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Rational Expression Calculator

Simplify · Add · Subtract · Multiply · Divide · Evaluate · Domain Restrictions

Enter polynomial numerators and denominators using standard notation (e.g. x^2-4, x+2). Get step-by-step factoring, cancellation, and the simplified result.

Simplify a Rational Expression

Factor & Cancel

Quick examples

Input notation guide
x^2 → x²
2x^2+3x-5 → 2x² + 3x − 5
x^2-9 → x² − 9 (difference of squares)
x^3-1 → x³ − 1 (difference of cubes)
4 → constant polynomial
x → linear monomial
-x+1 → −x + 1 (leading negative OK)

What Are Rational Expressions?

A rational expression is an algebraic fraction of the form P(x)/Q(x) where P and Q are polynomials and Q is not the zero polynomial. Rational expressions are the polynomial analog of ordinary numerical fractions — everything you can do with numeric fractions (add, subtract, multiply, divide, simplify) you can also do with rational expressions, following the exact same rules.

Everyday examples appear throughout algebra and precalculus: the slope formula, partial fraction decomposition in calculus, and asymptote analysis in graphing all involve rational expressions. Understanding how to simplify and operate on them is foundational for algebra, calculus, and engineering mathematics.

Because the denominator Q(x) cannot equal zero, rational expressions always carry domain restrictions — values of x that must be excluded. For instance, (x+1)/(x-3) is undefined at x = 3, so the domain is all real numbers except 3. These restrictions are a core part of every answer involving rational expressions.

How to Simplify Rational Expressions

Simplification reduces a rational expression to its lowest terms by cancelling common factors from numerator and denominator:

  • Step 1 — Factor the numerator completely (GCF, difference of squares, quadratic formula, grouping, etc.).
  • Step 2 — Factor the denominator completely using the same set of techniques.
  • Step 3 — Cancel common factors that appear in both numerator and denominator.
  • Step 4 — State domain restrictions from the original denominator (including any factors that were cancelled).

Worked example — simplify (x²-9)/(x+3):
Numerator: x²-9 = (x-3)(x+3). Denominator: (x+3). Cancel (x+3): result = (x-3), with restriction x ≠ -3.

Adding and Subtracting Rational Expressions

To add or subtract rational expressions you need a common denominator, just like with ordinary fractions. The process is:

  • Factor each denominator to find the LCD (Least Common Denominator).
  • Multiply each fraction top and bottom by the missing LCD factors.
  • Combine the numerators over the single LCD.
  • Simplify the result if possible.

Worked example — add 2/x + 3/(x+1):
LCD = x(x+1). Rewrite: 2(x+1)/[x(x+1)] + 3x/[x(x+1)] = (2x+2+3x)/[x(x+1)] = (5x+2)/[x(x+1)], with restrictions x ≠ 0, x ≠ -1.

Multiplying and Dividing Rational Expressions

Multiplication: multiply the numerators together and the denominators together — but factor first and cross-cancel common factors before multiplying to keep numbers small. Example: (x²-1)/(x+3) × (x+3)/(x-1) = (x-1)(x+1)(x+3) / [(x+3)(x-1)] = (x+1), with restrictions x ≠ ±1, x ≠ -3.

Division: multiply the first fraction by the reciprocal of the second (flip the second fraction). Then proceed exactly like multiplication. Remember: the numerator of the original divisor also becomes a denominator after the flip, so it contributes new domain restrictions.

Domain Restrictions Explained

The domain of a rational expression is the set of all x-values for which the expression is defined. Any value that makes any denominator in the problem zero must be excluded — even if it is cancelled during simplification. This is because the original expression was undefined at that point before simplification occurred.

To find restrictions: set each denominator equal to zero and solve. List every solution as an excluded value using "x ≠ ..." notation.

Frequently Asked Questions

What is a rational expression?
A rational expression is a fraction where both the numerator and denominator are polynomials. Examples include (x+3)/(x-2) and (x²-1)/(x²+5x+6). They are undefined wherever the denominator equals zero.
How do you simplify a rational expression?
Factor the numerator, factor the denominator, then cancel any common factors. Always record domain restrictions — excluded values from the original denominator apply even after cancellation.
Why must we exclude values from the domain?
Division by zero is undefined in mathematics. Any x that makes the denominator zero must be excluded from the domain. Even after simplification these restrictions carry over from the original unsimplified form.
How do you find the LCD of polynomial denominators?
Factor each denominator completely. Build the LCD by taking each unique factor to its highest power across all denominators. For example, x/(x-1) + 2/(x²-1): since x²-1 = (x-1)(x+1), the LCD = (x-1)(x+1).
What is the difference between domain and range?
The domain is the set of x-values for which the expression is defined (excludes denominator zeros). The range is the set of all output values. Finding the range requires solving for x in terms of y and checking what y-values are achievable.
How do you add rational expressions?
Find the LCD, rewrite each fraction over the LCD by multiplying numerator and denominator by the missing factors, combine numerators into one fraction over the LCD, then simplify the result.
How do you divide rational expressions?
Multiply the first expression by the reciprocal (flip) of the second. Factor all pieces, cross-cancel common factors, then multiply. Note new restrictions from the divisor's numerator becoming a denominator after flipping.
Does this calculator support x² and higher powers?
Yes. Enter polynomials using standard notation: x^2-4 for x²-4. The calculator factors quadratics and higher-degree polynomials, identifies common factors, and shows each cancellation step.

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