Section 6.2 / Expressing Fractions in Terms of the Least Common Multiple (LCM)
6.2
Objective A
297
Expressing Fractions in Terms of the
Least Common Multiple (LCM)
To find the least common multiple
(LCM) of two or more polynomials
Recall that the least common multiple (LCM) of two or more numbers is the
smallest number that contains the prime factorization of each number.
Video
12 2 2 3
18 2 3 3
The LCM of 12 and 18 is 36 because 36 contains the prime factors of 12 and the prime
factors of 18.





Factors of 12





LCM 36 2 2 3 3
Factors of 18
The least common multiple (LCM) of two or more polynomials is the polynomial of least degree that contains all the factors of each polynomial.
To find the LCM of two or more polynomials, first factor each polynomial completely. The LCM is the product of each factor the greatest number of times it
occurs in any one factorization.
The LCM of the
polynomials is the
product of the LCM
of the numerical
coefficients and each
variable factor the
greatest number of
times it occurs in any
one factorization.
Example 1
Factors of 4x2 4x
LCM 2 2 xx 1x 1 4xx 1x 1
Factors of x2 2x 1
You Try It 1
2
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4x2 4x 4xx 1 2 2 xx 1
x2 2x 1 x 1x 1









The LCM must contain
the factors of each polynomial. As shown with
the braces at the right, the
LCM contains the factors
of 4x 2 4x and the factors of x 2 2x 1.
Find the LCM of 4x2 4x and x2 2x 1.









HOW TO
TA K E N O T E
2
Find the LCM of 4x y and 6xy .
Find the LCM of 8uv2 and 12uw.
Solution
Your solution
4x2 y 2 2 x x y
6xy2 2 3 x y y
LCM 2 2 3 x x y y 12x2 y2
Example 2
You Try It 2
Find the LCM of x x 6 and 9 x .
2
2
Find the LCM of m2 6m 9 and
m2 2m 3.
Solution
x2 x 6 x 3x 2
9 x2 x2 9 x 3x 3
LCM x 3x 2x 3
Your solution
Solutions on p. S15
298
Chapter 6 / Rational Expressions
Objective B
Video
To express two fractions in terms
of the LCM of their denominators
When adding and subtracting fractions, it is frequently necessary to express two
or more fractions in terms of a common denominator. This common denominator is the LCM of the denominators of the fractions.
HOW TO
Write the fractions
x1
4x 2
and
x3
6x 2 12x
in terms of the LCM of the
denominators.
Find the LCM of the
denominators.
For each fraction,
multiply the
numerator and the
denominator by the
factors whose product
with the denominator
is the LCM.
Example 3
Write the fractions
The LCM is 12x2x 2.
x1
x 1 3x 2
3x2 3x 6
2
2
4x
4x
3x 2
12x2x 2
x3
x3
2x
2x2 6x
2
6x 12x
6xx 2 2x
12x2x 2
LCM
You Try It 3
x2
3x 2
and
x1
8xy
in
Write the fractions
x3
4xy 2
and
2x 1
9y 2z
terms of the LCM of the denominators.
of the LCM of the denominators.
Solution
Your solution
in terms
The LCM is 24x2 y.
x2
x 2 8y
8xy 16y
2 2 3x
3x
8y
24x2 y
x1
x 1 3x
3x2 3x
8xy
8xy
3x
24x2 y
Write the fractions
You Try It 4
2x 1
2x x 2
and
x
x2 x 6
in
Write the fractions
x4
x 2 3x 10
and
2x
25 x 2
terms of the LCM of the denominators.
in terms of the LCM of the denominators.
Solution
Your solution
2x 1
2x 1
2x 1
2
2x x2
x2 2x
x 2x
The LCM is xx 2x 3.
2x 1
2x 1 x 3
2x2 5x 3
2 2x x
xx 2 x 3
xx 2x 3
x
x
x
x2
x2 x 6
x 2x 3 x
xx 2x 3
Solutions on p. S15
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Example 4
Section 6.2 / Expressing Fractions in Terms of the Least Common Multiple (LCM)
299
6.2 Exercises
Objective A
To find the least common multiple
(LCM) of two or more polynomials
Copyright © Houghton Mifflin Company. All rights reserved.
For Exercises 1 to 33, find the LCM of the polynomials.
1.
8x3y
12xy2
2. 6ab2
18ab3
3. 10x4y2
15x3y
4. 12a2b
18ab3
5.
8x2
4x2 8x
6. 6y2
4y 12
7. 2x2y
3x2 12x
8. 4xy2
6xy2 12y2
9.
9xx 2
12x 22
10. 8x2x 12
10x3x 1
11. 3x 3
2x2 4x 2
12. 4x 12
2x2 12x 18
13. x 1x 2
x 1x 3
14.
2x 1x 4
2x 1x 4
15.
2x 32
2x 3x 5
16. x 7x 2
x 72
17.
x1
x2
x 1x 2
18.
x 4x 3
x4
x3
19. x2 x 6
x2 x 12
20.
x2 3x 10
x2 5x 14
21.
x2 5x 4
x2 3x 28
22. x2 10x 21
x2 8x 15
23.
x2 2x 24
x2 36
24.
x2 7x 10
x2 25
25. x2 7x 30
x2 5x 24
26.
2x2 7x 3
2x2 x 1
27.
3x2 11x 6
3x2 4x 4
28. 2x2 9x 10
2x2 x 15
29.
6 x x2
x2
x3
30.
15 2x x2
x5
x3
31. 5 4x x2
x5
x1
32.
x2 3x 18
3x
x6
33.
x2 5x 6
1x
x6
300
Chapter 6 / Rational Expressions
Objective B
To express two fractions in terms
of the LCM of their denominators
For Exercises 34 to 53, write the fraction in terms of the LCM of the denominators.
34.
4 3
,
x x2
35.
5 6
,
ab2 ab
36.
x z
,
3y2 4y
37.
5y 7
,
6x2 9xy
38.
y
6
,
xx 3 x2
39.
a
6
,
y2 yy 5
40.
9
6
,
x 12 xx 1
41.
a2
a
,
yy 7 y 72
42.
3
5
,
x 3 x3 x
43.
b
b2
,
yy 4 4 y
44.
3
2
,
x 52 5 x
45.
3
2
,
7 y y 72
46.
3
4
,
x2 2x x2
47.
2
3
,
y 3 y3 3y2
48.
x2
x
,
x3 x4
49.
x2
x1
,
2x 1 x 4
50.
3
x
,
x2 x 2 x 2
51.
3x
4
,
x 5 x2 25
52.
x
2x
, 2
x x6 x 9
53.
2
x1
x
, 2
x 2x 15 x 6x 5
2
54.
When is the LCM of two polynomials equal to their product?
For Exercises 55 to 60, write each fraction in terms of the LCM of the
denominators.
55.
8
9
3,
10 105
59.
c
d
,
6c2 7cd d2 3c2 3d2
56.
3,
2
n
57.
x,
x
2
x 1
58.
60.
x2 1
x1
1
3,
2,
x 1 x 1 x 1
1
1
,
ab 3a 3b b2 ab 3a 3b b2
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APPLYING THE CONCEPTS