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5.1
Simplifying Algebraic Fractions
5.1
OBJECTIVES
1. Find the GCF for two monomials and simplify a
fraction
2. Find the GCF for two polynomials and simplify a
fraction
Much of our work with algebraic fractions will be similar to your work in arithmetic. For
instance, in algebra, as in arithmetic, many fractions name the same number. You will
remember from Chapter 0 that
1
12
2
4
42
8
or
1
13
3
4
43
12
1 2
3
So , , and
all name the same number. They are called equivalent fractions. These
4 8
12
examples illustrate what is called the Fundamental Principle of Fractions. In algebra it
becomes
Rules and Properties: Fundamental Principle of Algebraic
Fractions
For polynomials P, Q, and R,
© 2001 McGraw-Hill Companies
PR
P
Q
QR
when Q 0 and R 0
This principle allows us to multiply or divide the numerator and denominator of a fraction
by the same nonzero polynomial. The result will be an expression that is equivalent to the
original one.
Our objective in this section is to simplify algebraic fractions by using the fundamental
principle. In algebra, as in arithmetic, to write a fraction in simplest form, you divide the
numerator and denominator of the fraction by their greatest common factor (GCF). The
numerator and denominator of the resulting fraction will have no common factors other
than 1, and the fraction is then in simplest form. The following rule summarizes this
procedure.
Step by Step:
NOTE Notice that step 2 uses
the Fundamental Principle of
Fractions. The GCF is R in the
rule above.
Step 1
Step 2
To Write Algebraic Fractions in Simplest Form
Factor the numerator and denominator.
Divide the numerator and denominator by the greatest common factor
(GCF). The resulting fraction will be in lowest terms.
395
396
CHAPTER 5
ALGEBRAIC FRACTIONS
Example 1
Writing Fractions in Simplest Form
(a) Write
18
in simplest form.
30
1
NOTE This is the same as
dividing both the numerator
18
and denominator of
by 6.
30
1
18
233
2 3 3
3
30
235
2 3 5
5
1
Divide by the GCF. The slash
lines indicate that we have
divided the numerator and
denominator by 2 and by 3.
1
3
4x
in simplest form.
6x
(b) Write
1
1
4x3
2 2 x x x
2x2
6x
2 3 x
3
1
1
3 2
15x y
in simplest form.
20xy4
(c) Write
1
1
1
1
15x3y2
3 5 x x x y y
3x2
20xy4
2 2 5 x y y y y
4y2
1
1
1
1
3a2b
in simplest form.
9a3b2
(d) Write
1
1
1
1
3a2b
3 a a b
1
9a3b2
3 3 a a a b b
3ab
1
1
1
1
5 4
(e) Write
10a b
in simplest form.
2a2b3
1
1
1
1
1
1
5 2 a a a a a b b b b
5a3b
10a5b4
5a3b
2 3 2a b
2 a a b b b
1
1
1
1
1
1
1
CHECK YOURSELF 1
this chapter build on our
factoring work of the last
chapter.
Write each fraction in simplest form.
(a)
30
66
(b)
5x4
15x
(c)
12xy4
18x3y2
(d)
5m2n
10m3n3
(e)
12a4b6
2a3b4
In simplifying arithmetic fractions, common factors are generally easy to recognize.
With algebraic fractions, the factoring techniques you studied in Chapter 4 will have to be
used as the first step in determining those factors.
© 2001 McGraw-Hill Companies
NOTE Most of the methods of
SIMPLIFYING ALGEBRAIC FRACTIONS
SECTION 5.1
397
Example 2
Writing Fractions in Simplest Form
Write each fraction in simplest form.
(a)
2x 4
2(x 2)
2
x 4
(x 2)(x 2)
Factor the numerator and
denominator.
1
2(x 2)
(x 2)(x 2)
Divide by the GCF x 2.
The slash lines indicate that
we have divided by that
common factor.
1
2
x2
1
3x2 3
3(x 1)(x 1)
(b) 2
x 2x 3
(x 3)(x 1)
1
3(x 1)
x3
1
2x2 x 6
(x 2)(2x 3)
(c)
2
2x x 3
(x 1)(2x 3)
1
C A U TI O N
Pick any value, other than 0, for
x and substitute. You will
quickly see that
x2
2
Z
x1
1
x2
x1
Be Careful! The expression
x2
is already in simplest form. Students are often tempted
x1
to divide as follows:
x 2
x 1
is not equal to
2
1
The x’s are terms in the numerator and denominator. They cannot be divided out. Only
factors can be divided. The fraction
x2
x1
is in its simplest form.
© 2001 McGraw-Hill Companies
CHECK YOURSELF 2
Write each fraction in simplest form.
(a)
5x 15
x2 9
(b)
a2 5a 6
3a2 6a
(c)
3x2 14x 5
3x2 2x 1
(d)
5p 15
p2 4
398
CHAPTER 5
ALGEBRAIC FRACTIONS
Remember the rules for signs in division. The quotient of a positive number and a
negative number is always negative. Thus there are three equivalent ways to write such a
quotient. For instance,
2
2
2
3
3
3
2
, with the negative
3
sign in the numerator, is the
most common way to write the
quotient.
NOTE
The quotient of two positive numbers or two negative numbers is always positive. For
example,
2
2
3
3
Example 3
Writing Fractions in Simplest Form
Write each fraction in simplest form.
1
quotient is written in the most
common way with the minus
sign in the numerator.
1
1
1
1
1
5a2b
(1) 5 a a b
a2
(b)
10b2
(1) 2 5 b b
2b
1
1
1
CHECK YOURSELF 3
Write each fraction in simplest form.
8x3y
4xy2
(a)
(b)
16a4b2
12a2b5
It is sometimes necessary to factor out a monomial before simplifying the fraction.
Example 4
Writing Fractions in Simplest Form
Write each fraction in simplest form.
(a)
6x2 2x
3x 1
2x(3x 1)
2x2 12x
2x(x 6)
x6
(b)
x2 4
x2
(x 2)(x 2)
x2 6x 8
(x 2)(x 4)
x4
© 2001 McGraw-Hill Companies
NOTE In part (a), the final
1
6x2
2 3 x x
2x
2x
(a)
3xy
(1) 3 x y
y
y
SIMPLIFYING ALGEBRAIC FRACTIONS
SECTION 5.1
399
CHECK YOURSELF 4
Simplify each fraction.
(a)
3x3 6x2
9x4 3x2
(b)
x2 9
x2 12x 27
Reducing certain algebraic fractions will be easier with the following result. First, verify
for yourself that
5 8 (8 5)
In general, it is true that
a b (b a)
or, by dividing both sides of the equation by b a,
ab
(b a)
ba
ba
So dividing by b a on the right, we have
NOTE Remember that a and b
cannot be divided out because
they are not factors.
ab
1
ba
Let’s look at some applications of that result in Example 5.
Example 5
Writing Fractions in Simplest Form
Write each fraction in simplest form.
(a)
2x 4
2(x 2)
4 x2
(2 x)(2 x)
© 2001 McGraw-Hill Companies
(b)
This is equal
to 1.
2(1)
2
2x
2x
9 x2
(3 x)(3 x)
2
x 2x 15
(x 5)(x 3)
(3 x)(1)
x5
x 3
x5
This is equal
to 1.
CHAPTER 5
ALGEBRAIC FRACTIONS
CHECK YOURSELF 5
Write each fraction in simplest form.
(a)
3x 9
9 x2
(b)
x2 6x 27
81 x2
CHECK YOURSELF ANSWERS
5
x3
2y2
1
; (b) ; (c) 2 ; (d)
; (e) 6ab2
11
3
3x
2mn2
5(p 3)
2x2
4a2
(d)
3. (a)
; (b) 3
(p 2)(p 2)
y
3b
3
x 3
5. (a)
; (b)
x3
x9
1. (a)
5
a3
x5
; (b)
; (c)
;
x3
3a
x1
x2
x3
4. (a) 2
; (b)
3x 1
x9
2. (a)
© 2001 McGraw-Hill Companies
400
Name
Exercises
5.1
Section
Date
Write each fraction in simplest form.
1.
16
24
2.
ANSWERS
56
64
1.
3.
80
180
4.
18
30
2.
3.
5.
4x5
6x2
6.
10x2
15x4
4.
5.
7.
9x3
27x6
8.
25w6
20w2
6.
7.
10a2b5
9.
25ab2
18x4y3
10.
24x2y3
8.
9.
10.
42x3y
11.
14xy3
18pq
12.
45p2q2
11.
12.
2
13.
2xyw
6x2y3w3
2 2
14.
3c d
6bc3d 3
13.
14.
5 5
15.
10x y
2x3y4
6 3
16.
3bc d
bc3d
15.
© 2001 McGraw-Hill Companies
16.
17.
4m3n
6mn2
18.
15x3y3
20xy4
17.
18.
19.
8ab3
16a3b
20.
14x2y
21xy4
19.
20.
401
ANSWERS
21.
21.
8r2s3t
16rs4t3
22.
10a3b2c3
15ab4c
23.
3x 18
5x 30
24.
4x 28
5x 35
25.
3x 6
5x 15
26.
x2 25
3x 15
27.
6a 24
a2 16
28.
5x 5
x2 4
29.
x2 3x 2
5x 10
30.
4w2 20w
w 2w 15
31.
x2 6x 16
x2 64
32.
y2 25
y2 y 20
33.
2m2 3m 5
2m2 11m 15
34.
6x2 x 2
3x2 5x 2
35.
p2 2pq 15q2
p2 25q2
36.
4r2 25s2
2r 3rs 20s2
37.
2x 10
25 x2
38.
3a 12
16 a2
39.
25 a2
a2 a 30
40.
2x2 7x 3
9 x2
41.
x2 xy 6y2
4y2 x2
42.
16z2 w2
2w 5wz 12z2
22.
23.
24.
25.
26.
27.
28.
29.
30.
2
31.
32.
33.
34.
35.
36.
2
38.
39.
40.
41.
42.
402
2
© 2001 McGraw-Hill Companies
37.
ANSWERS
x2 4x 4
43.
x2
43.
4x2 12x 9
44.
2x 3
44.
45.
xy 2y 4x 8
2y 6 xy 3x
46.
ab 3a 5b 15
15 3a2 5b a2b
47.
y7
7y
48.
5y
y5
45.
46.
47.
49. The area of the rectangle is represented by 6x2 19x 10. What is the length?
48.
49.
50.
3x 2
51.
50. The volume of the box is represented by (x2 5x 6)(x 5). Find the polynomial
52.
that represents the area of the bottom of the box.
53.
x2
54.
51. To work with algebraic fractions correctly, it is important to understand the difference
between a factor and a term of an expression. In your own words, write difinitions for
both, explaining the difference between the two.
55.
52. Give some examples of terms and factors in algebraic fractions, and explain how both
are affected when a fraction is reduced.
53. Show how the following algebraic fraction can be reduced:
© 2001 McGraw-Hill Companies
x2 9
4x 12
Note that your reduced fraction is equivalent to the given fraction. Are there other
algebraic fractions equivalent to this one? Write another algebraic fraction that you
think is equivalent to this one. Exchange papers with another student. Do you agree
that their fraction is equivalent to yours? Why or why not?
54. Explain the reasoning involved in each step of reducing the fraction
55. Describe why
42
.
56
3
27
and
are equivalent fractions.
5
45
403
ANSWERS
a.
Getting Ready for Section 5.2 [Section 0.2]
b.
Perform the indicated operations.
3
10
5
(c)
12
7
(e)
20
11
(g)
6
c.
(a)
d.
e.
f.
4
10
1
12
9
20
2
6
5
4
8
8
7
3
(d)
16
16
13
5
(f)
8
8
5
7
(h) 9
9
(b)
g.
Answers
h.
1.
2
3
1
2ab3
3x2
1
9.
11. 2
13.
3
3x
5
y
3xy2w
2m2
b2
r
3
3(x 2)
17.
19.
21.
23.
25.
2
2
3n
2a
2st
5
5(x 3)
x1
x2
m1
p 3q
29.
31.
33.
35.
5
x8
m3
p 5q
a 5
x 3y
(y 4)
39.
41.
43. x 2
45.
a6
2y x
y3
4
9
3.
15. 5x2y
6
a4
2
37.
x5
27.
47. 1
7
10
2x3
3
49. 2x 5
b.
1
8
c.
7.
51.
1
3
d.
53.
5
8
e.
4
5
55.
f. 1
g.
3
2
h.
4
3
© 2001 McGraw-Hill Companies
a.
5.
404