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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..

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