Mathematics » Rational Expressions and Equations » Simplify Rational Expressions

# Simplifying Rational Expressions With Opposite Factors

## Simplifying Rational Expressions With Opposite Factors

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say $$\frac{7}{-7}$$. We know this fraction simplifies to $$-1$$. We also recognize that the numerator and denominator are opposites.

In the tutorial on integers, we introduced opposite notation: the opposite of $$a$$ is $$\text{−}a$$. We remember, too, that $$\text{−}a=-1·a$$.

We simplify the fraction $$\frac{a}{\text{−}a}$$, whose numerator and denominator are opposites, in this way:

$$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{a}{\text{−}a}\hfill \\ \text{We could rewrite this.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1·a}{-1·a}\hfill \\ \text{Remove the common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1}{-1}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}-1\hfill \end{array}$$

So, in the same way, we can simplify the fraction $$\frac{x-3}{\text{−}\left(x-3\right)}$$:

$$\begin{array}{cccc}\text{We could rewrite this.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1·\left(x-3\right)}{-1·\left(x-3\right)}\hfill \\ \text{Remove the common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{1}{-1}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}-1\hfill \end{array}$$

But the opposite of $$x-3$$ could be written differently:

$$\begin{array}{cccc}& & & \hfill \phantom{\rule{12.5em}{0ex}}\text{−}\left(x-3\right)\hfill \\ \text{Distribute.}\hfill & & & \hfill \phantom{\rule{12.5em}{0ex}}\text{−}x+3\hfill \\ \text{Rewrite.}\hfill & & & \hfill \phantom{\rule{12.5em}{0ex}}3-x\hfill \end{array}$$

This means the fraction $$\frac{x-3}{3-x}$$ simplifies to $$-1$$.

In general, we could write the opposite of $$a-b$$ as $$b-a$$. So the rational expression $$\frac{a-b}{b-a}$$ simplifies to $$-1$$.

### Opposites in a Rational Expression

The opposite of $$a-b$$ is $$b-a$$.

$$\begin{array}{cccccc}\hfill \frac{a-b}{b-a}=-1\hfill & & & & & \hfill a\ne b\hfill \end{array}$$

An expression and its opposite divide to $$-1$$.

We will use this property to simplify rational expressions that contain opposites in their numerators and denominators.

## Example

Simplify: $$\frac{x-8}{8-x}.$$

### Solution

$$\begin{array}{cccc}& & & \frac{x-8}{8-x}\hfill \\ \text{Recognize that}\phantom{\rule{0.2em}{0ex}}x-8\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8-x\phantom{\rule{0.2em}{0ex}}\text{are opposites.}\hfill & & & -1\hfill \end{array}$$

Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.

## Example

Simplify: $$\frac{14-2x}{{x}^{2}-49}.$$

### Solution  Factor the numerator and denominator.  Recognize that $$7-x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}x-7\phantom{\rule{0.2em}{0ex}}\text{are opposites}$$.  Simplify.  ## Example

Simplify: $$\frac{{x}^{2}-4x-32}{64-{x}^{2}}.$$

### Solution  Factor the numerator and denominator.  Recognize the factors that are opposites.  Simplify.  