audio read-through Dividing More Than One Term by a Single Term

For all real numbers $\,A\,$ and $\,B\,,$ and for $\,C\ne 0\,$:

$$ \begin{gather} \cssId{s2}{\frac{A+B}{C} = \frac{A}{C} + \frac{B}{C}}\cr\cr \cssId{s3}{\text{ and }} \cr\cr \cssId{s4}{\frac{A-B}{C} = \frac{A}{C} - \frac{B}{C}} \end{gather} $$

Key idea: every term in the numerator must be divided by the denominator.

Examples

$\displaystyle \begin{align} &\cssId{s8}{\frac{6x^5 - 8x^2}{2x}}\cr\cr &\qquad\cssId{s9}{=\ \ \frac{6x^5}{2x} - \frac{8x^2}{2x}}\cr\cr &\qquad\cssId{s10}{=\ \ 3x^4 - 4x} \end{align} $
$\displaystyle \begin{align} &\cssId{s11}{\frac{2t - t^3 + 10t^4}{5t^3}}\cr\cr &\qquad\cssId{s12}{=\ \ \frac{2t}{5t^3} - \frac{t^3}{5t^3} + \frac{10t^4}{5t^3}}\cr\cr &\qquad\cssId{s13}{=\ \ \frac{2}{5t^2} - \frac{1}{5} + 2t} \end{align} $

The goal: go immediately from the original expression (like $\,\frac{2t - t^3 + 10t^4}{5t^3}\,$) to the final expression ($\,\frac{2}{5t^2} - \frac{1}{5} + 2t\,$), 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:   $\displaystyle \color{red}{+}\ \color{green}{2}\color{blue}{t}$

Concept Practice