method One– Do math on top to get a single fraction– Do math on bottom to get a single fraction– Divide top fraction by bottom fraction
• Method Two (Usually preferred)– Find the LCD of all of the “little fractions”– Multiply the complex fraction by “1” where “1”
is the LCD of the little fractions over itself
Method One Example ofSimplifying a Complex Fraction
• Do math on top to get single fraction:
• Do math on bottom to get single fraction:
• Top fraction divided by bottom:
y
x
65
231
x
xyy
5
122
23
1
x
1
2
3
1
x
x
x
x 3
6
3
1
x
x
3
61
y6
5 :fraction singlealready is bottom case, In this
yx
x
6
5
3
61
5
6
3
61 y
x
x2
Method Two Example ofSimplifying a Complex Fraction
• Find the LCD of all of the “little fractions”:
• Multiply the complex fraction by “1” where “1” is the LCD of the little fractions over itself
y
x
65
231
xy6
y
x
65
231
x
xyy
5
122
yxy
xyxxy
630
112
36
161
6
xy
xy
Homework Problems
• Section: 6.5
• Page: 431
• Problems: Odd: 7 – 35
• MyMathLab Homework Assignment 6.5 for practice
• MyMathLab Quiz 6.5 for grade
Other Types of Equations
• Thus far techniques have been discussed for solving all linear and some quadratic equations
• Now address techniques for identifying and solving “rational equations”
Rational Equations
• Technical Definition: An equation that contains a rational expression
• Practical Definition: An equation that has a variable in a denominator
• Example:
3
2
1
5
32
12
xxxx
Solving Rational Equations
1. Find “restricted values” for the equation by setting every denominator that contains a variable equal to zero and solving
2. Find the LCD of all the fractions and multiply both sides of equation by the LCD to eliminate fractions
3. Solve the resulting equation to find apparent solutions
4. Solutions are all apparent solutions that are not restricted
Example
3
2
1
5
32
12
xxxx
RV
01x 03 x
0322 xx
031 xx
01x
03 x
OR1x 3x
SolvedAlready
SolvedAlready 3
2
1
5
31
1
xxxx
LCD 31 xx 1
LCD
3
2
1
5
31
1
xxxx
12351 xx221551 xx
1731 x
x316
3
16x RV!Not
Example
1
1
2
1
1
22
mm
RV
01m 01m
012 m
011 mm
01m
OR1m 1m
SolvedAlready 1
1
2
1
11
2
mmm
LCD 112 mm
12114 mmm
11
1
2
1
11
2 LCD
mmm
2214 2 mm2214 2 mm
320 2 mm
130 mm
01 03 morm1 3 morm
Formula
• Any equation containing more than one variable
• To solve a formula for a specific variable we must use appropriate techniques to isolate that variable on one side of the equal sign
• The technique we use in solving depends on the type of equation for the variable for which we are solving
Example of Different Types of Equations for the Same Formula
• Consider the formula:
• What type of equation for A?Linear (variable to first power)
• What type of equation for B?Quadratic (variable to second power)
• What type of equation for C?Rational (variable in denominator)
1
43
2
C
BA
Solving Formulas Involving Rational Equations
• Use the steps previously discussed for solving rational equations:
1. Find “restricted values” for the equation by setting every denominator that contains the variable being solved for equal to zero and solving
2. Find the LCD of all the fractions and multiply both sides of equation by the LCD to eliminate fractions
3. Solve the resulting equation to find apparent solutions
4. Solutions are all apparent solutions that are not restricted
Solve the Formula for C:
Since the formula is rational for C, find RV:
Multiply both sides by LCD:
1
43
2
C
BA
01C 1C
1C
11
431
2
CC
BAC
2433 BACAC
Example Continued
Solve resulting equation and check apparent answer with RV:
2433 BACAC 343 2 ABCAC
343 2 ABCA
3
34
3
3 2
A
AB
A
CA
3
34 2
A
ABC RVNot
Cfor linear Now
Homework Problems
• Section: 6.6
• Page: 439
• Problems: Odd: 17 – 69, 73 – 87
• MyMathLab Homework Assignment 6.6 for practice
• ( No MyMathLab Quiz until we finish Section 6.7 )
Applications of Rational Expressions
• Word problems that translate to rational expressions are handled the same as all other word problems
• On the next slide we give an example of such a problem
Example
When three more than a number is divided by twice the number, the result is the same as the original number. Find all numbers that satisfy these conditions.
.
:Unknownsnumber The x
xx
x
2
3
:RV02 x0x
xxx
xx 2
2
32
223 xx
320 2 xx
1320 xx
01or 032 xx
1or 32 xx
1or 2
3 xx
Homework Problems
• Section: 6.7
• Page: 449
• Problems: Odd: 3 –9
• MyMathLab Homework Assignment 6.7 for practice
• MyMathLab Quiz 6.6 - 6.7 for grade