Section 14.3 Least Common Denominators

1. Slide - 1Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
2
Rational
Expressions and
Applications
14

2. Slide - 2Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
1. Find the least common denominator for a
list of fractions.
2. Write equivalent rational expressions.
Objectives
14.3 Least Common Denominators

3. Slide - 3Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Find the LCD for a List of Fractions
Finding the Least Common Denominator (LCD)
Step 1 Factor each denominator into prime factors.
Step 2 List each different denominator factor the
greatest number of times it appears in any of
the denominators.
Step 3 Multiply the denominator factors from Step 2
to find the LCD.

4. Slide - 4Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Find the LCD for the fractions.
2
7 5
,
12 18x x
Find the LCD for a List of Fractions
Step 1 12x = 2 · 2 · 3 · x = 22 · 3 · x
18x2 = 2 · 3 · 3 · x2 = 2 · 32 · x2
Step 2 The greatest number of times that the factor 2 appears is
twice. The greatest number of times both 3 and x appear
is twice.
Step 3 LCD = 22 · 32 · x2 = 36x2

5. Slide - 5Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Find the LCD for the fractions.
2 2
4 5
,
15 3 6y y y
Find the LCD for a List of Fractions
15y2 = 3 · 5 · y2
3y2 – 6y = 3 · y · (y – 2)
LCD = 3 · 5 · y2 · (y – 2) = 15y2(y – 2)

6. Slide - 6Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Find the LCD for fractions in the list.
2
2 2 2
2 5 1
, ,
12 9 7 12
z z
z z z z z    
Find the LCD for a List of Fractions
z2 + z – 12 = (z + 4)(z – 3)
z2 – 9 = (z + 3)(z – 3)
z2 + 7z + 12 = (z + 4)(z + 3)
LCD = (z + 4)(z – 3)(z + 3)

7. Slide - 7Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Write Equivalent Rational Expressions
Writing a Rational Expression with a Specified
Denominator
Step 1 Factor both denominators.
Step 2 Decide what factor(s) the denominator must
be multiplied by in order to equal the
specified denominator.
Step 3 Multiply the rational expression by that factor
divided by itself. (That is, multiply by 1.)

8. Slide - 8Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Write each rational expression as an equivalent expression with
the indicated denominator.
3
7 35
?

x
x
Write Equivalent Rational Expressions
First, factor the denominator on the right.
Then compare the denominator on the left
with the one on the right to decide what
factors are missing.
3 5
7 5
x x
x

2
15
35
x
x

Factors of 5 and x are missing.
Multiply the numerator and
denominator by 5x.
5
3
7 7
?

 
x
x

9. Slide - 9Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example
Write each rational expression as an equivalent expression with
the indicated denominator.
3 2
17
(a)
3 4 24 32
?

 y y y
Write Equivalent Rational Expressions
2
2
17
3 4
8
8
y
yy


2
3 2
136
24 32
y
y y


Factor the denominator on the right.
The missing factor is 8y2.
2
17
3 4 8 (3
?
4)

 y y y

10. Slide - 10Copyright © 2018, 2014, 2010 Pearson Education Inc.A L W A Y S L E A R N I N G
Example (cont)
Write each rational expression as an equivalent expression with
the indicated denominator.
2 3 2
3
(b)
16 16
?
16

   z z z z
Write Equivalent Rational Expressions
2
3
1
1
16z
z
z



Factor the denominator in
each rational expression.
The factor z + 1 is missing.
      
?3
4 4 1 4 4

    z z z z z
3 2
3 3
16 16
z
z z z


  