Rational Expression Calculator

About the Rational Expression Calculator

Rational expressions are fractions with polynomials on the top and bottom, such as $\frac{x^2 - 9}{x^2 + 5x + 6}$ . They behave like ordinary fractions, so everything you know about reducing, finding common denominators and dividing by a fraction still applies. The difference is that the numbers you factor are now polynomials, and you must keep track of values of $x$ that would make a denominator zero.

This calculator performs five operations on rational expressions in one variable: simplify, add, subtract, multiply and divide. It factors every numerator and denominator over the rationals, builds the least common denominator when one is needed, cancels common factors, and states the excluded values of the final answer, which many solutions forget.

What is a rational expression?

A rational expression is a quotient $\frac{P(x)}{Q(x)}$ of two polynomials where $Q$ is not the zero polynomial. It is undefined wherever $Q(x) = 0$ , so its domain is every real number except the roots of the denominator.

Two rational expressions are equivalent if they agree at every point where both are defined. Simplifying replaces an expression by an equivalent one in lowest terms, with no common factor shared by numerator and denominator. Because cancelling can remove a factor of the denominator, the excluded values of the original expression must be carried over explicitly.

How to use the Rational Expression Calculator

Choose simplify, add, subtract, multiply or divide.
Enter the first expression in A using parentheses around numerators and denominators, for example (x^2 - 9)/(x^2 + 5x + 6).
For the four arithmetic operations, enter the second expression in B, such as 2/(x + 1).
Press Calculate and follow the factoring, common denominator and cancelling steps.
Copy the result together with its excluded values.

Formulas

Reducing

\frac{P\,R}{Q\,R} = \frac{P}{Q},\quad R \neq 0

Adding with a common denominator

\frac{P}{D} + \frac{R}{D} = \frac{P + R}{D}

Multiplying

\frac{P}{Q} \cdot \frac{R}{S} = \frac{PR}{QS}

Dividing

\frac{P}{Q} \div \frac{R}{S} = \frac{P}{Q} \cdot \frac{S}{R},\quad R \neq 0

Worked example: Simplify (x² − 9)/(x² + 5x + 6)

The worked example reduces $\frac{x^2 - 9}{x^2 + 5x + 6}$ to lowest terms. The engine recognises a difference of squares on top and a factorable trinomial below, so both sides share the factor $x + 3$ .

After cancelling, the engine lists both excluded values, $-3$ and $-2$ . The value $-3$ no longer makes the reduced denominator zero, but it was excluded from the original expression, so it stays excluded.

The solution below is generated by the same engine that powers the calculator above.

Final answer

\frac{x - 3}{x + 2}

How we read your problem

Simplify the rational expression to lowest terms.

\frac{{x}^{2}-9}{{x}^{2}+5x+6}

Step-by-step solution

Write as a single fraction
$\frac{x^{2} - 9}{x^{2} + 5x + 6}$
Factor the numerator and denominator
$\frac{(x - 3)(x + 3)}{(x + 2)(x + 3)}$
Cancel the common factor
The factor x + 3 appears in both the numerator and the denominator. Cancelling it does not change the value wherever the original expression is defined.
$\frac{x - 3}{x + 2}$
State the excluded values
The original expression is undefined where any denominator (or divisor) is zero: x ≠ -3, x ≠ -2. These restrictions stay even if a factor cancelled.

Formulas used

Cancelling
$\frac{a\,c}{b\,c} = \frac{a}{b}\quad (c \neq 0)$

In lowest terms the result is (x - 3)/(x + 2). Excluded values: x ≠ -3, x ≠ -2.

Verification

The simplified form was compared with the original at four test points and agrees.

Detailed explanation

Adding and subtracting

To add rational expressions, rewrite them over a common denominator. The least common denominator (LCD) is the product of every distinct factor of the denominators, each raised to the highest power in which it appears. For $\frac{1}{x - 1} + \frac{2}{x + 1}$ the LCD is $(x - 1)(x + 1)$ ; each numerator is multiplied by the factor its denominator lacks, giving $\frac{(x + 1) + 2(x - 1)}{(x - 1)(x + 1)} = \frac{3x - 1}{(x - 1)(x + 1)}$ .

Subtraction follows the same pattern, but the whole second numerator is subtracted, so put it in parentheses before distributing the minus sign. After combining, factor the new numerator to see if anything cancels with the LCD.

Multiplying and dividing

Multiplication does not need a common denominator. Factor everything first, cancel any factor that appears in any numerator and any denominator, and multiply what remains. Cancelling before multiplying keeps the polynomials small.

To divide, multiply by the reciprocal of the second expression. This adds a new restriction: the numerator of the divisor cannot be zero, because it becomes a denominator after flipping. Dividing $\frac{x}{x + 2}$ by $\frac{x - 1}{x + 2}$ gives $\frac{x}{x - 1}$ , with $x \neq -2$ and $x \neq 1$ both excluded.

Why excluded values matter

Excluded values are not a formality. The graph of $\frac{x^2 - 9}{x^2 + 5x + 6}$ looks like the graph of $\frac{x - 3}{x + 2}$ , but it has a hole at $x = -3$ and a vertical asymptote at $x = -2$ . When solving equations with rational expressions, a candidate solution equal to an excluded value is extraneous and must be rejected.

Common mistakes

Cancelling terms instead of factors, such as crossing out $x$ in $\frac{x + 2}{x}$ .
Adding numerators and denominators separately: $\frac1x + \frac1y$ is not $\frac{2}{x + y}$ .
Forgetting to distribute the minus sign over the whole second numerator when subtracting.
Dropping the excluded values after cancelling a common factor.
Flipping the first fraction instead of the second when dividing.

Tips

Factor every numerator and denominator completely before doing anything else.
Write the LCD in factored form; multiplying it out early makes cancelling harder.
Collect the excluded values from every denominator, including the divisor's numerator, before you cancel.
Check a result by substituting a value such as $x = 5$ into the original and the simplified form.
Use the domain and range calculator to see the full domain of the result.

Frequently asked questions

How do I simplify a rational expression?

Factor the numerator and denominator completely, cancel any factors they share, and state the values of x that make the original denominator zero as excluded values.

What is an excluded value?

A value of the variable that makes a denominator zero, so the expression is undefined there. Excluded values from the original expression remain excluded after simplifying.

How do I find the least common denominator?

Factor each denominator and multiply together every distinct factor, using the highest power in which it appears in any one denominator.

Why can't I cancel the x in (x + 2)/x?

In the numerator x is a term being added, not a factor of the whole numerator. Only factors that multiply the entire numerator and the entire denominator can be cancelled.

How do I divide rational expressions?

Multiply the first expression by the reciprocal of the second, then factor and cancel. The numerator of the second expression adds new excluded values because it becomes a denominator.

Does the calculator handle more than one variable?

This calculator works with rational expressions in a single variable. For polynomial expressions in several variables, use the simplify calculator.

What is the difference between a hole and an asymptote?

A hole appears where a factor cancels from both numerator and denominator. A vertical asymptote appears where a factor remains in the denominator after simplifying.

Simplify or combine your rational expressions

Enter one or two algebraic fractions above to see the factoring, the common denominator and the excluded values.

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