Rational Expressions: Simplifying, Operations, and Domain Restrictions

What Is a Rational Expression?

A rational expression is any expression that can be written as a ratio of two polynomials P(x)/Q(x), where Q(x) is not the zero polynomial. Examples:

Unlike polynomial expressions (which are always defined for every real number), rational expressions have restricted domains — wherever the denominator is zero, the expression is undefined.

Understanding rational expressions is prerequisite to working with rational functions, performing polynomial long division, applying partial fraction decomposition, and integrating rational functions in calculus.

Domain Restrictions (Excluded Values)

To find excluded values, set the denominator equal to zero and solve. Those x-values must be excluded from the domain.

Example: Find the excluded values of \dfrac{x+2}{x^2-9}.

Set the denominator to zero and solve: x^2-9 = 0 \implies x = \pm 3. So x = 3 and x = -3 are excluded.

Why this matters: Excluded values stay restricted even after simplification. If a factor cancels, the original restriction remains — the simplified expression is only equivalent to the original on the unrestricted domain.

Simplifying Rational Expressions

To simplify, factor numerator and denominator completely using the techniques from the factoring guide, then cancel common factors.

Example: Simplify \dfrac{x^2-4}{x^2+5x+6}.

Factor both top and bottom, then cancel:

Critical rule: you can only cancel factors (multiplied terms), not terms (added/subtracted pieces). (x+2)/(x+3) cannot be simplified further because x+2 is a single term, not a product containing x+3.

Multiplying and Dividing Rational Expressions

To multiply: factor everything, then cancel common factors across any numerator with any denominator, and multiply what remains.

Example: \dfrac{x^2-1}{x+3} \cdot \dfrac{x+3}{x-1}.

To divide: flip the second fraction (take reciprocal) and multiply. Same "factor then cancel" rule applies.

Example: \dfrac{x^2}{x+1} \div \dfrac{x}{x^2-1}.

Adding and Subtracting Rational Expressions

Finding the Least Common Denominator (LCD)

The LCD is the smallest polynomial divisible by all denominators. Construction steps:

1. Factor each denominator completely.
2. Identify each distinct factor appearing in any denominator.
3. For each distinct factor, take the highest power in which it appears.
4. Multiply these highest-power factors together — that's the LCD.

Building Equivalent Fractions

Multiply the numerator and denominator of each fraction by whatever is needed to turn its denominator into the LCD. This preserves the value while creating a common denominator.

Combining and Simplifying

With a common denominator, add or subtract numerators, then simplify.

Example: \dfrac{2}{x-1} + \dfrac{3}{x+2}.

The denominators share no factors, so the LCD is (x-1)(x+2). Build equivalent fractions and combine:

Subtraction warning: when subtracting, distribute the negative sign to every term of the second numerator. This is one of the most common error sources.

Complex Fractions

A complex fraction has a fraction in its numerator or denominator (or both). The fastest simplification method: multiply top and bottom by the LCD of all inner fractions.

Example: Simplify \dfrac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}.

The inner fractions have denominators x and y, so the LCD is xy. Multiply top and bottom by xy:

This method turns complex fractions into simple rational expressions in one step — much cleaner than the "combine, then divide" approach.

Common Mistakes

Practice Problems

Simplify each expression; state the domain restrictions.

1. \dfrac{x^2-25}{x^2+3x-10}
2. \dfrac{2x+6}{x^2-9}
3. \dfrac{x^2-1}{x+1} \cdot \dfrac{x}{x-1}
4. \dfrac{x^2+x}{x+2} \div \dfrac{x}{x^2-4}
5. \dfrac{1}{x} + \dfrac{2}{x+1}
6. \dfrac{3}{x-2} - \dfrac{1}{x+2}
7. \dfrac{\frac{1}{x}}{\frac{1}{x}+\frac{1}{2}}

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