﻿ Dividing More Than One Term by a Single Term
DIVIDING MORE THAN ONE TERM BY A SINGLE TERM

by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!
• PRACTICE (online exercises and printable worksheets)

For all real numbers $\,A\,$ and $\,B\,$, and for $\,C\ne 0\,$: $$\cssId{s2}{\frac{A+B}{C} = \frac{A}{C} + \frac{B}{C}} \qquad \cssId{s3}{\text{ and }} \qquad \cssId{s4}{\frac{A-B}{C} = \frac{A}{C} - \frac{B}{C}}$$ Key idea: every term in the numerator must be divided by the denominator.

## EXAMPLES

$\displaystyle \cssId{s8}{\frac{6x^5 - 8x^2}{2x}} \ \ \cssId{s9}{=\ \ \frac{6x^5}{2x} - \frac{8x^2}{2x}} \ \ \cssId{s10}{=\ \ 3x^4 - 4x}$

$\displaystyle \cssId{s11}{\frac{2t - t^3 + 10t^4}{5t^3}} \ \ \cssId{s12}{=\ \ \frac{2t}{5t^3} - \frac{t^3}{5t^3} + \frac{10t^4}{5t^3}} \ \ \cssId{s13}{=\ \ \frac{2}{5t^2} - \frac{1}{5} + 2t}$

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

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

CONCEPT QUESTIONS EXERCISE: