Rational Expressions

🍕 Rational Expressions: The Pizza Fraction Story

Imagine you’re at a pizza party. Everything is about sharing pizzas fairly—but sometimes the slices get weird! Welcome to the world of rational expressions.

📖 What is a Rational Expression?

Think about this: What’s a fraction?

A fraction is one number sitting on top of another, like: $\frac{3}{4}$

That’s 3 pieces out of 4. Simple!

Now, a rational expression is the SAME thing, but with algebra letters (like x) instead of just numbers.

🍕 The Pizza Analogy

Simple Example: $\frac{x + 5}{x - 3}$

This is a rational expression! The top (numerator) is x + 5. The bottom (denominator) is x - 3.

✨ The Magic Rule

A rational expression = polynomial ÷ polynomial

(As long as the bottom is NOT zero!)

Real Life Connection:

• Speed = Distance ÷ Time → could be $\frac{100}{t}$ miles per hour
• Cost per person = Total ÷ People → could be $\frac{50}{x}$ dollars each

🚫 Domain of Rationals: Where Can X Play?

Imagine you have a pizza cut into x slices. What if x = 0?

DISASTER! You can’t divide by zero pieces!

The Golden Rule of Fractions

You can NEVER divide by zero. EVER. NEVER EVER.

So when we have $\frac{x + 2}{x - 3}$, we need to ask:

“What value of x would make the bottom equal zero?”

🔍 Finding the Forbidden Values

Example 1: $\frac{5}{x - 4}$

Ask: When does x - 4 = 0?

• Solve it: x = 4
• So x cannot be 4

Domain: All numbers EXCEPT 4

Example 2: $\frac{x + 1}{x^2 - 9}$

The bottom factors: $x^2 - 9 = (x + 3)(x - 3)$

When is this zero?

• When x = 3 OR x = -3

Domain: All numbers EXCEPT 3 and -3

🎯 Quick Steps to Find Domain

1. Look at the BOTTOM only
2. Set bottom = 0
3. Solve for x
4. Those x values are FORBIDDEN
5. Domain = everything else

✂️ Simplifying Rationals: Cancel the Twins!

Remember how $\frac{6}{8}$ simplifies to $\frac{3}{4}$?

You cancel the common factor of 2: $\frac{6}{8} = \frac{2 \times 3}{2 \times 4} = \frac{3}{4}$

Rational expressions work the SAME way!

The Twin Rule

If something appears on both top AND bottom, you can cancel it!

🍕 Pizza Example

You ordered pizzas and now you’re sharing:

$\frac{x(x + 2)}{x(x - 5)}$

Look! There’s an x hiding on both top and bottom.

Cancel it: $\frac{\cancel{x}(x + 2)}{\cancel{x}(x - 5)} = \frac{x + 2}{x - 5}$

Step-by-Step Simplifying

Example: Simplify $\frac{x^2 - 4}{x + 2}$

Step 1: Factor everything you can

Top: $x^2 - 4 = (x + 2)(x - 2)$ ← difference of squares!

Step 2: Rewrite $\frac{(x + 2)(x - 2)}{x + 2}$

Step 3: Cancel matching twins $\frac{\cancel{(x + 2)}(x - 2)}{\cancel{x + 2}} = x - 2$

Answer: $x - 2$ (but x ≠ -2, always remember the original restriction!)

⚠️ Warning: Only Cancel FACTORS!

You can only cancel things that are multiplied.

❌ WRONG: $\frac{x + 3}{x + 5}$ does NOT simplify (nothing cancels!)

✅ RIGHT: $\frac{3(x + 5)}{x + 5} = 3$ (the factor x + 5 cancels)

✖️ Multiplying Rationals: Straight Across!

This is the easiest operation! Remember how to multiply regular fractions?

$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$

Just multiply top × top and bottom × bottom!

The Simple Recipe

$\frac{A}{B} \times \frac{C}{D} = \frac{A \times C}{B \times D}$

🍕 Pizza Story

You eat $\frac{1}{2}$ of $\frac{3}{4}$ of a pizza.

That’s $\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}$ of the pizza!

Example with Variables

Multiply: $\frac{x}{x + 1} \times \frac{x - 2}{x}$

Step 1: Multiply straight across $\frac{x \times (x - 2)}{(x + 1) \times x} = \frac{x(x - 2)}{x(x + 1)}$

Step 2: Cancel common factors $\frac{\cancel{x}(x - 2)}{\cancel{x}(x + 1)} = \frac{x - 2}{x + 1}$

🎯 Pro Tip: Cancel BEFORE You Multiply!

It’s easier to cancel first, then multiply what’s left.

Example: $\frac{3x^2}{4} \times \frac{8}{9x}$

Look for twins:

• 3 and 9 share factor 3
• 8 and 4 share factor 4
• $x^2$ and $x$ share factor $x$

$\frac{\cancel{3}x^{\cancel{2}}}{\cancel{4}} \times \frac{\cancel{8}^2}{\cancel{9}^3 \cancel{x}} = \frac{2x}{3}$

➗ Dividing Rationals: Flip and Multiply!

Here’s a secret: Division is just multiplication in disguise!

Remember: $6 ÷ 2 = 6 \times \frac{1}{2} = 3$

To divide fractions, you flip the second one and multiply.

The Magic Trick

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$

“Keep, Change, Flip!”

• Keep the first fraction
• Change ÷ to ×
• Flip the second fraction

🍕 Pizza Story

You have $\frac{3}{4}$ of a pizza. You divide it among groups that each get $\frac{1}{8}$ of the whole pizza.

How many groups?

$\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1} = \frac{24}{4} = 6$

Six groups can eat!

Example with Variables

Divide: $\frac{x^2 - 1}{x + 3} \div \frac{x - 1}{x}$

Step 1: Keep, Change, Flip $\frac{x^2 - 1}{x + 3} \times \frac{x}{x - 1}$

Step 2: Factor (top of first = difference of squares) $\frac{(x + 1)(x - 1)}{x + 3} \times \frac{x}{x - 1}$

Step 3: Cancel twins $\frac{(x + 1)\cancel{(x - 1)}}{x + 3} \times \frac{x}{\cancel{x - 1}}$

Step 4: Multiply what’s left $\frac{x(x + 1)}{x + 3}$

🎉 Summary: The Rational Expression Journey

graph TD
    A["🍕 Rational Expression"] --> B["Polynomial ÷ Polynomial"]
    B --> C{Is bottom = 0?}
    C -->|Yes| D["❌ UNDEFINED!"]
    C -->|No| E["✅ Valid!"]
    E --> F["Simplify by factoring"]
    E --> G["Multiply: Top × Top, Bottom × Bottom"]
    E --> H["Divide: Flip second, then multiply"]

Your New Superpowers

💪 You’ve Got This!

Rational expressions are just fractions wearing algebra costumes.

Every rule you knew about regular fractions still works—you just need to watch out for that sneaky zero in the denominator!

Remember our pizza party: Share fairly, never divide by zero, and always cancel the twins when you can.

Go forth and conquer those rationals! 🎯