Transcript Slide 1

§ 6.3
Complex Rational Expressions
Simplifying Complex Fractions
Complex rational expressions, also called complex
fractions, have numerators or denominators containing
one or more fractions.
x 5

5 x.
1 1

5 x
Woe is me,
for II am
amcurrently suffering from…
a feeling of complexity
complex.
Blitzer, Algebra for College Students, 6e – Slide #2 Section 6.3
Complex Rational Expressions
Simplifying a Complex Rational
Expression by Multiplying by One in
the Form LCD T
LCD
1) Find the LCD of all rational expressions within the
complex rational expression.
2) Multiply both the numerator and the denominator of
the complex rational expression by this LCD.
3) Use the distributive property and multiply each term in
the numerator and denominator by this LCD. Simplify.
No fractional expressions should remain.
4) If possible, factor and simplify.
Blitzer, Algebra for College Students, 6e – Slide #3 Section 6.3
Simplifying Complex Fractions
EXAMPLE
x 5

Simplify: 5 x .
1 1

5 x
SOLUTION
The denominators in the complex rational expression are 5 and
x. The LCD is 5x. Multiply both the numerator and the
denominator of the complex rational expression by 5x.
 x 5
x 5
  

Multiply the numerator and
5 x  5x   5 x 
denominator by 5x.
1 1 5x  1 1 

  
5 x
5 x
Blitzer, Algebra for College Students, 6e – Slide #4 Section 6.3
Simplifying Complex Fractions
CONTINUED
x
5
5x   5x 
5
x

1
1
5x   5x 
5
x
x
5
5x   5x 
5
x

1
1
5x   5x 
5
x
x 2  25

x5

x  5x  5
1   x  5
Use the distributive
property.
Divide out common factors.
Simplify.
Factor and simplify.
Blitzer, Algebra for College Students, 6e – Slide #5 Section 6.3
Simplifying Complex Fractions
CONTINUED
x5
1
Simplify.
 x5
Simplify.

Blitzer, Algebra for College Students, 6e – Slide #6 Section 6.3
Simplifying Complex Fractions
EXAMPLE
1
1

Simplify: x  6 x .
6
SOLUTION
The denominators in the complex rational expression are x + 6
and x. The LCD is (x + 6)x. Multiply both the numerator and
the denominator of the complex rational expression by (x + 6)x.
1
 1
1
1
 


x  6 x  x  6x   x  6 x 
x  6x
6
6
Multiply the numerator and
denominator by (x + 6)x.
Blitzer, Algebra for College Students, 6e – Slide #7 Section 6.3
Simplifying Complex Fractions
CONTINUED
1
1
x  6x 
  x  6 x 
x6
x

x  6x  6
Use the distributive
property.
x  6x  1  x  6x  1
x6
x

x  6x  6
Divide out common factors.
x  x  6

x  6x  6
Simplify.
x x6
 2
6 x  36 x
Simplify.
Blitzer, Algebra for College Students, 6e – Slide #8 Section 6.3
Simplifying Complex Fractions
CONTINUED
6
 2
6 x  36 x
Subtract.

6   1
6 x x  6 
Factor and simplify.

1
x x  6 
Simplify.
Blitzer, Algebra for College Students, 6e – Slide #9 Section 6.3
Simplifying Complex Fractions
Simplifying a Complex Rational
Expression by Dividing
1) If necessary, add or subtract to get a single rational
expression in the numerator.
2) If necessary, add or subtract to get a single rational
expression in the denominator.
3) Perform the division indicated by the main fraction
bar: Invert the denominator of the complex rational
expression and multiply.
4) If possible, simplify.
Blitzer, Algebra for College Students, 6e – Slide #10 Section 6.3
Simplifying Complex Fractions
EXAMPLE
Simplify:
m
2

m 2  9 m 2  4m  4 .
3
m
 2
2
m  5m  6 m  m  6
SOLUTION
1) Subtract to get a single rational expression in the
numerator.
m
2
m
2



m2  9 m2  4m  4 m  3m  3 m  22
mm  2
2m  3m  3
mm  2  2m  3m  3



2
2
m  3m  3m  2 m  2 m  3m  3
m  3m  3m  22
2
Blitzer, Algebra for College Students, 6e – Slide #11 Section 6.3
2
Simplifying Complex Fractions
CONTINUED
m3  4m2  4m  2m2  18 m3  6m2  4m  18


2
m  3m  3m  2
m  3m  3m  22
2) Add to get a single rational expression in the denominator.
3
m
3
m



m 2  5m  6 m 2  m  6 m  3m  2 m  2m  3
3m  3
mm  3
3m  3  mm  3



m  3m  2m  3 m  2m  3m  3 m  3m  2m  3
3m  9  m2  3m
m2  9


m  3m  2m  3 m  3m  2m  3
Blitzer, Algebra for College Students, 6e – Slide #12 Section 6.3
Simplifying Complex Fractions
CONTINUED
3) & 4) Perform the division indicated by the main fraction
bar: Invert and multiply. If possible, simplify.
3
2
m

6
m
 4m  18
m
2
 2
2
2




m

3
m

3
m

2
m  9 m  4m  4 
2
3
m
m
9

m 2  5m  6 m 2  m  6
m  3m  2m  3
m3  6m2  4m  18 m  3m  2m  3


2
m2  9
m  3m  3m  2
m3  6m2  4m  18
m3  6m2  4m  18 m  3m  2m  3



2
2
m  2m2  9
m 9
m  3m  3m  2
Blitzer, Algebra for College Students, 6e – Slide #13 Section 6.3
Simplifying Complex Fractions
Important to Remember:
Complex rational expressions have numerators or denominators
containing one or more fractions.
Complex rational expressions can be simplified by one of two
methods presented in this section:
(a) multiplying the numerator and denominator by the LCD
(b) obtaining single expressions in the numerator and
denominator and then dividing, using the
definition of division for fractions
Blitzer, Algebra for College Students, 6e – Slide #14 Section 6.3