Factoring Polynomials

Common Factor

Factor out the GCD of all terms: 6x³ + 9x² = 3x²(2x + 3). Always check for a common factor first.

Grouping

Rearrange and group terms: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y).

Special Formulas

Difference of squares: a² − b² = (a − b)(a + b). Example: x² − 9 = (x − 3)(x + 3).

Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²).

Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²). Example: 8x³ − 27 = (2x − 3)(4x² + 6x + 9).

Completing the Square

Converts ax² + bx + c into a(x − h)² + k: ax² + bx + c = a(x + b/2a)² + (c − b²/4a).

Example: x² + 6x + 5 = (x + 3)² − 4 = (x + 1)(x + 5).

Factoring Quadratic Trinomials

Find two numbers whose sum equals the coefficient of x and whose product equals the constant term. Example: x² − 5x + 6 = (x − 2)(x − 3).