Simplify Rational Expressions

By the end of this section, you will be able to: determine the values for which a rational expression is undefined, evaluate rational expressions, simplify rational expressions, and simplify rational expressions with opposite factors.

In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.

Rational expression. A rational expression is an expression of the form

\tfrac{p(x)}{q(x)}

, where

p

and

q

are polynomials and

q \neq 0

Remember, division by $0$ is undefined. Here are some examples of rational expressions:

-\frac{13}{42} \qquad \frac{7y}{8z} \qquad \frac{5x+2}{x^2-7} \qquad \frac{4x^2+3x-1}{2x-8}

Notice that the first rational expression, $-\tfrac{13}{42}$ , 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 perform the same operations with rational expressions that we do with fractions — simplify, add, subtract, multiply, divide, and use them in applications.

Determine the values for which a rational expression is undefined

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

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be $0$ — but not the denominator. So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero.

Determine the values for which a rational expression is undefined.

Set the denominator equal to zero.
Solve the equation in the set of reals, if possible.

Example. Determine the values for which each rational expression is undefined: (a) $\tfrac{9y}{x}$ , (b) $\tfrac{4b-3}{2b+5}$ , (c) $\tfrac{x+4}{x^2+5x+6}$ .

The expression will be undefined when the denominator is zero.

(a) Set the denominator equal to zero and solve for the variable:

x = 0

So $\tfrac{9y}{x}$ is undefined for $x = 0$ .

(b) Set the denominator equal to zero and solve:

\begin{array}{rcl} 2b + 5 &=& 0 \\ 2b &=& -5 \\ b &=& -\tfrac{5}{2} \end{array}

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

\begin{array}{rcl} x^2 + 5x + 6 &=& 0 \\ (x+2)(x+3) &=& 0 \\ x = -2 \ \text{or}\ x &=& -3 \end{array}

So $\tfrac{x+4}{x^2+5x+6}$ is undefined for $x = -2$ or $x = -3$ . Saying that this rational expression is undefined for $x = -2$ or $x = -3$ is similar to writing the phrase “void where prohibited” in contest rules.

Determine the value for which $\frac{3y}{x}$ is undefined.

Determine the value for which $\frac{8n-5}{3n+1}$ is undefined. Enter the value of $n$ .

Determine the values for which $\frac{a+10}{a^2+4a+3}$ is undefined. Enter the values of $a$ from least to greatest, separated by commas.

Evaluate rational expressions

To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.

Example. Evaluate $\tfrac{2x+3}{3x-5}$ for each value: (a) $x = 0$ , (b) $x = 2$ , (c) $x = -3$ .

(a) Substitute $0$ for $x$ and simplify:

\frac{2(0)+3}{3(0)-5} = -\frac{3}{5}

(b) Substitute $2$ for $x$ and simplify:

\frac{2(2)+3}{3(2)-5} = \frac{4+3}{6-5} = \frac{7}{1} = 7

\frac{2(-3)+3}{3(-3)-5} = \frac{-6+3}{-9-5} = \frac{-3}{-14} = \frac{3}{14}

Evaluate $\frac{5x-1}{2x+1}$ for $x = 1$ .

Evaluate $\frac{5x-1}{2x+1}$ for $x = 0$ .

Remember that a fraction is simplified when it has no common factors, other than $1$ , in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction.

Example. Evaluate $\tfrac{x^2+8x+7}{x^2-4}$ for each value: (a) $x = 0$ , (b) $x = 2$ , (c) $x = -1$ .

(a) Substitute $0$ for $x$ and simplify:

\frac{(0)^2+8(0)+7}{(0)^2-4} = \frac{7}{-4} = -\frac{7}{4}

(b) Substitute $2$ for $x$ and simplify:

\frac{(2)^2+8(2)+7}{(2)^2-4} = \frac{4+16+7}{4-4} = \frac{27}{0}

This rational expression is undefined for $x = 2$ .

\frac{(-1)^2+8(-1)+7}{(-1)^2-4} = \frac{1-8+7}{1-4} = \frac{-7+7}{-3} = \frac{0}{-3} = 0

Evaluate $\frac{y^2+1}{2y-3}$ for $y = 1$ . Give a fraction.

Evaluate $\frac{a^2+2ab+b^2}{3ab^2}$ for $a = 1$ and $b = 2$ .

Simplify rational expressions

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

Simplified rational expression. A rational expression is considered simplified if there are no common factors in its numerator and denominator.

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

We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here, as we will also use it to simplify rational expressions.

Equivalent Fractions Property. If $a$ , $b$ , and $c$ are numbers where $b \neq 0$ , $c \neq 0$ , then

\frac{a}{b} = \frac{a \cdot c}{b \cdot c} \qquad \text{and} \qquad \frac{a \cdot c}{b \cdot c} = \frac{a}{b}

Notice that the values that would make the denominators zero are specifically disallowed. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples. Throughout this chapter we assume that all values that would make a denominator zero are excluded.

Let’s start by reviewing how we simplify numerical fractions.

Example. Simplify $-\tfrac{36}{63}$ .

Rewrite the numerator and denominator showing the common factors, then simplify using the Equivalent Fractions Property:

-\frac{36}{63} = -\frac{4 \cdot 9}{7 \cdot 9} = -\frac{4}{7}

The fraction $-\tfrac{4}{7}$ is simplified because there are no more common factors.

Example. Simplify $\tfrac{3xy}{18x^2y^2}$ (with $x \neq 0$ and $y \neq 0$ ).

Rewrite the numerator and denominator showing the common factors, then simplify:

\frac{3xy}{18x^2y^2} = \frac{1 \cdot 3xy}{6xy \cdot 3xy} = \frac{1}{6xy}

These are the same steps we took when we divided monomials.

To simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.

Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.

\frac{2 \cdot 3 \cdot 7}{3 \cdot 5 \cdot 7} = \frac{2}{5} \qquad\qquad \frac{3x(x-9)}{5(x-9)} = \frac{3x}{5} \qquad\qquad \frac{x+5}{x} \ \text{— no common factors}

In the first two, we removed a common factor from a product. But while there is an $x$ in both the numerator and denominator of $\tfrac{x+5}{x}$ , the $x$ in the numerator is a term of a sum, so it cannot be removed. Removing the $x$ ’s there would be like cancelling the $2$ ’s in the fraction $\tfrac{2+5}{2}$ !

Simplify a rational expression.

Factor the numerator and denominator completely.
Simplify by dividing out common factors.

Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors.

Example. Simplify $\tfrac{2x+8}{5x+20}$ .

Factor the numerator and denominator completely, then divide out the common factor $x + 4$ :

\frac{2x+8}{5x+20} = \frac{2(x+4)}{5(x+4)} = \frac{2}{5}

Simplify: $\frac{3x-6}{2x-4}$ .

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

Factor the numerator and denominator, then remove the common factor $x + 2$ :

\frac{x^2+5x+6}{x^2+8x+12} = \frac{(x+2)(x+3)}{(x+2)(x+6)} = \frac{x+3}{x+6}

Simplify: $\frac{x^2-x-2}{x^2-3x+2}$ .

Example. Simplify $\tfrac{y^2+y-42}{y^2-36}$ .

Factor the numerator and denominator (the denominator is a difference of squares), then remove the common factor $y - 6$ :

\frac{y^2+y-42}{y^2-36} = \frac{(y+7)(y-6)}{(y+6)(y-6)} = \frac{y+7}{y+6}

Simplify: $\frac{x^2+x-6}{x^2-4}$ .

Example. Simplify $\tfrac{p^3-2p^2+2p-4}{p^2-7p+10}$ .

Factor the numerator by grouping and the denominator into a product of binomials, then remove the common factor $p - 2$ :

\frac{p^3-2p^2+2p-4}{p^2-7p+10} = \frac{p^2(p-2)+2(p-2)}{(p-5)(p-2)} = \frac{(p^2+2)(p-2)}{(p-5)(p-2)} = \frac{p^2+2}{p-5}

Simplify: $\frac{p^3-p^2+2p-2}{p^2+4p-5}$ .

When the numerator and denominator share a greatest common factor, factor it out first.

Example. Simplify $\tfrac{2n^2-14n}{4n^2-16n-48}$ .

Factor the numerator and denominator, first factoring out the GCF, then remove the common factor $2$ :

\frac{2n^2-14n}{4n^2-16n-48} = \frac{2n(n-7)}{4(n-6)(n+2)} = \frac{n(n-7)}{2(n-6)(n+2)}

Simplify: $\frac{2n^2-10n}{4n^2-16n-20}$ .

Example. Simplify $\tfrac{m^3+8}{m^2-4}$ .

Factor the numerator using the sum of cubes and the denominator using the difference of squares, then remove the common factor $m + 2$ :

\frac{m^3+8}{m^2-4} = \frac{(m+2)(m^2-2m+4)}{(m+2)(m-2)} = \frac{m^2-2m+4}{m-2}

Simplify: $\frac{x^3+8}{x^2-4}$ .

Simplify 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 $\tfrac{7}{-7}$ . We know this fraction simplifies to $-1$ . We also recognize that the numerator and denominator are opposites.

Recall that the opposite of $a$ is $-a$ , and that $-a = -1 \cdot a$ . So the fraction $\tfrac{a}{-a}$ , whose numerator and denominator are opposites, simplifies like this:

\frac{a}{-a} = \frac{1 \cdot a}{-1 \cdot a} = \frac{1}{-1} = -1

In the same way, we can simplify $\tfrac{x-3}{-(x-3)} = -1$ . But the opposite of $x - 3$ can also be written differently: $-(x-3) = -x + 3 = 3 - x$ . This means the fraction $\tfrac{x-3}{3-x}$ also simplifies to $-1$ . In general, the opposite of $a - b$ is $b - a$ .

Opposites in a rational expression. The opposite of $a - b$ is $b - a$ .

\frac{a-b}{b-a} = -1 \qquad a \neq b

An expression and its opposite divide to $-1$ .

Example. Simplify $\tfrac{x-8}{8-x}$ .

Recognize that $x - 8$ and $8 - x$ are opposites, so the expression divides to:

\frac{x-8}{8-x} = -1

Simplify: $\frac{y-2}{2-y}$ .

Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.

Example. Simplify $\tfrac{14-2x}{x^2-49}$ .

Factor the numerator and denominator. Recognize that $7 - x$ and $x - 7$ are opposites, so $\tfrac{7-x}{x-7} = -1$ :

\frac{14-2x}{x^2-49} = \frac{2(7-x)}{(x+7)(x-7)} = -\frac{2}{x+7}

The factor $\tfrac{7-x}{x-7}$ divides to $-1$ , leaving $-\tfrac{2}{x+7}$ .

Simplify: $\frac{10-2y}{y^2-25}$ .

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

Factor the numerator and denominator, recognize the opposite factors $x - 8$ and $8 - x$ , and simplify:

\frac{x^2-4x-32}{64-x^2} = \frac{(x-8)(x+4)}{(8-x)(8+x)} = -\frac{x+4}{x+8}

The factor $\tfrac{x-8}{8-x}$ divides to $-1$ , leaving $-\tfrac{x+4}{x+8}$ .

Simplify: $\frac{x^2-4x-5}{25-x^2}$ .

Key terms

rational expression — an expression of the form $\tfrac{p(x)}{q(x)}$ , where $p$ and $q$ are polynomials and $q \neq 0$ . undefined — a rational expression is undefined for any value of the variable that makes its denominator zero. simplified rational expression — a rational expression that has no common factors, other than $1$ , in its numerator and denominator. opposites — two expressions such as $a - b$ and $b - a$ ; a factor and its opposite divide to $-1$ .

This section is adapted from Elementary Algebra 2e, Section 8.1: Simplify Rational Expressions by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recast the two-column worked examples as prose with display equality chains; omitted the Be Prepared quiz, Self Check checklist, media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

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