Rational Expression Calculator

The Rational Expression Calculator helps you work with algebraic fractions that contain variables, such as \(\frac{x+1}{x-2}\). Enter a rational expression to simplify, add, subtract, multiply, or divide it, or enter a rational equation to solve for a variable. The tool can show solution steps, but denominator restrictions still matter because division by zero is never allowed. Enter your problem above, choose the kind of help you need, and use the steps to compare each algebra move with your own work.

How to Use the Rational Expression Calculator

1. Add Your Problem

   Use Calculator for the on-screen keypad, Math Input to type notation, or Canvas to draw, handwrite, or upload a clear photo. Put parentheses around any multi-term numerator or denominator, such as \( \frac{(x+3)}{(x-2)} \). For operations, enter each fraction clearly, such as \( \frac{x+1}{x-2} + \frac{3}{x+2} \).

2. Choose Your Output

   Step-by-step answers are the default choice. You can also choose Explain Like I’m 10, Find My Mistake, Create Practice Test, Create Study Guide, or Create Flashcards. Use Create Graph only when you entered a suitable rational function.

3. Choose a Language

   Use the language dropdown labeled “Default” if you want the explanation presented in another available language.

4. Get the Result

   Select Solve to submit the problem. Depending on what you entered, you may receive solution steps, a simplified form, an equation solution, or a graph when applicable.

What This Calculator Can Help You Do

Simplify Rational Expressions

Enter an expression when you need to reduce it to simpler factors. The calculator can show step-by-step explanations that factor polynomials, identify common factors, and simplify the fraction.

For example,

\[ \frac{(x-2)(x+5)}{(x-2)(x+1)} = \frac{x+5}{x+1} \]

You may cancel the common factor \(x-2\), not separate terms joined by addition or subtraction. The original restriction remains: \(x \ne 2,-1\). If factoring is the sticking point, use the factoring steps tool before simplifying.

Add or Subtract Algebraic Fractions

Enter both fractions with clear parentheses when you need to combine them. The calculator can show how each fraction is rewritten using a least common denominator, or LCD, before the numerators are added or subtracted.

For instance,

\[ \frac{1}{x}+\frac{1}{x+1} = \frac{x+1}{x(x+1)}+\frac{x}{x(x+1)} = \frac{2x+1}{x(x+1)} \]

The restrictions are \(x\ne0\) and \(x\ne-1\). Do not add denominators together. First build the LCD, then combine only the numerators over that shared denominator.

Multiply or Divide Rational Expressions

Use the tool for products and quotients of rational expressions, especially when factoring reveals factors that can cancel. For multiplication, factor when useful, cancel common factors, and multiply what remains.

For division, keep the first expression, change division to multiplication, and multiply by the reciprocal of the second expression:

\[ \frac{x}{x+2}\div\frac{x-1}{x} = \frac{x}{x+2}\cdot\frac{x}{x-1} \]

The restrictions are \(x\ne-2\), \(x\ne0\), and \(x\ne1\). The second fraction must be defined and cannot equal zero before you take its reciprocal. Keep every restriction from the original problem visible.

Solve Rational Equations

Include an equals sign when your goal is to solve a rational equation. The calculator can provide step-by-step help for supported rational equations. Review the displayed restrictions, denominator-clearing steps, and final candidate solutions against the original equation.

A rational equation may turn into a linear or quadratic equation after denominators are cleared. If the result is quadratic, you can solve a resulting quadratic after you have preserved the original restrictions. Always substitute candidate answers into the original equation, because a value that makes a denominator zero must be rejected.

For example,

\[ \frac{1}{x-1}=\frac{2}{x+1} \]

The restrictions are \(x\ne1\) and \(x\ne-1\). Multiply both sides by \((x-1)(x+1)\):

\[ x+1=2(x-1) \]

\[ x=3 \]

Since \(3\) does not violate either restriction, \(\boxed{x=3}\).

How Rational Expression Results Work

A rational expression is a quotient of two polynomials, such as \(\frac{x+1}{x-2}\). Without an equals sign, your entry is usually an expression to simplify or calculate. A rational equation contains one or more rational expressions joined by an equals sign, so the tool treats it as a problem to solve

Find denominator restrictions before simplifying. In \(\frac{(x-2)(x+5)}{(x-2)(x+1)}\), the factor \(x-2\) can cancel, but \(x=2\) is still excluded because the original denominator was zero there.

Clear formatting helps the tool read your intent. Type \( \frac{(x+1)}{(x-2)} \), not \( \frac{x+1}{x-2} \), when you mean the whole numerator divided by the whole denominator. Create Graph can help you inspect a suitable rational function. Denominator-zero values may correspond to holes or vertical asymptotes, depending on whether a factor cancels. For equations that are not primarily rational, use broader equation help.

Worked Calculator Example

Simplify:

\[ \frac{x^2-5x+6}{x^2-4} \]

1. Find the denominator restrictions.

   \[ x^2-4 \ne 0 \]

   \[ (x-2)(x+2)\ne0 \]

   \[ x\ne2,-2 \]

   These values would make the original denominator zero. The solution steps should keep these exclusions visible.

2. Factor the numerator.

   \[ x^2-5x+6=(x-2)(x-3) \]

   We need factors because cancellation works with factors, not with individual terms.

3. Factor the denominator.

   \[ x^2-4=(x-2)(x+2) \]

   This difference of squares creates the factor shared with the numerator.

4. Rewrite using factors.

   \[ \frac{x^2-5x+6}{x^2-4} = \frac{(x-2)(x-3)}{(x-2)(x+2)} \]

   A step-by-step result should show this form before cancellation.

5. Cancel the common factor and keep restrictions.

   \[ \frac{(x-2)(x-3)}{(x-2)(x+2)} = \frac{x-3}{x+2}, \qquad x\ne2,-2 \]

   The simplified expression is \(\frac{x-3}{x+2}\), but \(x=2\) remains excluded even though \(x-2\) canceled.

Common Mistakes

1. Canceling terms instead of common factors

   Students may try to cancel the \(x\) in \(\frac{x+2}{x}\), even though \(x+2\) is a sum, not a factor containing \(x\). Factor first, then cancel only factors that multiply the entire numerator and denominator.

2. Forgetting original denominator restrictions after cancellation

   A canceled factor can make it look as if a value is allowed. Write restrictions from the original denominators first, because canceling a factor does not restore a value where the original expression was undefined.

3. Adding or subtracting before finding an LCD

   Students sometimes combine \(\frac{1}{x}+\frac{1}{x+1}\) as \(\frac{2}{2x+1}\). The rule is to find an LCD, rewrite both fractions, and then combine the numerators.

4. Forgetting to multiply by the reciprocal when dividing

   Dividing by \(\frac{a}{b}\) is not the same as multiplying by \(\frac{a}{b}\). Change division to multiplication and use \(\frac{b}{a}\), while ensuring the divisor is not zero.

5. Accepting an extraneous solution from a rational equation

   Clearing denominators can produce a candidate that was not valid in the original equation. Substitute each answer back into the original equation and reject any value that makes an original denominator zero.

6. Incorrectly factoring rational expressions

   Students may factor the numerator or denominator incorrectly, which can lead to invalid cancellations or incorrect simplification. For example, \( x^2 - 9 \) should be factored as \( (x-3)(x+3) \), not \( (x-3)^2 \). Always verify the factoring before canceling common factors.

7. Failing to simplify completely

   Students may stop after finding a common denominator or canceling one common factor, even when the resulting rational expression can still be reduced. After performing the required operations, check whether any additional common factors remain.