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NAME
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Simplifying Rational
Expressions (Pages 660–665)
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A rational expression is an algebraic fraction whose numerator and
denominator are polynomials. Any values of the variable that result in a
denominator of zero must be excluded from the domain of the variable. These
are called excluded values of the rational expression. To simplify a rational
expression, eliminate (by dividing) any common factors of the numerator and
denominator using the GCF.
EXAMPLE
b3
Simplify and state the excluded values of b.
2
b 2b 3
b3
b2 2b 3
b3
Factor the denominator.
(b 3)(b 1)
b30
b3
b10
b 1
and
Exclude the values for which b 3 0 and b 1 0.
Therefore, b cannot equal 3 or 1.
b3
(b 3)(b 1)
1
b3
(b 3)(b 1)
1
1
, b 1,
b1
Simplify the fraction by dividing by the GCF, b 3.
3
Try These Together
Simplify and state the excluded values of the variables.
x2 3x 2
x 4x 5
7a3
14a
1. 2. 2
HINT: Find the exclude values before you
simplify the expression.
HINT: Factor both the numerator and the
denominator.
PRACTICE
Simplify and state the excluded values of the variables.
6x2y
3. 9x4y2z
4. 6
30x
x y
y 5y 6
2x2 98
15. Standardized Test Practice Simplify the rational expression .
8x 56
A 4(x 7)
49
C x7
D x2
B 4(x 7)
x7
4
y1
x2
2x
x1
4
a2
2
x5
5
3
a3
9yz
xy
a2
x2
1
8
Answers: 1. , a 0 2. , x 1, 5 3. , x 0 4. x 0, y 0 5. x, x 0, y 0, z 0 6. , a 0, 3
x2
C
B
A
6x 24
4x
10 5x
b2
3
1
7. , x 2 8. , x 0 9. x 5, x 5 10. , b 2 11. , x 1, 1 12. , a 7, 2
8.
y2 7y 6
14. 2
y1
C
A
7.
x2 6x 8
13. 6
C
B
B
6.
a 9a 14
4b 8
x2
13. , x 4 14. , y 6, 1 15. D
B
A
5.
b2 4
10. x5
a7
12. 2
x 1
4.
25
9. 2x
3x 3
11. 2
a 3a
x2
10x
8. 2
3x 6
8a
6. 2
60x yz
5x2
12x
7. 3.
20xyz3
5. 2 3
© Glencoe/McGraw-Hill
91
CA Parent and Student Study Guide, Algebra 1
NAME
DATE
Multiplying Rational
Expressions (Pages 667–670)
12-2
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To multiply rational expressions, you can divide by the common factors
either before or after you multiply the expressions. From this point on, you
may assume that no denominator has a value of 0.
EXAMPLE
1
2x2(3x 2)
Multiply .
2
3x x 2
2x2(3x 2)
3x2 x 2
1
4x
4x
2x2(3x 2)
(3x 2)(x 1)
1
Factor the denominator.
4x
1
1 x
1
2x2(3x 2)
Divide by the GCF of 2x(3x 2) before multiplying.
x
x
or Multiply. Then, simplify the denominator.
(3x 2)(x 1)
1
2(x 1)
4x
2 1
2x 2
Try These Together
ab2
12
6
b
4
x 64
1. Multiply .
2. Multiply (x 8) .
2
x8
HINT: Write x 8 as .
HINT: Divide both numerator and denominator by
the same quantity—their greatest common factor.
1
PRACTICE
Find each product. Assume that no denominator has a value of 0.
15a
2b4
3. 3
3x 4 yz2
4
4. 2
ab
5. 16abc 2
25mn2
10n3
6. 7
7. (2x 8) a2
12(a 1)
8. b
3
4n
24y
6x2
2x 10
11. 2
x2 16
x
12. 4x 2
6
13. 2
x2
x2 2x 15
14. 2
36
3x 12
2x 2
16. 2
6x 9
x 4
17. 2
x 2x
x4
x
y2
y4
15. 2
y3
x x2
x 4x
x5
3x2
6x 24
2
x x2
6x 12
x 14x 49
x7
18. Standardized Test Practice Multiply .
2
2
x7
x8
4
2. y3
x x2
x1
16. 2
2y
x 3z 2
3. 10ab 4. c
16a b
5. 2
2
x3
17. 2
25n
6. 4
18. C
5x
2
7. 14 8. 4a 9. 92
y6
x5
4x
10. 3x2 7x 6 11. x4
x 16
12. 2
© Glencoe/McGraw-Hill
D x2
C 1
x7
2
1
B A x7
ab
Answers: 1. x 49
15. C
B
A
2x 7x 3
x 25
2
8.
2x 6
3x
x4
A
7.
3
x 3x
14. C
B
B
6.
x3
x 6x 9
C
A
5.
y 2y 24
x2
6
13. 2
B
4.
a1
3a
9
9x 6
10. 5
3.
bc
x4
5m
x2
2
9. 2
x
CA Parent and Student Study Guide, Algebra 1
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Dividing Rational Expressions
(Pages 671–674)
To divide algebraic rational expressions, multiply by the reciprocal of the
divisor (the second fraction).
EXAMPLE
x2
x2 4
Find .
x2
5x
x2 4
5x
x2
x2
x2
x2
The reciprocal of is .
x2 4
x2
x2
5x
1
(x 2)(x 2)
5x
x2
x2
x2 1
Factor. Then divide by the common factor x 2.
(x 2)(x 2)
x2 4x 4
or 5x
x2
5x
Multiply.
Try These Together
5m2
10
3a 15
a4
3m5
12
1. Find .
2. Find (a 5).
HINT: First rewrite, multiplying by the reciprocal of
the second fraction. Then divide by the greatest
common factor.
1
HINT: The reciprocal of a 5 is .
a5
PRACTICE
Find each quotient. Assume that no denominator has a value of 0.
8x
4xy
3. 2
2abc
4. 10bc2 x5
x5
5. x8
x2
6. 4x2 4
x3 x
7. b2 25
8. (b 5)
3yz
3yz
x3
x2
n2 1
n1
9. x1
4
13. 2
x3
x2 x 6
14. 2
n2 9
n3
15. 2
x1
x3
16. 2
4m
m 2m
17. 2
x 8x 7
k3
2x 14
x 2x 1
2x
k3
4x 8x
2
x1
m6
m 4m 12
x1
x2 6x 5
18. Standardized Test Practice Find the quotient .
a
40bc
4. x3
x8
5. 4 6. 4
b5
7. 2 8. 93
2
yz
2
1
9. n2 1 10. 5b3 11. k 5 12. 8 13. © Glencoe/McGraw-Hill
x5
D 5)
2
3. C
a4
B 2(x 5)
3
2. A
4
1
(x
2
m
2
2
x5
2
Answers: 1. 3
C
B
A
n 7n 12
5b 15
17. 4 18. A
C
B
8.
y 4
b3
x3
C
B
A
7.
2
1
14. 2x2 8 15. n2 7n 12 16. B
A
6.
5
4
8
y2
12. 2
n3
5.
x
32
2
2k 10
11. 3n 3
y2
4.
2
8
4b
4b
10. 3
3.
8b
CA Parent and Student Study Guide, Algebra 1
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Dividing Polynomials (Pages 675–680)
To divide a polynomial by a monomial, divide each term of the polynomial by
the monomial. To divide a polynomial by a binomial, first try factoring the
dividend. If you cannot factor the dividend, use long division.
EXAMPLES
A Find (5x2 3xy 2y2 ) 2xy.
5x2 3xy 2y 2
2xy
5x
2y
2xy
3
2
Since the dividend, t2 5t 10, cannot be
factored, use long division.
Rewrite as a fraction.
2y 2
5x 2
3xy
2xy
B Find ( t2 5t 10) (t 3).
2xy
y
x
t
t 3
t2 5t 10
() t 2 3t
8t
Divide each term by 2xy.
Simplify each term.
y
5x
3
The quotient is .
2y
2
t2 t t
Multiply t and t 3.
Subtract.
t 8
t 3
t2 5t 10
() t2 3t
8t 10
() 8t 24
34
x
Multiply 8 and t 3.
Subtract.
The quotient is t 8 with remainder 34 or
34
t8
.
t3
Try These Together
1. Find (x3 4x 8) 2x.
2. Find ( y2 7y 10) (y 2).
HINT: Factor the dividend, y 2 7y 10.
HINT: Divide each term of the dividend by 2x.
PRACTICE
Find each quotient.
3. (k2 12k 6) 3k
4. (x2 7x 10) (x 2)
5. (x2 5x 6) (x 3)
6. (a2 3a 4) (a 1)
7. (2y2 10y 8) (y 4)
8. (x2 8x 14) (x 1)
9. (2b2 5b 8) (b 2)
10. (2x2 9x 3) (x 3)
t2 6t 16
11. 2n2 6n 3
12. x2 5x 6
13. x 10
14. 2y 8
15. x2
16. n3
8t
6x2
y3
2x 3
B
C
17. Standardized Test Practice Find (3x2 6x 9) 3x.
A 3x 3
3
C x3
B 3x 2
3
D x2
x
x
2
t
x
x2
4
Answers: 1. 2
n3
3
12. 2n 3
k
k
2
2. y 5 3. 4
x1
2
13. x 4 2x 3
5
14. 3x 5 y1
1
15. y2 5y 7 x 1
7
4. x 5 5. x 2 6. a 4 7. 2y 2 8. x 7 94
4
b2
6
9. 2b 1 © Glencoe/McGraw-Hill
8
B
A
3
2
11. t C
B
8.
x3
B
A
7.
x1
C
A
5.
6.
y1
6
10. 2x 3 4.
x3
16. x2 x 2 17. D
3.
x1
4y2
CA Parent and Student Study Guide, Algebra 1
NAME
DATE
Rational Expressions with
Like Denominators (Pages 681–684)
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To add or subtract rational expressions with like denominators, add or
subtract the numerators and then write the sum or difference over the
common denominator. To subtract a quantity, add its additive inverse.
Remember to simplify your answer, if necessary, by dividing by the GCF.
EXAMPLES
7t
2t 1
A Find .
9
7t
9
6y 3
5y 1
B Find .
2y 1
9
The denominator of the second expression can be
rewritten. 1 2y (1 2y) or (2y 1).
2t 1
7t (2t 1)
9
1 2y
9
6y 3
2y 1
5t 1
9
5y 1
6y 3
5y 1
1 2y
2y 1
2y 1
6y 3 (5y 1)
2y 1
y4
2y 1
PRACTICE
Find each sum or difference. Express in simplest form.
9
12
1. 5x
12x
2. t2
4. t
y3
4y 6
5. 14x
2x
6. 3c
c1
7. 7k
6k
8. x3
2
9. 3n
n6
10. 3d 2
d4
11. 8a
a
12. 2n
n1
13. x4
2x 5
14. 2x 12
x9
15. 3m
3m
4
4
4c 1
2
4c 1
2n 3
B
k2
2
5n 5
1x
8
8
x5
x5
a4
2
x1
a4
x2
x2
C
16. Standardized Test Practice Which of the following is an
expression for the perimeter of the rectangle?
15ab
A 20a 10b
B 15ab
C 10a 5b
D 3a 4b
9a 8b
10a
3a – 4b
5b
3a – 4b
3a 4b
3a 4b
16. B
3
x
2. x
6
3. 2
1
4. 2
5y 3
5. k2
k
6. 2x 7. 1 8. 95
m
2n 3
2n 6
9. 1 10. a4
8
11. 2d 1 12. 5
1
13. 14. 1
© Glencoe/McGraw-Hill
1
Answers: 1. B
A
x2
B
8.
2
x
C
B
A
7.
x
C
A
5.
6.
9
3
3. 21
x 3
15. 4.
k2
3 2n
5n 5
3.
21
CA Parent and Student Study Guide, Algebra 1
NAME
DATE
Rational Expressions with
Unlike Denominators (Pages 685–689)
12-6
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The least common multiple (LCM) of two or more positive whole numbers is
the least positive number that is a common multiple of all the numbers. To add
or subtract rational expressions with unlike denominators, first rename the
fractions so the denominators are alike, using the least common denominator
of the fractions. You may need to factor one or both of the denominators first.
The least common denominator (LCD) is the LCM of the denominators.
EXAMPLES
x
5
B Find .
5
4
A Find .
2
2y
x1
3y
List the prime factors of 2y and
3y2
LCD: (x 1)(x 2)
to find the LCD.
3y2 3 y y
2y 2 y
x
x1
Use each prime factor the greatest number of times
it appears in each of the factorizations.
LCD: 2 3 y y or 6y2
2y(3y)
15y
6y 2
5(x 1)
(x 1)(x 2)
(x 1)(x 2)
3y (2)
8
6y 2
x2
x(x 2)
(x 1)(x 2)
2x
5x 5
5(3y)
4(2)
4
2
2
3y
5
x2
Change each rational expression into an equivalent
expression with the LCD.
5
2y
x2
(x 1)(x 2)
x 2 2x (5x 5)
(x 1)(x 2)
x 2 7x 5
15y 8
(x 1)(x 2)
or 2
6y
PRACTICE
Find each sum or difference. Express in simplest form.
1
2
1. 2
1
2. 10
5
3. 2
2
9
7
4. 3
2
5
5. 2
7
6. 5
3
8. 5x
4
9. 2x
10x
a
7x
3x 6
a
2x
x
7. x1
4x 4
x6
7x
2
10. 2
x 16
B
x3
x
4x
12. 2
x
3
11. 2
x 10
x3
x 100
x1
x 5x 6
2
3
13. Standardized Test Practice Find .
2
2x 5
D 2
2x 5
C x4
x x 20
2.
3.
13. D
a
7a 9
4. 3
2
3x 6
17
5. 2x 8
3
6. 96
5x 10
xy2
4x 4
9x
7. x 9x 18
8x 33
8. 2
x 9
5x 11x 12
9. 2
2
x 16
9x 8
10. 2
© Glencoe/McGraw-Hill
2
x x 20
Answers: 1.
5
B 2
x4
15
7x
5
A x5
3
10x
x x 20
x 5x 6
B
A
4x 23x
12. 2
C
B
8.
x3
x4
x 100
C
B
A
7.
2x 8
C
A
5.
6.
x2
y
2
4.
x4
xy
x 10x 3
11. 2
3.
x
CA Parent and Student Study Guide, Algebra 1
NAME
DATE
Mixed Expressions and
Complex Fractions (Pages 690–695)
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A mixed expression is an algebraic expression that contains a monomial
and a rational expression. Simplifying a mixed expression is similar to the
process used in rewriting a mixed number as an improper fraction.
Simplifying
a Complex
Fraction
Any complex fraction
a
b
,
c
d
where b 0, c 0, and d 0, can be expressed as
ad
.
bc
EXAMPLE
6
3 x
Simplify x2 .
4
6
3(x)
6
3 x
x
x
x2 x2
4
4
3x 6
x
x2
4
The LCD of the numerator is x.
Add to simplify the numerator.
3x 6
x2
x
Multiply by the reciprocal of the divisor.
3(x 2)
x2
x
Factor to simplify before multiplying.
4
4
1
3(x 2)
4
x
x2
Divide by the common factor of x 2.
1
12
x
Multiply.
PRACTICE
Write each mixed expression as a rational expression.
4
1. x x
2
2. 4 x7
n4
3. 9 x5
4. 3 2
m 5
m
7. m7
t 3
t2
8. 2 4
n1
x 25
Simplify.
C
C
9. Standardized Test Practice Simplify
x2 5x
B x 3x 10
4
y
x
6. 4
3
97
2
m7
m 5
7. 5
2t
t 2t 3
8. 2
9. B
© Glencoe/McGraw-Hill
x
C 2
x2
b
5. 2x 3
x5
x1
A x
x2
.
1
x5
3x 14
4. 8.
B
A
n1
A
7.
8n 13
3. B
B
6.
2x 5
D x3
x7
C
A
5.
t2
4x 26
2. 4.
m
x
B
3.
6.
xyz
x2
y5z
x4
x2 4
Answers: 1. 5.
a
b
2a
b5
CA Parent and Student Study Guide, Algebra 1
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DATE
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Solving Rational Equations
(Pages 696–702)
A rational equation is an equation that contains rational expressions. To
solve a rational equation, multiply each side of the equation by the LCD of
the rational expressions in the equation. Doing so can yield results that are
not solutions to the original equation, called extraneous solutions or
“false” solutions. To eliminate extraneous solutions, be sure no solution is an
excluded value of the original equation.
EXAMPLE
3a 4
a
Solve 3.
a1
a1
a
a1
3a 4
3
a1
3a 4
(a 1)3
(a 1) a1
a1 a
3a 4
a
(a 1) (a 1) (a 1)3
a1
a1
Multiply each side by the LCD, a 1.
Use the distributive property.
a 3a 4 3a 3
4a 4 3a 3
a 4 3
a 1
Since 1 is an excluded value of the original equation, 1 is an extraneous solution.
Thus, this equation has no solution.
PRACTICE
Solve each equation.
B
4x
6. 2x x2
k8
k4
7. 3
k
k
a1
a1
8. a
a4
n3
2n
9. 2
n1
n1
w5
w
1
10. w6
4
4
x
1
11. x2
x
n1
n1
12. n
n3
x
2
x
13. 8
x
4
y3
y1
14. 1 y2
y2
c
c4
15. 3 c2
4
x
1
2
16. Standardized Test Practice Solve .
3
x
x
A x 3
B x3
C x 3, 3
4. 3 5. 7, 1 6. 0, 4 7. 4 8. 1 9. no solution 10. 7, 2 11. 1, 2 12. 3
B
A
D no solution
© Glencoe/McGraw-Hill
98
1
2
C
B
8.
x2
9
5. (x 7) x
x
C
B
A
7.
1
3
4. t t6
C
A
5.
6.
3
3. 7 2x
x
Answers: 1. 14 2. 1, 5 3. 3, 4.
5
2. n 4 n
13. 4, 4 14. no solution 15. 10, 4 16. C
3.
2
4
1
1. 3y
y
3
CA Parent and Student Study Guide, Algebra 1
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Chapter 12 Review
Connect the Dots
Imagine that you have just won the vacation of a lifetime in a raffle.
Complete this puzzle to find out how you will be traveling to your
destination. First simplify each expression completely. Then connect the dots
following the instructions in the box at the right.
9x
1. 3xy
2.
x2 5x
3x 15
x2
x4
3. 2
x2 4
4.
x2 x
x2 1
5. (x3 5x2 5x 3) (x 3)
6.
18
3x
1
3
7. 5x
7x
8.
x2 3x
x3
2
8
35x
Connect the
answers to each
problem in the
following order:
x
x1
Connect #1 to #2.
Connect #3 to #4.
Connect #5 to #6.
Connect #2 to #7.
Connect #5 to #3.
Connect #7 to #8.
Connect #4 to #6.
6x
3x
1
x3
3
x
x2
x2 2x 1
x2
2
x5
3
x2 3
x2 9
2
35x
x2
2x 8
x2 2x 8
2x2 8
x
3x
xy
x4
2x 4
x2
22
35x
3x
y
x
3
1
x2 3x 1
x3
x2 2x 1
x
3y
2x 1
x2 2x 1
x2 2x 1
6
Answers are located on page 113.
© Glencoe/McGraw-Hill
99
CA Parent and Student Study Guide, Algebra 1