1.4: Dealing with Rational Expressions

Back in Section 1.1, we defined five fraction properties we can use when manipulating fractions. They are copied here for convenience:

Properties of Fractions
Suppose $a,b,c,d$ are real numbers.

$\dfrac{a}{b}\cdot \dfrac{c}{d} = \dfrac{ac}{bd}$
$\dfrac{a}{b}\div \dfrac{c}{d} = \dfrac{a}{b}\cdot \dfrac{d}{c}$
$\dfrac{a}{c} \pm \dfrac{b}{c} = \dfrac{a \pm b}{c}$
$\dfrac{a}{b} + \dfrac{c}{d}\qquad $ Find the LCD, then use Fraction Law 3.
$\dfrac{ac}{bc} = \dfrac{a}{b}$

This section is all about manipulating fractional expressions.

Simplifying (property 5)

Property 5 says you can only cancel a global factor in both the numerator and denominator.

To simplify, convert numerator/denominator to factors, then cancel.

If you cannot use either of the four factoring techniques, just leave your answer in terms.

Simplify the expression \[\dfrac{x^2 - 1}{x^2 + x - 2}\]

Sometimes you need to first expand to combine like terms in order to factor.

The next problem appears in Calculus.

Simplify the expression \[\dfrac{(x+h)^2 - 1 - (x - 1)}{h}\]

Multiplying (property 1)

When multiplying, you should simplify first, then multiply.

Multiply and simplify: \[\dfrac{x^2 + 2x - 3}{x^2 + 8x + 16}\cdot \dfrac{3x + 12}{x - 1}\]

Dividing (property 2)

Divide and simplify: \[\dfrac{x-4}{x^2 - 4} \div \dfrac{x^2 - 3x - 4}{x^2 + 5x + 6}\]

Adding (property 3 + 4)

Adding fractions required you to find the LCD.

LCD requires factors.

You can group terms together with parentheses to create a factor.

Perform the indicated operation and simplify:

$\dfrac{3}{x-1} + \dfrac{x}{x+2}$
$\dfrac{1}{x^2 - 1} - \dfrac{2}{(x + 1)^2}$

Compound Fractions

A compound fraction is a fraction nested within another fraction, for example \[\dfrac{\dfrac{x}{y} + 1}{1 - \dfrac{y}{x}}\] When you're asked to "simplify a compound fraction," this means getting rid of the nested fraction by removing the internal denominators.

Simplify the following:

$\dfrac{\dfrac{x}{y} + 1}{1 - \dfrac{y}{x}}$
$\dfrac{\dfrac{1}{a+h} - \dfrac{1}{a}}{h}$

The second problem will appear in calculus again! Take note of the $h$'s cancelling through fraction law #5.

Rationalizing

Rationalizing means to remove square roots. There are two types of problems you will encounter: one factor or two term.

One Factor

Rationalize the denominator: $\dfrac{2}{\sqrt{x}}$

Remember: when you are asked to simplify a fractional expression, remember to simplify nested fractions!

Rationalize the denominator and simplify: $\dfrac{\frac{1}{\sqrt{x}}}{x}$

Two term

Rationalize the denominator: $\dfrac{1}{\sqrt{x} + 1}$

Rationalization appears in calculus. Here's a rationalization problem from Calculus.

Rationalize the numerator: $\dfrac{\sqrt{4 + h} - 2}{h}$

What does "Simplify" mean?

Upon completing this section, we see that "simplify" means:

Answer is one fraction.
Compound fractions have been eliminated.
Answer has no negative exponents.
Expand if you can create like terms. Then combine them.
Try to keep existing factors if you have them. If you have terms, try to factor using the four methods to factor.
- If you are unable to factor or don't know how to, leave answer as terms.

General advice:

Use all exponent laws until you cannot.
Use all fraction properties until you cannot. Never leave your answer as two or more independent fractions.
If you can expand a local term to create like terms, do so.