DIVIDING MORE THAN ONE TERM BY A SINGLE TERM

For all real numbers $\,A\,$ and $\,B\,$, and for $\,C\ne 0\,$: $$ \frac{A+B}{C} = \frac{A}{C} + \frac{B}{C} \qquad \text{ and } \qquad \frac{A-B}{C} = \frac{A}{C} - \frac{B}{C} $$ Key idea: every term in the numerator must be divided by the denominator.

EXAMPLES

$\displaystyle\frac{6x^5 - 8x^2}{2x} \ \ =\ \ \frac{6x^5}{2x} - \frac{8x^2}{2x} \ \ =\ \ 3x^4 - 4x$


$\displaystyle\frac{2t - t^3 + 10t^4}{5t^3} \ \ =\ \ \frac{2t}{5t^3} - \frac{t^3}{5t^3} + \frac{10t^4}{5t^3} \ \ =\ \ \frac{2}{5t^2} - \frac{1}{5} + 2t $

The goal: go immediately from the original expression (like $\displaystyle\,\frac{2t - t^3 + 10t^4}{5t^3}\,$) to the final expression ($\displaystyle\,\frac{2}{5t^2} - \frac{1}{5} + 2t\,$),
without writing down any intermediate step(s).

To do this, use the ‘three-pass’ system (sign/size/variable), illustrated next:

$\displaystyle \frac{\class{highlight}{2t} - t^3 + 10t^4}{\class{highlight}{5t^3}}$
  1. sign:
    positive over positive = positive;
    don't write down the ‘$+$’ sign since it's the leading term
  2. size:
    $\,2\,$ over $\,5\,$ cannot be simplified
  3. variable:
    There are more factors of $\,t\,$ downstairs.
    How many more? $\,3 - 1 = 2\,$.
    Put $\,t^2\,$ downstairs.
result: $\displaystyle\frac{\color{green}{2}}{\color{green}{5}\color{blue}{t^2}}$
$\displaystyle \frac{2t\class{highlight}{ - t^3} + 10t^4}{\class{highlight}{5t^3}}$
  1. sign:
    negative over positive = negative;
    write down the minus sign;
    for the next pass, ignore the minus sign
  2. size:
    $\,1\,$ over $\,5\,$ cannot be simplified
  3. variable:
    the factors of $\,t\,$ completely cancel
result: $\displaystyle \color{red}{-} \frac{\color{green}{1}}{\color{green}{5}}$
$\displaystyle \frac{2t - t^3\class{highlight}{ + 10t^4}}{\class{highlight}{5t^3}}$
  1. sign:
    positive over positive = positive;
    write down the plus sign
  2. size:
    $\,10\,$ divided by $\,5\,$ is $\,2\,$
  3. variable:
    There are more factors of $\,t\,$ upstairs.
    How many more? $\,4 - 3 = 1\,$
    Put $\,t\,$ upstairs.
result: $\displaystyle \color{red}{+}\color{green}{2}\color{blue}{t}$

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

When you're done practicing, move on to:
Practice with the Mathematical Words
‘and’, ‘or’ ‘is equivalent to’

 
 
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
AVAILABLE MASTERED IN PROGRESS