HFCC Math Lab
Intermediate Algebra - 7
FINDING THE LOWEST COMMON DENOMINATOR (LCD)
Adding or subtracting two rational expressions require the rational expressions to have the
same denominator.
Example 1: Adding two rational expressions with the common denominator 2x+1
5
3
2x 1 2x 1
5 3
=
2x 1
8
=
2x 1
When the denominators of two rational expressions are not the same we write equivalent
rational expression for both or one of the rational expressions so that they both have same
denominator.
To find the equivalent rational expressions we multiply the numerator and denominator of a
rational function with same non-zero factor.
2x 1
5x 1
10(2 x 1)
=>
10(5 x 1)
Example 2: A list of equivalent rational expressions to
2 x 1 10
5 x 1 10
2x 1 x
5x 1 x
=>
x(2 x 1)
x(5 x 1)
assuming x
2 x 1 (2 x 3)
5 x 1 (2 x 3)
=>
(2 x 1)(2 x 3)
(5 x 1)(2 x 3)
assuming 2x 3 0
0
2 x 1 10 x 2 (3x 5)
10 x 2 (3 x 5)(2 x 1)
=>
assuming 10 x 2 (3x 5) 0
5 x 1 10 x 2 (3x 5)
10 x 2 (3 x 5)(5 x 1)
2x 1
Remark: Try to reason out why every rational expression in the list above is equivalent to
5x 1
We write equivalent rational expressions so that the two rational expressions have common
denominator the natural choice is the product of the two denominators.
Remark: Try to reason why the above statement makes sense. (Hint: You may have to use your
knowledge on equivalent rational expressions to do so).
Revised 11/09
1
Example 3: Some of the multiples of the polynomial 6 x(5x 1)2 are
16 x(2 x 1)2 6 x(2 x 1) 2
36 x(2 x 1)2 18 x(2 x 1) 2
x5 6 x(2 x 1)2
6 x6 (2 x 1)2
5(2 x 1)6 x(2 x 1)2 30 x(2 x 1)3
7 x 2 (2 x 1)3 6 x(2 x 1)2
42 x3 (2 x 1)5
Remark: Try to reason out why every expression in the above list is a multiple of 6 x(5x 1)2
The best choice for the denominator is the Least Common Multiple (LCM) of the
denominators of two expressions. We also refer to this value as the Least Common
Denominator (LCD)
Remark: Try to reason why the above statement makes sense. (Hint: Least means smallest multiple
of a polynomial. A smallest multiple of a polynomial has the least degree and yet can be written as a
product of both the polynomials under consideration.)
To find the least common denominator (LCD) of two or more fractional expression you
can follow this procedure:
1. Factor each denominator completely, showing repeated factors as powers; i.e., as
bases with exponents.
2. Write the product of all the different bases without exponents.
3. Raise each base to the highest exponent to which it is raised in any single
denominator
Remark: A base and its opposite (negative) can be considered the same when find the
LCD; any (-1) factor can be attached to the numerator.
Example 4: Simplifying rational expression with a -1 factor in the denominator
Example 5: Find the Least Common Denominator (LCD) of the expressions
1
( x 2)
1
x 2
x 1 x 4 x
,
,
20
6 10
Solution: Using the above procedure to find the LCD
20
6
10
Revised 11/09
22 5
23
25
Step 1: Factor each denominator completely,
showing repeated factors as powers
235
Step 2: Write the product of all the different
bases without exponents.
22 35 60
Step 3: Raise each base to the highest exponent
to which it is raised in any single denominator
2
Example 6: Find the Least Common Denominator (LCD) of the expressions
1
5x y
7
,
,
2 2
3
18 x y 10 xy z 15 yz
Solution: Using the above procedure to find the LCD
18x 2 y
232 x 2 y 2
10xy 2 z
25xy 3 z
15yz 2
35yz
Step 1: Factor each denominator completely,
showing repeated factors as powers
235xyz
232 5x 2 y 3 z
Step 2: Write the product of all the different
bases without exponents.
Step 3: Raise each base to the highest exponent
to which it is raised in any single denominator
90 x 2 y 3 z
Example 7: Find the Least Common Denominator (LCD) of the expressions
4
a
2
3
7
,
1 2a 2 (a 1) 2
,
Solution: Using the above procedure to find the LCD
a 2 1 (a 1)(a 1)
2a 2 2(a 1)
(a 1)2
Step 1: Factor each denominator completely,
showing repeated factors as powers
(a 1)2
2(a 1)(a 1)
Step 2: Write the product of all the different
bases without exponents.
2(a 1)2 (a 1)
Step 3: Raise each base to the highest exponent
to which it is raised in any single denominator
Example 8: Find the LCD of the expressions
4
8x
3
,
27 x
2
3
7
, 2
6 x 9 2 x 3x 9
Solution: Using the above procedure to find the LCD
8 x3 27 (2 x 3)(4 x 2 6 x 9)
Step 1: Factor each denominator completely,
showing repeated factors as powers
x 2 6 x 9 ( x 3) 2
2 x 2 3x 9 (2 x 3)( x 3)
Revised 11/09
( x 3)(2 x 3)(4 x 2 6 x 9)
Step 2: Write the product of all the different
bases without exponents.
( x 3)2 (2 x 3)(4 x 2 6 x 9)
Step 3: Raise each base to the highest exponent
to which it is raised in any single denominator
3
Example 9: Find the LCD of the expressions
4
3
,
( x 2) (2 x)
Solution: Denominators are ( x 2) and its opposite are (2 x)
4
3
negative to the numerator
,
( x 2) ( x 2)
1( x 2) . Hence we move the
We do not have to use any procedure to find the common denominator because both the fractions
have the common denominator
Example 10: Find the LCD of the expressions
7
6( x 5)
2
,
1
4(5 x)
Solution: (5 x) is the negative of ( x 5) hence rewriting the expressions we have
7
1
,
2
6( x 5) 4( x 5)
6( x 5) 2
23( x 5) 2
4( x 5)
2
Step 1: Factor each denominator completely,
showing repeated factors as powers
2 ( x 5)
23( x 5)
Step 2: Write the product of all the different
bases without exponents.
22 3( x 5) 2
Step 3: Raise each base to the highest exponent
to which it is raised in any single denominator
Exercise: Find the Least Common Denominator (LCD) in each of the following exercise
(Hint: follow the procedure suggested above)
1.
2 3
,
x xy
2.
3x 2 x 4
,
8
12
3.
3
x 2 6 3x
4.
5
4x 2 y y 2x
5.
1 3
,
2xy x 2
6.
3
5
, 2
2
4a b 8b
7.
5
4
1
,
,
2
8 x 3xy 16 y 3
8.
1
2
, 2
2
3n m nm
9.
1
,
9
4 3
, ,
a b a b
Revised 11/09
10.
4
3
3
,
,
8
5 x 20 9 x 36
11.
5
9
, 2
2c 15 c 6c 9
c2
2y
y2
12.
,
( y 3) 2 5( y 3)( y 1)
13.
1
2
3
,
, 2
6x 6 x 1 2x 2
14.
15.
3x
1
, 2
4 2x x 4
16.
17.
y 3
,
8 y 16 y 3 y 2
19.
x 4
x
, 2
3x 11x 6 2 x x 15
2
5
7y
,
21y 36 4 y 12
18.
20.
2
1
,
a 1 a
1
y
x
,
2
2
3y y
2
3
, 2
2a 1 2 a 2
1
2
3y
3
1
, 2
x 2 x 4
3
10 x
2
3
,
270 x
2
x
1
,
6 x 9 6 x 18
Solutions to the odd-numbered exercise and answers to the even- numbered exercise:
1. Denominators: x, xy
Step1: x x, xy x y
Step 2 : x y
Step3 : LCD
2. LCD : 24
x y
3. Denominators x 2, 6 3x
Step1: x 2 ( x 2), 6 3 x
Step 2 : 3( x 2)
4. LCD : 2( y 2 x) or 2(2 x y)
3( x 2)
Step3 : LCD 3( x 2)
Remark: opposite of (x-2) is (2-x)
Hence we can rewrite the rational expression as
1
3
and find the LCD
,
x 2 3x 6
5. Denominators 2 xy, x 2
Step1: 2 xy 2xy, x 2
Step 2 : 2xy
Step3 : LCD
6. LCD : 8a 2b2
x2
2x 2 y
7. Denominators 8 x 2 ,3xy,16 y 3
Step1: 8 x 2 23 x 2 ,3 xy 3xy,16 y 3
Step 2 : 23xy
Step3 : LCD
Revised 11/09
24 3x 2 y 3
8. LCD : 3n2 m2
24 y 3
48x 2 y 3
5
10. LCD : 325( x 4) or 45( x 4)
9. Denominators a b, a, b
Step1: a b (a b), a a, b b
Step 2 : a b(a b)
Step3 : LCD
a b(a b)
11. Denominators c 2 2c 15, c 2 6c 9
Step1: c 2 2c 15 (c 5)(c 3), c 2 6c 9 (c 3) 2
Step 2 : (c 5)(c 3)
Step3 : LCD
12. LCD : 5( y 3)2 ( y 1)
(c 5)(c 3) 2
13. Denominators 6 x 6, x 1, 2 x 2 2
Step1: 6 x 6 23( x 1),
14. LCD : 2(a 1)(a 1)2
x 1 ( x 1), 2 x 2 2 2( x 1)( x 1)
Step 2 : 23( x 1)( x 1)
Step3 : LCD
23( x 1)( x 1)
15. Denominators 4 2 x, x 2 4
Step1: 4 2 x
2( x 2)y, x 2 4 ( x 2)( x 2)
Step 2 : 2( x 2)( x 2)
16. LCD : y ( y 3)( y 3)
Step3 : LCD 2( x 2)( x 2)
Remark: opposite of (x-2) is (2-x)
Hence we can rewrite the fraction as
3x
1
and find the LCD
, 2
2x 4 x 4
17. Denominators 8 y 2 16,3 y 2 21y 36, 4 y 12
Step1: 8 y 2 16 y
18. LCD : ( x 1)( x 2)2
23 y ( y 2),
3 y 2 21y 36 3( y 3)( y 4), 4 y 12 22 ( y 3)
Step 2 : 2y ( y 2)( y 3)
Step3: LCD
2y ( y 2)( y 3)
19. Denominators 3x 2 11x 6, 2 x 2
20. LCD : 30( x 3)2 ( x 3)( x 2 3x 9)
x 15
Step1: 3x 2 11x 6 (3x 2)( x 3),
2 x 2 x 15 (2 x 5)( x 3)
Step 2 : (3x 2)(2 x 5)( x 3)
Step3 : LCD (3x 2)(2 x 5)( x 3)
Revised 11/09
6
Note: You can get additional instructions and practice for solving these
problems by going to the following websites:
http://www.purplemath.com/modules/lcm_gcf.htm This website has step-by-step
instruction on how to find the least common multiple of integers and
polynomials. Finding least common multiples is same as finding the least
common denominators.
http://www.youtube.com/watch?v=drZopvFRa7s This website has a you tube
video on how to find the least common multiple which is same as find the least
common denominator.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_t
ut10_addrat.htm#lcd This website provides video demonstration and step-bystep instruction on how to add two rational expressions. This website also has
information on how to find the least common multiples
http://www.regentsprep.org/Regents/math/algtrig/ATO2/addfrac.htm Student
friendly notes on adding and subtracting rational functions. This web site also
has information on how to find the least common denominator of two or more
fractions
Revised 11/09
7