Mr. Integral
Math notes/Pre-Calculus/Factoring

Factoring by Grouping — Cheat sheet

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Key method

ab+ac+db+dc=a(b+c)+d(b+c)=(a+d)(b+c).ab+ac+db+dc=a(b+c)+d(b+c)=(a+d)(b+c).

If the two parentheses differ by a sign, factor a negative from one group. If they genuinely differ, try another grouping.

Example

Factor x3+2x2−3x−6x^3+2x^2-3x-6:

x2(x+2)−3(x+2)=(x+2)(x2−3).x^2(x+2)-3(x+2)=(x+2)(x^2-3).

Over the reals this can continue to (x+2)(x−3)(x+3)(x+2)(x-\sqrt3)(x+\sqrt3). Over integer coefficients, the first product is the natural stopping point.

Avoid this mistake

Factoring −3x−6-3x-6 as −3(x−2)-3(x-2) changes the constant term. Distribute the negative to check both signs.