Factoring Polynomials: A Summary by Type

—8x³

Prime Factorization

—1*2*2*2*x*x*x

BINOMIAL, not Difference of Squares

5x² — 20x

UN-distribute GCF

5x(x — 4)

BINOMIAL, Difference of Squares

4x² — 25

Conjugate Binomials

(2x + 5)(2x — 5)

PST, middle term positive

4x² + 20x + 25

Square of a sum:
(ax + b)²

(2x + 5)²

PST, middle term negative

9x² — 42x + 49

Square of a difference:
(ax — b)²

(3x — 7)²

TRINOMIAL,

a = 1; last term positive

x² + 12x + 35

or

x² — 12x + 35

Sign of L means add factors of L to get middle term. Factors have same sign as middle term.

(x + 7)(x + 5)

or

(x — 7)(x — 5)

TRINOMIAL,

a = 1; last term negative

x² + 2x — 35

or

x² — 2x — 35

Sign of L means subtract factors of L to get middle term. Larger Factor has same sign as middle term.

(x + 7)(x — 5)

or

(x — 7)(x + 5)

TRINOMIAL,

a not equal to 1

6x² + x — 35

Identify: a, b, and c 
factor Bottoms Up! to the front.

a*c = 6*—35 = —210
Y1 = —210/X
Y2= —210/X + X

POLYNOMIAL of Four Terms

6x² + 3x + 2x + 1

Factor by Grouping
(2 X 2); then 
UN-distribute up to three times.

(6x² + 3x) + (2x + 1)
3x(2x + 1) + 1(2x + 1)
(2x + 1)(3x + 1)