💠 Rational Expressions
Master algebraic fractions! Learn to simplify, add, subtract, multiply, and divide rational expressions like a pro.
🎯 What are Rational Expressions?
A rational expression is a fraction where both the numerator and denominator are polynomials!
Examples of Rational Expressions
- $\frac{x + 2}{x - 3}$
- $\frac{2x^2 + 5x - 1}{x^2 - 4}$
- $\frac{1}{x}$
Important: The denominator cannot equal zero! For $\frac{x + 2}{x - 3}$, we need $x \neq 3$.
✂️ Simplifying Rational Expressions
Factor and cancel common terms - just like with numerical fractions!
Example: Simplify $\frac{x^2 - 9}{x^2 + 6x + 9}$
Step 1: Factor numerator and denominator
$$\frac{(x + 3)(x - 3)}{(x + 3)(x + 3)}$$
Step 2: Cancel common factors
$$\frac{x - 3}{x + 3}$$
Restriction: $x \neq -3$ (from original denominator)
✖️ Multiplying Rational Expressions
Multiply numerators together and denominators together, then simplify!
Example: $\frac{x + 2}{x - 1} \cdot \frac{x - 1}{x + 3}$
Step 1: Multiply
$$\frac{(x + 2)(x - 1)}{(x - 1)(x + 3)}$$
Step 2: Cancel common factors
$$\frac{x + 2}{x + 3}$$
➗ Dividing Rational Expressions
Flip the second fraction and multiply!
Example: $\frac{x}{x + 1} \div \frac{x^2}{x + 1}$
Step 1: Flip and multiply
$$\frac{x}{x + 1} \cdot \frac{x + 1}{x^2}$$
Step 2: Simplify
$$\frac{1}{x}$$
➕ Adding and Subtracting
Find a common denominator first!
Example: $\frac{2}{x} + \frac{3}{x + 1}$
LCD: $x(x + 1)$
$$\frac{2(x + 1)}{x(x + 1)} + \frac{3x}{x(x + 1)}$$
$$= \frac{2x + 2 + 3x}{x(x + 1)} = \frac{5x + 2}{x(x + 1)}$$
🌟 Real-World Applications
- ⚡ Physics: Ohm's Law uses rational expressions: $R = \frac{V}{I}$
- 💧 Chemistry: Concentration formulas
- 🏃 Rate Problems: Speed = $\frac{\text{distance}}{\text{time}}$
- 💰 Finance: Investment return ratios
🎯 Practice Questions
Master rational expressions!
🔥 Challenge Questions
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