11-4
Rational Expressions
TEKS FOCUS
VOCABULARY
ĚRational expression – the
TEKS (7)(F) Determine the sum, difference,
product, and quotient of rational expressions
with integral exponents of degree one and
of degree two.
quotient of two polynomials
ĚSimplest form of a rational
expression – A rational
expression is in simplest form
when its numerator and
denominator have no common
divisors.
TEKS (1)(G) Display, explain, and justify
mathematical ideas and arguments using
precise mathematical language in written or
oral communication.
ĚArgument – a set of statements
put forth to show the truth or
falsehood of a mathematical
claim
ĚJustify – To justify is to explain
with logical reasoning. You
can justify a mathematical
argument.
Additional TEKS (1)(A), (6)(H)
ESSENTIAL UNDERSTANDING
You can use much of what you know about multiplying and dividing fractions to
multiply and divide rational expressions.
Problem 1
P
TEKS Process Standard (1)(G)
Simplifying a Rational Expression
2
What is x2 + 7x + 10 in simplest form? State any restrictions on the variable.
x − 3x − 10
x 2 + 7x + 10 (x + 2)(x + 5)
=
x 2 - 3x - 10 (x + 2)(x - 5)
Is there more than
one restriction?
Yes, before you divided
out the common factors,
(x + 2) was one of
the factors of the
denominator, so x ≠ -2.
476
=
(x + 2)(x + 5)
(x + 2)(x - 5)
x+5
=x-5
Factor the numerator and denominator.
Divide out common factors.
Simplify.
The simplified form is xx +- 55 for x ≠ 5 and x ≠ -2. The restriction x ≠ -2 is not
T
evident
from the simplified form, but is needed to prevent the denominator of the
e
original
expression from being zero.
o
Lesson 11-4 Rational Expressions
Problem 2
P
Multiplying Rational Expressions
What are the following products in simplest form? State any restrictions on
the variables.
3x
A 35y
~
49y
12z
3x
35y
3x # 7 y
# 49y
12z = 5 # 7y 3 # 2 z
7 # 7y
3x
= 5 # 7y # 3 # 4z
2
2
7x
= 20z
Factor all polynomials.
Divide out common factors.
Simplify.
7x
The product is 20z
for y ≠ 0 and z ≠ 0.
B
x−2
5x
~
10x
3x2 − 12
x-2
5x
# 5x
# 3x 10x- 12 = x 5x- 2 # 3(x -2 2)(x
+ 2)
x-2
2 # 5x
= 5x #
3(x - 2)(x + 2)
2
=
2
3(x + 2)
Factor all polynomials.
Divide out common factors.
Simplify.
The product is 3(x 2+ 2) for x ≠ 0, x ≠ 2, and x ≠ -2.
How is multiplying
rational expressions
like multiplying
fractions?
To multiply rational
expressions, you
multiply the numerators
and multiply the
denominators.
C
x2 + x − 6
x−5
~
x2
x2 + x - 6
x-5
x2 − 25
+ 4x + 3
# x x+ -4x25+ 3
2
2
=
(x + 3)(x - 2)
x-5
- 5)
# (x(x ++ 5)(x
3)(x + 1)
=
(x + 3)(x - 2)
x-5
- 5)
# (x(x ++ 5)(x
3)(x + 1)
=
(x - 2)(x + 5)
x+1
The product is
(x - 2)(x + 5)
x+1
Factor all polynomials.
Divide out common factors.
Simplify.
for x ≠ -3, x ≠ 5, and x ≠ -1.
PearsonTEXAS.com
477
Problem 3
P
Rewriting Quotients of Rational Expressions
Rewrite each quotient as a product. Use this to find the quotient of the rational
expressions in simplest form. What are the restrictions on the variables?
3x
6x
A 4y ÷ 5z
How did you divide
fractions without
variables?
You multiplied by the
reciprocal.
# 6x5z
3x
5z
= 2 # 2y # 2 # 3x
3x
5z
= 2 # 2y # 2 # 3x
3x 6x 3x
4y , 5z = 4y
5z
= 8y
Multiply by the reciprocal.
Factor.
Cancel common factors.
Simplify.
5z
The quotient is 8y
. When you identify the restrictions, you need to include both
the denominator of the divisor, 5z, and its reciprocal, 6x. The restrictions are
x ≠ 0, y ≠ 0, and z ≠ 0.
B
7x + 14
7x2 − 28
x − 1 ÷ 5x2 − 5
# 7x5x -- 285
Multiply by the reciprocal.
7(x + 2)
+ 1)(x - 1)
# 5(x
7(x + 2)(x - 2)
Factor.
7(x + 2)
+ 1)(x - 1)
# 5(x
7(x + 2)(x - 2)
7x + 14
7x 2 - 28 7x + 14
x - 1 , 5x 2 - 5 = x - 1
= x-1
= x-1
2
2
Cancel common factors.
5(x + 1)
= x-2
To identify any restrictions on the variable x, determine when x - 1, 7x 2 - 28,
5(x + 1)
and 5x 2 - 5 are equal to 0. The quotient is x - 2 for x ≠ -1, x ≠ 1, x ≠ -2
and x ≠ 2.
478
Lesson 11-4 Rational Expressions
Problem 4
P
Dividing Rational Expressions
What is the quotient
2−x
x 2 + 2x + 1
restrictions on the variable.
How do you start?
Think of division
as multiplying by
the reciprocal.
To divide, you multiply by
the reciprocal.
2
÷ x +23x − 10 in simplest form? State any
x −1
x2
x2 + 3x − 10
2−x
÷
+ 2x + 1
x2 − 1
#x
x2 − 1
+ 3x − 10
=
2−x
x2 + 2x + 1
=
2−x
(x + 1)(x + 1)
1)(x − 1)
# (x(x ++ 5)(x
− 2)
Factor -1 from (2 - x)
to get a second (x - 2).
=
− 1(x − 2)
(x + 1)(x + 1)
1)(x − 1)
# (x(x ++ 5)(x
− 2)
Divide out common
factors.
=
−1(x − 2)
(x + 1)(x − 1)
(x + 1)(x + 1) (x + 5)(x − 2)
Rewrite the remaining
factors.
=
− 1(x − 1)
(x + 1)(x + 5)
The expressions may
have common factors.
So, factor the numerators
and denominators.
Identify the restrictions
from the denominator of
the simplified expression
and from any other
denominator used.
2
⋅
x ≠ −1, x ≠ −5, x ≠ 1, and x ≠ 2
PearsonTEXAS.com
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Problem 5
P
TEKS Process Standard (1)(A)
Using Rational Expressions to Solve a Problem
Construction Your community is building a park. It wants to fence in a play space
for toddlers. It wants the maximum area for a given amount of fencing. Which
shape, a square or a circle, provides a more efficient use of fencing?
One measure of efficiency is the ratio of area fenced to fencing used, or area to
perimeter. Which of the two shapes has the greater ratio?
Square
Area =
Circle
s2
Area = pr 2
Define area and perimeter.
Perimeter = 4s
Perimeter = 2pr
s=4
Express s and r in terms of a
common variable, P.
r = 2p
Area
s2
Perimeter = P
Write the ratios.
Area
pr 2
Perimeter = P
P
How can you compare
P
P
16 and 4P without
evaluating P?
Since the numerators
are the same, the
fraction with the smaller
denominator is the larger
fraction.
=
( P4 )
2
P
P
= 16
P
4p
Since
S
7
P
16,
( )
P
p 2p
Substitute for s and r.
=
Simplify.
P
= 4p
2
P
a circle provides a more efficient use of fencing.
40 2
Check Assume P = 40 ft. The area of the circle is p 2p
C
≈ 127 ft2 .
( )
The area of the square is
40 2
4
( )
=
100 ft2.
The area of the circle is greater.
NLINE
HO
ME
RK
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P
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Simplify each rational expression. State any restrictions on the variables.
1. For additional support when
completing your homework,
go to PearsonTEXAS.com.
5x 3y
15xy 3
49 - z 2
4. z + 7
2.
2x
4x 2 - 2x
3.
6c2 + 9c
3c
5.
x 2 + 8x + 16
x 2 - 2x - 24
6.
12 - x - x 2
x 2 - 8x + 15
Multiply. State any restrictions on the variables.
x-1
x
9.
20x 2
3x 2 - 75
11.
480
#x
7.
x2 - 4
x2 - 1
2
x2
- 3x + 2
# 2x25 -+ x18
2
2
# xx ++ 2x1
2
Lesson 11-4 Rational Expressions
# 15b
14c
2x + 12 6 - 2x
10. 3x - 9 # 3x + 8
x - 5x + 6 # x + 3x + 2
12.
x -4
x - 2x - 3
4a
8. 9b
2
2
2
2
Divide. State any restrictions on the variables.
35y
7y
13. 18z , 36x
6x + 6y
18
15. y - x , 5x - 5y
17.
x2
3x - 9
9 - x2
3y - 12
6y - 24
14. 25x , 2
5x + 20
x2
3x
,
+ 2x + 1 x 2 - 1
16. 2y + 4 , 8 + 4y
18.
y 2 - 5y + 6 y 2 + 3y - 10
,
y3
4y 2
19. Apply Mathematics (1)(A) A kitchen storage container will have a circular
base of radius r and a height of r. The container can be either cylindrical or
hemispherical (half a sphere).
a. Write and simplify an expression for the ratio of
the volume of the hemispherical container to
its surface area (including the base). For a
sphere, V = 43pr 3 and SA = 4pr 2.
r
r
r
r
b. Write and simplify an expression for the ratio
of the volume of the cylindrical container to its
surface area (including the bases).
c. Compare the ratios of volume to surface area
for the two containers.
d. Compare the volumes of the two containers.
e. Describe how you used these ratios to compare the volumes of the two
containers. Which measurement of the containers determines the volumes?
20. Create Representations to Communicate Mathematical Ideas (1)(E) Write
three rational expressions that simplify to x +x 1.
21. Apply Mathematics (1)(A) A cereal company wants to use the most efficient
packaging for their new product. They are considering a cylindrical-shaped box
and a cube-shaped box. Compare the ratios of the volume to the surface area of
the containers to determine which packaging will be more efficient.
Multiply or divide. State any restrictions on the variables.
22.
6x 3 - 6x 2 3x 2 - 15x + 12
, 2
x 4 + 5x 3
2x + 2x - 40
23.
24.
x2 - x - 2
x 2 - x - 12
,
2x 2 - 5x + 2 2x 2 + 5x - 3
25.
x2
2x 2 - 6x
+ 18x + 81
2x 2 + 5x + 2
4x 2 - 1
# 9xx +- 819
2
# 2xx ++ xx -- 21
2
2
26. a. Explain Mathematical Ideas (1)(G) Write a simplified expression
for the area of the rectangle at the right.
b. Which parts of the expression do you analyze to determine the
restrictions on a? Explain.
c. State all restrictions on a.
4a 4
a3
3a 9
2a 6
PearsonTEXAS.com
481
Decide whether the given statement is always, sometimes, or never true.
27. Rational expressions contain exponents.
28. Rational expressions contain logarithms.
29. Rational expressions are undefined for values of the variables that make the
denominator 0.
30. Restrictions on variables change when a rational expression is simplified.
Simplify. State any restrictions on the variables.
31.
(x 2 - x)2
x(x - 1)-2(x 2 + 3x - 4)
32.
54x 3y -1
3x -2y
33.
2x + 6
(x - 1)-1(x 2 + 2x - 3)
34. a. Use Representations to Communicate Mathematical Ideas (1)(E) Simplify
(2xn)2 - 1
, where x is an integer and n is a positive integer. (Hint: Factor the
2xn - 1
numerator.)
b. Use the result from part (a). Which part(s) of the expression can you use to
show that the value of the expression is always odd? Explain.
a
b
Use the fact that c = ab ÷ dc to simplify each rational expression. State any
d
restrictions on the variables.
8x 2y
x+1
35.
6xy 2
x+1
3a3b3
a-b
36. 4ab
b-a
9m + 6n
m2n2
37. 12m + 8n
5m2
x2 - 1
x2 - 9
38. 2
x + 3x - 4
x 2 + 8x + 15
TEXAS Test Practice
T
39. Which function is graphed at the right?
8
A. y = (x + 4)(x - 1)(x + 2)
B. y = (x - 4)(x - 1)(x + 2)
y
O
2
x
4
8
C. y = (x - 4)(x + 1)(x - 2)
D. y = (x + 4)(x + 1)(x - 2)
40. Which function generates the table of values at the right?
F. y = log 1x
2
G. y = -log 2x
41. Which expression equals
2x - 1
A.
(x - 1)(x + 3)(x + 1)
2x + 1
B.
(x - 1)(x + 1)(x - 3)
x
x2 - 2x - 3
H. y = log 2x
# x 2x- 4x- 6+ 3 ?
2
2x
(x - 1)(x + 1)(x - 3)
2x
D.
(x + 3)(x - 1)(x + 1)
C.
1
42. What is the solution of the equation 3-x = 243
?
482
Lesson 11-4 Rational Expressions
( 1 )x
J. y = 2
x
y
1
2
1
1
2
0
1
4
2