A rational expression is a fraction involving polynomials, where the polynomial in the denominator is not zero. Just like a fraction involving numbers, a rational expression can be simplified, multiplied, and divided. The rules for performing these operations often mirror the rules for simplifying, multiplying, and dividing fractions. Performing these operations on rational expressions often involves factoring polynomial expressions out of the numerator and denominator.

Simplifying a Rational Expression

Rational expressions can be simplified by factoring the numerator and denominator where possible, and canceling terms. 

As a first example, consider the rational expression $\frac { 3x^3 }{ x }$. This can be simplified by canceling out one factor of $x$ in the numerator and denominator, which gives the expression $3x^2$.

Note that the domain of the equation $f(x) = \frac{3x^3}{x}$ does not include $x=0$, as this would cause division by $0$. The latter form is a simplified version of the former graphically.

Consider a more complicated example:

$\displaystyle \frac { x^2+5x+6 }{ 2x^2+5x+2 }$

This expression must first be factored to provide the expression

$\displaystyle \frac {(x+2)(x+3)}{(2x+1)(x+2)}$

which, after canceling the common factor of $(x+2)$ from both the numerator and denominator, gives the simplified expression

$\displaystyle \frac {x+3}{2x+1}$

Multiplying Rational Expressions

Rational expressions can be multiplied and divided in a similar manner to fractions. Recall that when two fractions are multiplied together, their numerators are multiplied to yield the numerator of their product, and their denominators are multiplied to yield the denominator of their product.

For a simple example, consider the following, where a rational expression is multiplied by a fraction of whole numbers:

$\displaystyle \frac {x^2+3}{2x-3} \times \frac{2}{3}$

Following the rule for multiplying fractions, simply multiply their respective numerators and denominators: 

$\displaystyle \frac {2(x^2+3)}{3(2x-3)}$

This can be multiplied through to yield $\displaystyle \frac {2x^2+6}{6x-9}$

Notice that we multiplied the numerators together and the denominators together, but we did not multiply the numerator by the denominator or vice-versa.

We follow the same rules to multiply two rational expressions together. The operations are slightly more complicated, as there may be a need to simplify the resulting expression.

Example 1

Consider the following:

$\displaystyle \frac {x+1}{x-1} \times \frac {x+2}{x+3}$  

Multiplying these two expressions, we have the product:

 $\displaystyle\frac {(x+1)(x+2)}{(x-1)(x+3)}$

Multiplying out the numerator and denominator, this can be written as:

$\displaystyle \frac {x^2+3x+2}{x^2+2x-3}$ 

Notice that this expression cannot be simplified further.

Dividing Rational Expressions

Dividing rational expressions follows the same rules as dividing fractions. Recall the rule for dividing fractions: the dividend is multiplied by the reciprocal of the divisor. The same applies to dividing rational expressions; the first expression is multiplied by the reciprocal of the second.

Example 2

Consider the following:

 $\displaystyle \frac {x+1}{x-1} \div \frac {x+2}{x+3}$

Rather than divide the expressions, we multiply $\displaystyle \frac {x+1}{x-1}$ by the reciprocal of $\displaystyle \frac {x+2}{x+3}$:

$\displaystyle \frac{x+1}{x-1} \times \frac {x+3}{x+2}$

Then, multiplication is carried out in the same way as described above:

$\displaystyle \frac{(x+1)(x+3)}{(x-1)(x+2)} = \frac{x^2 + 3x +3}{x^2 + x - 2}$

The expression cannot be simplified further.