Algebra of rational functions summarized

The following rules summarize the various rules for manipulating rational expressions. To practice any particular ones or see examples, open a practice session, and under "Practice options", choose the topic you want to concentrate on.

Cancellation rule

If $R$ is any nonzero expression that is a factor of both the numerator and denominator, then you can cancel it to simplify the rational expression:

$\dfrac{P\color{indianred}{R}}{Q\color{indianred}{R}} = \dfrac{P}{Q} \qquad \quad \ \ $ Cancel the R.

Caution $R$ must be a factor and not a summand; For instance,

$\dfrac{P+\color{indianred}{R}}{Q+\color{indianred}{R}} \neq \dfrac{P}{R} \qquad$ You cannot cancel a summand.

Multiplying rational expressions

As is the case with ordinary fractions, we multiply two rational expressions by simply multiplying their numerators and denominators:

$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} \times \dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \dfrac{\color{#026fc1}{P}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad $ \t Product of numerators over product of denominators

Dividing rational expressions

As with ordinary fractions, division by a rational expression means multiplication by its reciprocal:

$\dfrac{\left(\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}\right)}{\left(\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}\right)} = \dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} \times \dfrac{\color{#c1026f}{S}}{\color{#c1026f}{R}} \qquad$ \t

Flip the denominator and multiply.

\\ $\qquad = \dfrac{\color{#026fc1}{P}\color{#c1026f}{S}}{\color{#026fc1}{Q}\color{#c1026f}{R}} \qquad \quad$ \t Calculate the product.

Addition and subtraction with common denominator:

$\dfrac{\color{#026fc1}{P}}{\color{#c1026f}{Q}} + \dfrac{\color{#026fc1}{R}}{\color{#c1026f}{Q}} = \dfrac{\color{#026fc1}{P} + \color{#026fc1}{R}}{\color{#c1026f}{Q}} \qquad $ \t Sum of numerators divided by common denominator \\ \\ $\dfrac{\color{#026fc1}{P}}{\color{#c1026f}{Q}} - \dfrac{\color{#026fc1}{R}}{\color{#c1026f}{Q}} = \dfrac{\color{#026fc1}{P} - \color{#026fc1}{R}}{\color{#c1026f}{Q}} \qquad $ \t Difference of numerators divided by common denominator

Addition and subtraction: General case:

$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} + \dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \dfrac{\color{#026fc1}{P}\color{#c1026f}{S} + \color{#026fc1}{Q}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad \quad \ \ $	Cross multiply to get the numerator:	$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$
	Multiply across to get the denominator:	$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$

$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}} - \dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}} = \dfrac{\color{#026fc1}{P}\color{#c1026f}{S} - \color{#026fc1}{Q}\color{#c1026f}{R}}{\color{#026fc1}{Q}\color{#c1026f}{S}} \qquad \quad \ \ $	Cross multiply to get the numerator:	$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$
	Multiply across to get the denominator:	$\dfrac{\color{#026fc1}{P}}{\color{#026fc1}{Q}}$ $\dfrac{\color{#c1026f}{R}}{\color{#c1026f}{S}}$