Rational Expressions - Examples & Applications

This section covers operations involving rational expressions (algebraic fractions) and solving rational equations, including partial fraction decomposition. The most critical step in working with rational expressions is identifying domain restrictions.

Domain Restrictions

A rational expression is undefined when its denominator equals zero. These values must be excluded from the domain.

Example

Case Study 1: Finding Domain Restrictions (Basic) Determine the domain of the rational expression: 3xx2−4\frac{3x}{x^2 - 4}

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Example

Case Study 2: Domain Restrictions after Simplifying (Intermediate) Simplify the rational expression and state its domain: x2+5x+6x2+2x−3\frac{x^2 + 5x + 6}{x^2 + 2x - 3}

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Operations on Rational Expressions

Adding and subtracting require finding a common denominator, whereas multiplication and division involve factoring and canceling common terms across numerators and denominators.

Example

Example 1: Multiplication and Division (Intermediate) Multiply the expressions: x2−9x2+x⋅xx−3\frac{x^2 - 9}{x^2 + x} \cdot \frac{x}{x - 3}

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Example

Example 2: Addition with Different Denominators (Intermediate) Add the rational expressions: 2x−1+3x+2\frac{2}{x - 1} + \frac{3}{x + 2}

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Example

Example 3: Complex Fractions (Advanced) Simplify the complex rational expression: 1x+1y1x2−1y2\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x^2} - \frac{1}{y^2}}

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Solving Rational Equations

To solve an equation with rational expressions, multiply every term by the LCD to clear the fractions. Always check the final solutions against the domain restrictions to eliminate extraneous solutions.

Checking Extraneous Solutions

The process of multiplying by an LCD containing variables can introduce solutions that make the original denominators zero. These are invalid and must be discarded.

Example

Example 1: Solving a Rational Equation (Advanced) Solve for xx: 1x−2+3x+2=4x2−4\frac{1}{x - 2} + \frac{3}{x + 2} = \frac{4}{x^2 - 4}

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Example

Example 2: Partial Fraction Decomposition (Advanced) Decompose the rational expression into partial fractions: 5x−4x2−x−2\frac{5x - 4}{x^2 - x - 2}

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