Transforming Rational Expressions
Definition: A rational expression is a fraction where the numerator and/or denominator are polynomials. Transforming rational expressions means rewriting them in equivalent, often simpler forms using operations like simplifying, adding, subtracting, multiplying, and dividing.
📐 What Is a Rational Expression?
Just like a rational number is a ratio of two integers, a rational expression is a ratio of two polynomials:
$$ \frac{P(x)}{Q(x)} \quad \text{where } Q(x) \neq 0 $$
Examples:
- $ \frac{x+3}{x-1} $
- $ \frac{2x^2 + 5x}{x^2 - 4} $
- $ \frac{7}{x+2} $
⚠️ Key Rule: The denominator can never equal zero! We must always state restrictions on the variable.
🔑 The Fundamental Principle
The most important tool for transforming rational expressions is:
Fundamental Principle of Fractions:
$$ \frac{a \cdot c}{b \cdot c} = \frac{a}{b} \quad (c \neq 0) $$
$$ \frac{a \cdot c}{b \cdot c} = \frac{a}{b} \quad (c \neq 0) $$
This means we can:
- Cancel common factors (simplify)
- Multiply by a form of 1 to create common denominators
The key is factoring — it unlocks everything!