Factoring
Always check for GCF first. 6x² + 9x = 3x(2x + 3).
Difference of squares: a² − b² = (a + b)(a − b). Example: 4x² − 25 = (2x + 5)(2x − 5).
Factoring trinomials (leading coefficient = 1): x² + bx + c = (x + p)(x + q) where p × q = c and p + q = b. Example: x² − 5x + 6 — need two numbers that multiply to 6 and add to −5: (−2)(−3) → (x − 2)(x − 3).
Factoring trinomials (leading coefficient ≠ 1): 2x² + 7x + 3 — multiply a × c = 6; find factors of 6 that add to 7: (6 + 1). Rewrite: 2x² + 6x + x + 3. Group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²). Example: x³ + 8 = (x + 2)(x² − 2x + 4).
Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²).
Simplifying rational expressions: Factor numerator and denominator fully, cancel common factors. (x² − x − 6)/(x² − 4) = (x − 3)(x + 2) / (x − 2)(x + 2) = (x − 3)/(x − 2), where x ≠ ±2.