1. Standard Binomial & Polynomial Square Expansions
Square identities: (a + b)² = a² + 2ab + b²; (a - b)² = a² - 2ab + b²; Difference of squares: a² - b² = (a - b)(a + b); Combined symmetric relations: (a + b)² + (a - b)² = 2(a² + b²), and (a + b)² - (a - b)² = 4ab. Three-variable expansion: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca).
2. Essential Cubic Identities & Factorization Forms
Cubic expansions: (a + b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a + b); (a - b)³ = a³ - 3a²b + 3ab² - b³ = a³ - b³ - 3ab(a - b). Sum and difference of cubes: a³ + b³ = (a + b)(a² - ab + b²); a³ - b³ = (a - b)(a² + ab + b²). Special 3-variable cubic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). Crucial exam corollary: If a + b + c = 0, then a³ + b³ + c³ = 3abc.
3. Quadratic Equation Formula & Nature of Roots
For quadratic equation ax² + bx + c = 0 (with a ≠ 0), roots are given by x = [-b ± √(b² - 4ac)] / (2a). Sum of roots: α + β = -b/a; Product of roots: αβ = c/a. Discriminant D = b² - 4ac: If D > 0 and a perfect square, roots are real, rational, and unequal; if D > 0 and not a perfect square, roots are real, irrational, and unequal (occurring in conjugate pairs); if D = 0, roots are real and equal (α = β = -b/2a); if D < 0, roots are complex/imaginary conjugate pairs.
4. Exponent & Power Rules
Fundamental laws of indices: aᵐ · aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿ bⁿ; (a/b)ⁿ = aⁿ / bⁿ; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a; a⁰ = 1 (for any a ≠ 0).