Multiply and Divide Rational Expressions

We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.

Examples of rational expressions include

The first expression, $-\tfrac{24}{56}$, is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero. We will do the same operations with rational expressions that we did with fractions: simplify, add, subtract, multiply, divide, and use them in applications.

Determine the values for which a rational expression is undefined

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be zero—but not the denominator.

When we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. To avoid dividing by zero in a rational expression, we must not allow values of the variable that make the denominator zero.

Before beginning any operation with a rational expression, first examine it to find the values that would make the denominator zero. That way, when we solve a rational equation, we will know whether the algebraic solutions are allowed.

Example. Determine the value for which each rational expression is undefined: (a) $\tfrac{8a^2b}{3c}$, (b) $\tfrac{4b-3}{2b+5}$, and (c) $\tfrac{x+4}{x^2+5x+6}$.

The expression is undefined when its denominator is zero.

Thus $\tfrac{8a^2b}{3c}$ is undefined for $c=0$.

Thus $\tfrac{4b-3}{2b+5}$ is undefined for $b=-\tfrac{5}{2}$.

Thus $\tfrac{x+4}{x^2+5x+6}$ is undefined for $x=-2$ or $x=-3$.

Simplify rational expressions

A fraction is simplified if there are no common factors, other than $1$, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than $1$, in its numerator and denominator.

For example, $\tfrac{x+2}{x+3}$ is simplified because there are no common factors of $x+2$ and $x+3$. The expression $\tfrac{2x}{3x}$ is not simplified because $x$ is a common factor of $2x$ and $3x$.

We use the Equivalent Fractions Property to simplify numerical fractions, and we use it to simplify rational expressions as well.

The values that would make the denominators zero are specifically disallowed. To simplify rational expressions, first write the numerator and denominator in factored form. Then remove the common factors using the Equivalent Fractions Property.

Be careful when removing common factors. Factors are multiplied to make a product. You can remove a factor from a product, but you cannot remove a term from a sum. For example, common factors can be removed from $\tfrac{2\cdot3\cdot7}{3\cdot5\cdot7}$ and from $\tfrac{3x(x-9)}{5(x-9)}$ (where $x\ne9$). But the $x$ in $\tfrac{x+5}{x}$ cannot be removed: the numerator is a sum. Doing so would be like cancelling the $2\text{s}$ in $\tfrac{2+5}{2}$.

Example. Simplify $\tfrac{x^2+5x+6}{x^2+8x+12}$.

The original denominator gives the restrictions $x\ne-2$ and $x\ne-6$.

Usually, we leave the simplified rational expression in factored form. This makes it easy to check that we have removed all common factors. Every time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. To focus on the work at hand, the examples below omit those restrictions.

Example. Simplify $\tfrac{3a^2-12ab+12b^2}{6a^2-24b^2}$.

Sometimes the numerator and denominator have opposite factors. The opposite of $a$ is $-a=-1\cdot a$. For example, $\tfrac{-7}{7}=-1$, and in the same way $\tfrac{a}{-a}=-1$. Rewriting $b-a$ as $-a+b=-1(a-b)$ shows that $b-a$ is the opposite of $a-b$.

Be careful not to treat $a+b$ and $b+a$ as opposites. Addition is commutative, so $a+b=b+a$. Thus, if $a\ne-b$, then $\tfrac{a+b}{b+a}=1$.

Example. Simplify $\tfrac{x^2-4x-32}{64-x^2}$.

Multiply rational expressions

To multiply rational expressions, we do what we did with numerical fractions: multiply the numerators and multiply the denominators. Then remove common factors to simplify the result.

Throughout this chapter, we assume that all numerical values that would make a denominator zero are excluded. In the next example, $x\ne0$, $x\ne3$, and $x\ne4$.

Example. Simplify $\tfrac{2x}{x^2-7x+12}\cdot\tfrac{x^2-9}{6x^2}$.

Example. Multiply $\tfrac{3a^2-8a-3}{a^2-25}\cdot\tfrac{a^2+10a+25}{3a^2-14a-5}$.

Divide rational expressions

As with numerical fractions, to divide rational expressions, multiply the first fraction by the reciprocal of the second.

Once the division is rewritten as multiplication of the first expression by the reciprocal of the second, factor everything and look for common factors.

Example. Divide $\tfrac{p^3+q^3}{2p^2+2pq+2q^2}\div\tfrac{p^2-q^2}{6}$.

A complex fraction is a fraction that contains a fraction in the numerator, the denominator, or both. A fraction bar means division, so a complex fraction is another way of writing division of two fractions.

Example. Divide the complex fraction

When there are more than two rational expressions, follow the same procedure. First rewrite any division as multiplication by the reciprocal. Then factor and multiply.

Example. Perform the indicated operations: $\tfrac{3x-6}{4x-4}\cdot\tfrac{x^2+2x-3}{x^2-3x-10}\div\tfrac{2x+12}{8x+16}$.

Multiply and divide rational functions

A rational expression has the form $\tfrac{p}{q}$, where $p$ and $q$ are polynomials and $q\ne0$. Similarly, we define a rational function as a function of the form $R(x)=\tfrac{p(x)}{q(x)}$.

The domain of a rational function is all real numbers except values that would cause division by zero. We must eliminate any values that make $q(x)=0$.

Example. Find the domain of $R(x)=\tfrac{2x^2-14x}{4x^2-16x-48}$.

The domain is all real numbers except those values that make the denominator zero.

The domain of $R(x)$ is all real numbers where $x\ne6$ and $x\ne-2$.

To multiply rational functions, multiply the resulting rational expressions on the right side of the equation using the same techniques used to multiply rational expressions.

Example. Find $R(x)=f(x)\cdot g(x)$ where $f(x)=\tfrac{2x-6}{x^2-8x+15}$ and $g(x)=\tfrac{x^2-25}{2x+10}$.

To divide rational functions, divide the resulting rational expressions on the right side of the equation using the same techniques used to divide rational expressions.

Example. Find $R(x)=\tfrac{f(x)}{g(x)}$ where $f(x)=\tfrac{3x^2}{x^2-4x}$ and $g(x)=\tfrac{9x^2-45x}{x^2-7x+10}$.

Key terms

rational expression — an expression of the form $\tfrac{p}{q}$, where $p$ and $q$ are polynomials and $q\ne0$. simplified rational expression — a rational expression with no common factors in its numerator and denominator. complex fraction — a fraction containing a fraction in its numerator, denominator, or both. rational function — a function of the form $R(x)=\tfrac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomial functions and $q(x)$ is not zero.

This section is adapted from Intermediate Algebra 2e, Section 7.1: Multiply and Divide Rational Expressions by Lynn Marecek and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: reformatted the worked solutions as accessible typeset step arrays; omitted the Be Prepared quiz, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.