Notes on Rational Expressions, Multiplication, Division, Addition and Subtraction

Multiplication Steps:
- Step 1: Completely factor both numerators and denominators of all fractions.
- Step 2: Cancel or reduce the fractions.
- You can cancel something in the numerator with something in the denominator if the two factors are EXACTLY the same.
- Step 3: Rewrite the remaining factors without needing to multiply those factors.
- Example:
- $\frac{3}{4} \times \frac{1}{6} = \frac{3 \times 1}{4 \times 6} = \frac{3}{24} = \frac{1}{8}$

Adding and Subtracting with Like Denominators:
- Example:
- $\frac{3 + x + 4}{x} = \frac{x + 7}{x}$
- $\frac{4x + 5 - (x - 2)}{x + 3} = \frac{3x + 7}{x + 3}$
Adding and Subtracting with Unlike Denominators:
- To add or subtract, find the Least Common Denominator (LCD).
- Example:
- Add:
  - $\frac{4a^2}{10a} + \frac{15b}{10a} = \frac{4a^2 + 15b}{10a}$
- Find LCD while combining like terms for more complex expressions.

When faced with expressions such as $\frac{3x^2 + 3xy}{(x + y)(x - y)} + \frac{(2 - 3x)(x + y)}{(x + y)(x - y)}$, you first determine the LCD:
- $LCD = (x + y)(x - y)$
Combine by adding and simplifying the numerators, ensuring you factor and cancel where applicable.

Division Steps:
- Step 1: Factor both numerators and denominators.
- Step 2: Change division to multiplication and flip the second fraction (take the reciprocal).
- Step 3: Cancel or reduce the fractions, similar to multiplication.
- Step 4: Rewrite the remaining factors without multiplying them.
- Example:
- $\frac{2}{3} \div \frac{6}{5} \rightarrow \frac{2}{3} \times \frac{5}{6} = \frac{10}{18} = \frac{5}{9}$

Always fully factor expressions whenever simplifying.
Cancel terms must be exactly the same to reduce fractions.
Ensure all operations result in fully factored results for cleaner simplification.
Look for opportunities to factor during addition, subtraction, and multiplication to simplify complex expressions.