🔢 Algebra Basics — Order of Operations & Factoring
The foundation of all algebraic manipulation, these skills appear in virtually every AMC problem.
🎯 Key Ideas
Order of Operations (PEMDAS) — The universal rule for evaluating expressions: Parentheses, Exponents, Multiplication/Division (left-to-right), Addition/Subtraction (left-to-right). This prevents ambiguity and ensures consistent results.
Factoring Patterns — Recognizing and applying standard algebraic identities is crucial for simplifying expressions, solving equations, and revealing hidden structure in problems.
📚 Essential Factoring Patterns
Basic Identities
| Pattern | Formula | Example |
|---|---|---|
| Difference of squares | $a^2 - b^2 = (a-b)(a+b)$ | $x^2 - 9 = (x-3)(x+3)$ |
| Perfect square | $(a \pm b)^2 = a^2 \pm 2ab + b^2$ | $(x+2)^2 = x^2 + 4x + 4$ |
| Sum of cubes | $a^3 + b^3 = (a+b)(a^2-ab+b^2)$ | $x^3 + 8 = (x+2)(x^2-2x+4)$ |
| Difference of cubes | $a^3 - b^3 = (a-b)(a^2+ab+b^2)$ | $x^3 - 27 = (x-3)(x^2+3x+9)$ |
Advanced Patterns
| Pattern | Formula | Example |
|---|---|---|
| Sophie Germain | $a^4 + 4b^4 = (a^2+2ab+2b^2)(a^2-2ab+2b^2)$ | $x^4 + 4 = (x^2+2x+2)(x^2-2x+2)$ |
| Trinomial | $x^2 + (a+b)x + ab = (x+a)(x+b)$ | $x^2 + 5x + 6 = (x+2)(x+3)$ |
| Grouping | $ax + ay + bx + by = a(x+y) + b(x+y) = (a+b)(x+y)$ | $2x + 2y + 3x + 3y = 5(x+y)$ |
🎯 Micro-Examples
Example 1: Factor $x^2 - 16$
- Recognition: Difference of squares pattern
- Solution: $x^2 - 16 = x^2 - 4^2 = (x-4)(x+4)$
Example 2: Factor $x^2 + 6x + 9$
- Recognition: Perfect square pattern
- Solution: $x^2 + 6x + 9 = (x+3)^2$
Example 3: Factor $x^3 - 8$
- Recognition: Difference of cubes pattern
- Solution: $x^3 - 8 = x^3 - 2^3 = (x-2)(x^2+2x+4)$
⚠️ Common Traps & Fixes
Trap: Forgetting to check for common factors first
- Fix: Always factor out GCF before applying patterns
- Example: $2x^2 - 8 = 2(x^2 - 4) = 2(x-2)(x+2)$
Trap: Incorrect signs in perfect squares
- Fix: Remember $(a-b)^2 = a^2 - 2ab + b^2$ (middle term is negative)
- Example: $(x-3)^2 = x^2 - 6x + 9$, not $x^2 + 6x + 9$
Trap: Confusing sum and difference of cubes
- Fix: Sum: $a^3 + b^3 = (a+b)(a^2-ab+b^2)$ (middle term negative)
- Fix: Difference: $a^3 - b^3 = (a-b)(a^2+ab+b^2)$ (middle term positive)
🎯 AMC-Style Worked Example
Problem: Factor completely: $x^4 + 4x^2 + 4$
Solution:
- Recognize pattern: This looks like a perfect square trinomial
- Apply identity: $x^4 + 4x^2 + 4 = (x^2)^2 + 2(x^2)(2) + 2^2 = (x^2 + 2)^2$
- Check if further factoring possible: $x^2 + 2$ doesn’t factor over reals
- Final answer: $(x^2 + 2)^2$
Key insight: Treat $x^2$ as a single variable to recognize the pattern.
🔗 Related Topics
- Polynomial Theory — These patterns extend to higher-degree polynomials
- Rational Expressions — Factoring is essential for simplifying fractions
- Quadratic Equations — Factoring is often the fastest solution method
- Systems of Equations — Factoring can reveal substitution opportunities
📝 Practice Checklist
- Master all basic factoring patterns
- Practice recognizing patterns in complex expressions
- Learn to factor by grouping
- Understand when to use each pattern
- Practice with negative signs and coefficients
- Work on speed and accuracy
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