Rational Expressions
Rational expressions are algebraic fractions where both the numerator and denominator are polynomials. The denominator cannot be zero, so every rational expression has a domain.
The above expression is defined if x ≠ 3, because then the denominator would be zero.
Domain
The domain is the set of x values for which the denominator is not zero. This must always be determined separately.
- If the denominator is x - 5, then x ≠ 5.
- If the denominator is x² - 4, then x ≠ ±2.
- If the denominator is (x + 1)(x - 7), then x ≠ -1 and x ≠ 7.
Simplification
We simplify rational expressions by factoring the numerator and denominator, then canceling common factors. It is important to still indicate the excluded values.
Operations with Rational Expressions
- Addition: bring to a common denominator then add the numerators.
- Subtraction: bring to a common denominator then subtract the numerators.
- Multiplication: multiply numerators together, denominators together.
- Division: multiply by the reciprocal of the second fraction.
Example of multiplication:
Addition Example
Rational Expressions in Practice
Rational expressions often appear in physics, economics, and engineering calculations. For example:
- Speed = distance / time → if distance and time are described by polynomials, the result is a rational expression.
- When calculating average speed, the denominator is always the total time, which cannot be zero.
- In chemical reaction rate laws, rational ratios often appear.
Practice Exercise
We have reviewed and checked the materials, but errors may still occur. The content is provided for educational purposes only, so use it at your own responsibility and verify with other sources if needed.
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