Section 8 – 3:
Finding The Least Common Denominator
Selected Worked Homework Problems
How to find the Least Common Denominator (LCD) of two Rational Expressions
1. Factor each Denominator.
2. The LCD is a product of the LCD of the coefficients and the LCD of the variable terms.
2a. The LCD for the coefficients is the smallest number each coefficient divides into evenly.
2b. The LCD for the variable part of the denominator is the product of each of the different factors
raised to the highest power it appears in the different denominators.
Find The LCD for each pair of fractions. Do not combine the fractions.
1.
2
1
and
!
3x
6x 2
Find the LCD for 6xy 2 and 8x 2
Find the LCD for 3x and 6x 2
1.
5
3
and
!
2
6xy
8x 2
2.
3• x and 6 • x 2 • y are already factored
2A. The LCD of 3 and 6 is the smallest
number 3 and 6 BOTH divide INTO
The LCD of 3 and 6 is 6
1.
!
6 • x • y 2 and 8 • x 2 are already factored
2A. The LCD of 6 and 8 is the smallest
number 6 and 8 BOTH divide INTO
The LCD of 6 and 8 is 24
2B. The higest power of x is x 2
2B. The higest power of x is x 2
The higest power of y is y 2
3. The LCD of 3x and 6x 2 is 6x 2
3. The LCD of 6xy 2 and 8x 2 y is 24 x 2 y 2
Math 100 !
Section 8 – 3
HW WKD!
© 2016 Eitel
3.
1
2
6x y
and
5
4x
3
!
Find the LCD for 6(2x + 5) and 4 (2x + 5)
Find the LCD for 6x 2 y and 4 x 3
1.
x
1
and
6(2x + 5)
4(2x + 5)
4.
1. 6(2x + 5) and 4 (2x + 5) are already factored
6 • x 2 • y and 4 • x 3 are already factored
2A. The LCD of 6 and 4 is the smallest
number 6 and 4 BOTH divide INTO
The LCD of 6 and 4 is 12
2A. The LCD of 6 and 4 is 12
2B. The higest power of (2x + 5) is (2x + 5)
!
3. The LCD = 12(2x + 5)
2B. The higest power of x is x 3
The higest power of y is y
3. The LCD of 6x 2 y and 4 x 3 is 12x 3 y
5.
x
x −1
and
!
3(x + 7)
8(x + 7)(x + 7)
8.
Find the LCD for
3(x + 7) and 8(x + 7)(x + 7)
3
2
and
!
2(x + 3)(x + 3)
4(x + 3)
Find the LCD for
2(x + 3)(x + 3) and 4(x + 3)
= 3(x + 7) and 8(x + 7)2
= 2(x + 3)2 and 4(x + 3)
1. 3x(x + 7) and 8(x + 7)2
are already factored
1. 2(x + 3)2 and 4(x + 3)
are already factored
!
2A. The LCD of 3 and 8 is 24
2B. The higest power of (x + 7) and (x + 7)2
is (x + 7)2
2A. The LCD of 2 and 4 is 4
2B. The higest power of (x + 3)2 and (x + 3)
is (x + 3)2
3. The LCD of 3(x + 7) and 8(x + 7)2
3. The LCD of 2(x + 3)2 and 4(x + 3)
= 24(x + 7)2
or = 24 (x + 7)(x + 7)
Math 100 !
= 4(x + 3)2
or = 4 (x + 3)(x + 3)
Section 8 – 3
HW WKD!
© 2016 Eitel
3x
9.
x(x − 1)
2
and
5
4(x − 1) !
x
x −1
and
2x + 6
3x + 9 !
10.
Find the LCD for
2x + 6 and 3x + 9
Find the LCD for
x(x − 1) 2 and 4(x − 1)
1.
1. 2x + 6 and 3x+9 factor as
2(x + 3) and 3(x + 3)
1x(x − 1) 2 and 4(x − 1)
are already factored
!
2A. The LCD of 1 and 4 is 4
2B. The higest power of x is x
2C. The higest power of (x − 1) is (x − 1)
3. The LCD is 4x(x − 1)
11.
−x
2
x − 25
and
2B. The higest power of (x + 3) is (x + 3)1
2
3. The LCD is 6(x + 3)
2
x −1
2
x + 10x + 25
!
12.
−2
3
5x − 5x
2
and
4
2
x −x
Find the LCD for
Find the LCD for
x 2 − 25 and x 2 + 10x + 25
5x 3 − 5x 2 and x 2 − x
1. x 2 − 25 and x 2 + 10x + 25 factor as
(x + 5)(x − 5) and (x − 5)
1.
2
5x 3 − 5x 2 and x 2 − x factor as
5x 2 (x − 1) and x(x − 1)
!
2A. The LCD of 1 and 1 is 1
2B. The higest power of (x + 5) is (x + 5)1
3.
2A. The LCD of 2 and 3 is 6
2A. The LCD of 5 and 1 is 5
The higest power of (x − 5) is (x − 5)2
2B. The higest power of x is x 2
The higest power of (x − 1) is (x − 1)
The LCD = (x + 5)(x − 5)2
or=(x + 5)(x − 5)(x − 5)
3.
Math 100 !
Section 8 – 3
The LCD = 5x 2 (x − 1)
HW WKD!
© 2016 Eitel
15.
x −2
7
and
!
2x − 8
2x − 2
16.
Find the LCD for
Find the LCD for
2x − 8 and 2x − 2
1.
x 2 + 2x and 4 x + 8
2x − 8 and 2x − 2 factor as
2(x − 4) and 2(x − 1)
1.
!
2A. The LCD of 2 and 2 is 2
2B. The higest power of (x − 4) is (x − 4)1
The LCD = 2(x − 4)(x − 1)
Math 100 !
2B. The higest power of x is x1
The higest power of (x + 2) is (x + 2)1
3.
Section 8 – 3
x 2 + 2x and 4x + 8 factor as
1x(x + 2) and 4(x + 2)
2A. The LCD of 1 and 4 is 4
The higest power of (x − 1) is (x − 1)1
3.
1
x
and
4x + 8
x + 2x
2
The LCD = 4x(x + 2)
HW WKD!
© 2016 Eitel