Tips For Subtracting Rational Expressions

A rational number is any number that you can express as a fraction ​p​/​q​ where ​p​ and ​q​ are integers and ​q​ does not equal 0. To subtract two rational numbers, they must have a common denomination, and to do this, you have to multiply each of them by a common factor. The same is true when subtracting rational expressions, which are polynomials. The trick to subtracting polynomials is to factor them to get them in their simplest form before giving them a common denominator.

In a general way, you can express one rational number by ​p​/​q​ and another by ​x​/​y​, where all numbers are integers and neither ​y​ nor ​q​ equals 0. If you want to subtract the second from the first, you would write:

\(\frac{p}{q} – \frac{x}{y}\)

Now multiply the first term by ​y​/​y​ (which equals 1, so it doesn't change its value), and multiply the second term by ​q​/​q​. The expression now becomes:

\(\frac{py}{qy} – \frac{qx}{qy}\)

which can be simplified to

\(\frac{py -qx}{ qy}\)

The term ​qy​ is called the least common denominator of the expression

\(\frac{p}{q} – \frac{x}{y}\)

​1. Subtract 1/4 from 1/3​

Write the subtraction expression:

\(\frac{1}{3} – \frac{1}{4}\)

Now, multiply the the first term by 4/4 and the second by 3/3, then subtract the numerators:

\(\frac{4}{12} – \frac{3}{12} = \frac{1}{12}\)

​2. Subtract 3/16 from 7/24​

The subtraction is

\(\frac{7}{24} – \frac{3}{16}\)

​Notice that the denominators have a common factor, 8​. You can write the expressions like this:

\(\frac{7}{8 × 3} \text{ and } \frac{3}{8 × 2}\)

This makes the subtraction easier. Because 8 is common to both expressions, you only have to multiply the first expression by 2/2 and the second expression by 3/3.

\(\begin{aligned}
\frac{7}{24} – \frac{ 3}{16} &= \frac{14 – 9}{48} \
\,\
&= \frac{5}{48}
\end{aligned}\)

If you factor polynomial fractions, subtracting them becomes easier. This is called reducing to lowest terms. Sometimes you'll find a common factor in both the numerator and denominator of one of the fractional terms that cancels and produces an easier-to-handle fraction. For example:

\(\begin{aligned}
\frac{x^2 – 2x – 8}{x^2 – 9x + 20} &= \frac{(x – 4) (x + 2)}{(x – 5) (x – 4)} \
\,\
&= \frac{x + 2}{x – 5}
\end{aligned}\)

​Perform the following subtraction:​

\(\frac{2x}{x^2 – 9} – \frac{1}{x + 3}\)

Start by factoring ​x​² – 9 to get (​x​ + 3) (​x​ −3).

Now write

\(\frac{2x}{(x + 3) (x – 3)} – \frac{1}{x + 3}\)

The lowest common denominator is (​x​ + 3) (​x​ −3), so you only need to multiply the the second term by (​x​ − 3) / (​x​ − 3) to get

\(\frac{2x – (x – 3)}{(x + 3) (x – 3)}\)

which you can simplify to

\(\frac{x + 3}{x^2 – 9}\)