Rational Expressions Calculator
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Rational expressions are fractions of polynomials. To add \(\frac{2}{x+1} + \frac{3}{x-2}\), rewrite both over the least common denominator \((x+1)(x-2)\): \(\frac{2(x-2) + 3(x+1)}{(x+1)(x-2)} = \frac{5x – 1}{x^{2} – x – 2}\), for x ≠ −1, 2.
\(A = \frac{2}{x + 1}\)
\(B = \frac{3}{x – 2}\)
| Simplified | \(\frac{5x – 1}{x^{2} – x – 2}\) |
| Factored | \(\frac{5x – 1}{\left(x + 1\right)\left(x – 2\right)}\) |
| Excluded values | \(x \neq -1,\ 2\) |
Show steps
- Factor each expression: \(\frac{2}{x + 1}\) and \(\frac{3}{x – 2}\)
- Least common denominator: \(\left(x + 1\right)\left(x – 2\right)\)
- Rewrite over the LCD and add the numerators: \(\frac{2x – 4 + \left(3x + 3\right)}{x^{2} – x – 2}\)
- Simplify: \(\frac{5x – 1}{x^{2} – x – 2}\)
How to use the rational expressions calculator
- Choose the operation.
- Enter A (and B), using one variable, for example (x^2 – 4)/(x + 3).
- Press Calculate to see the factoring, the LCD or the reciprocal, the simplified answer and the excluded values.
Rules for each operation
- Multiply: \(\frac{a}{b}\cdot\frac{c}{d} = \frac{ac}{bd}\); factor first and cancel common factors.
- Divide: multiply by the reciprocal: \(\frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\cdot\frac{d}{c}\).
- Add or subtract: rewrite over the least common denominator, then combine the numerators.
- Simplify: factor the numerator and denominator and cancel common factors.
Excluded values
A rational expression is undefined where any denominator in the problem is 0. When dividing, the numerator of B is also excluded, because it becomes a denominator. These values stay excluded even if a factor cancels.
Worked example: subtraction
\(\frac{x}{x^{2}-9} – \frac{1}{x-3}\). Since \(x^{2} – 9 = (x-3)(x+3)\), the LCD is \((x-3)(x+3)\): \(\frac{x – (x+3)}{(x-3)(x+3)} = -\frac{3}{x^{2}-9}\), with \(x \neq \pm 3\).
Rational expressions practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you add rational expressions?
Factor the denominators, find the least common denominator, rewrite each fraction with it, add the numerators, and simplify.
How do you multiply rational expressions?
Factor everything, cancel factors that appear in any numerator and any denominator, then multiply what is left.
What are excluded values?
Values of the variable that make a denominator zero. The expression is undefined there, even after simplifying.
Can I use more than one variable?
This calculator works with one variable. For several variables, use the simplify calculator.
Why is my answer in a different form?
Equivalent answers can be factored or expanded. The calculator shows both the expanded and factored forms.
Sources
- Paul’s Online Math Notes: Rational Expressions
- OpenStax College Algebra 2e: Rational Expressions
- OpenStax Intermediate Algebra 2e: Rational Expressions and Functions
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