Factoring cheat sheet

Factoring is pattern recognition applied in a fixed order. Pull out the common factor first, then count terms: two terms suggests a squares or cubes pattern, three suggests a trinomial, four suggests grouping.

Practise algebra

Order of attack

$1. GCF$ Always first: 6x^2+9x = 3x(2x+3)
$2. Two terms$ Difference of squares or cubes
$3. Three terms$ Trinomial: unfoil, or AC method
$4. Four terms$ Group in pairs
$5. Re-check$ Factor each factor again if possible

Two-term patterns

$a^{2} - b^{2} = (a - b) (a + b)$
$a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$
$a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$
$a^{2} + b^{2}$ Prime over the reals, do not factor

Trinomials

$x^{2} + (p + q) x + pq = (x + p) (x + q)$
$x^{2} + 7 x + 12 = (x + 3) (x + 4)$ Sum 7, product 12
$a x^{2} + b x + c$ AC method: split bx using two factors of ac
$a^{2} \pm 2 ab + b^{2} = (a \pm b)^{2}$ Perfect square trinomial

Grouping

$x^{3} + 3 x^{2} + 2 x + 6 = x^{2} (x + 3) + 2 (x + 3)$
$= (x + 3) (x^{2} + 2)$

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