MULTIPLYING AND DIVIDING FRACTIONS WITH VARIABLES

LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
 

To multiply and divide fractions with variables:

EXAMPLE:

Multiply, and write your answer in simplest form:     $$ \cssId{s16}{\frac{x^2-9}{5x^2+20x+15}} \cssId{s17}{\cdot} \cssId{s18}{\frac{x+1}{x+4}} $$

SOLUTION:

$\displaystyle \cssId{s20}{\frac{x^2-9}{5x^2+20x+15}} \cssId{s21}{\cdot} \cssId{s22}{\frac{x+1}{x+4}} $ $=$ $\displaystyle \cssId{s24}{\frac{(x-3)(x+3)}{5(x^2+4x+3)}} \cssId{s25}{\cdot} \cssId{s26}{\frac{x+1}{x+4}}$ factor: difference of squares (numerator), common factor (denominator)
  $=$ $\displaystyle \cssId{s31}{\frac{(x-3)(x+3)}{5(x+3)(x+1)}} \cssId{s32}{\cdot} \cssId{s33}{\frac{x+1}{x+4}} $ factor the trinomial in the denominator
  $=$ $\displaystyle \cssId{s36}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}} $ multiply, re-order
  $=$ $\displaystyle \cssId{s39}{\frac{(x-3)}{5(x+4)}} $ cancel the two extra factors of $\,1\,$

It is interesting to compare the original expression (before simplification), and the simplified expression (after cancellation).
Although they are equal for almost all values of $\,x\,$, they do differ a bit, because of the cancellation:

VALUES
OF $\,x\,$
ORIGINAL EXPRESSION:
$\displaystyle \cssId{s47}{\frac{x^2-9}{5x^2+20x+15} \cdot \frac{x+1}{x+4}} $
in factored form:
$\displaystyle \cssId{s49}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}}$
SIMPLIFIED EXPRESSION:
$\displaystyle \cssId{s51}{\frac{(x-3)}{5(x+4)}}$
COMPARISON
$x = -4$ not defined
(division by zero)
not defined
(division by zero)
behave the same:
both are not defined
$x = -1$ not defined
(division by zero)
$\displaystyle \cssId{s60}{\frac{-1-3}{5(-1+4)} = -\frac{4}{15}}$ the presence of $\,\frac{x+1}{x+1}\,$ causes a
puncture point at $\,x = -1\,$;

see the first graph below
$x = -3$ not defined
(division by zero)
$\displaystyle \frac{-3-3}{5(-3+4)} = -\frac{6}{5}$ the presence of $\,\frac{x+3}{x+3}\,$ causes a
puncture point at $\,x = -3\,$;

see the first graph below
all other
values of $\,x\,$
both defined;
values are equal
behave the same:
values are equal

GRAPH OF:
$\displaystyle \cssId{s72}{\frac{\hphantom{5}(x+1)(x+3)(x-3)}{5(x+1)(x+3)(x+4)}}$
GRAPH OF:
$\displaystyle \cssId{s74}{\frac{(x-3)}{5(x+4)}}$

Master the ideas from this section
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Adding and Subtracting Fractions With Variables

 
 

For more advanced students, a graph is displayed.
For example, the expression $\,\frac{x+1}{x+2}\cdot\frac{x+3}{x+1}\,$ is optionally accompanied by the graph of $\,y = \frac{x+1}{x+2}\cdot\frac{x+3}{x+1}\,$.
A puncture point occurs at $\,x = -1\,$, due to the presence of $\,\frac{x+1}{x+1}\,$.
The graph of the simplified expression would not have this puncture point.
Horizontal/vertical asymptote(s) are shown in light grey.
Note: A puncture point may occasionally occur outside the viewing window;
use the arrows in the lower-right graph corner to navigate left/up/down/right.
Click the “show/hide graph” button if you prefer not to see the graph.

CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18
AVAILABLE MASTERED IN PROGRESS

Perform the indicated operation and write in simplest form:
(MAX is 18; there are 18 different problem types.)