Factoring Strategies (GCF & Grouping)

Factoring reverses multiplication: given a polynomial, find the product of polynomials that expands to it. There are five patterns to master in order; try them in this exact sequence.

1. Greatest Common Factor (GCF)

Always your first step. Find the largest factor (numerical and variable) shared by every term, and pull it out front.

6x² − 9x = 3x(2x − 3)

The numerical GCF of 6 and 9 is 3; the variable GCF is x (the lowest power present).

2. Difference of squares

a² − b² = (a + b)(a − b). Both terms must be perfect squares, separated by a minus sign.

x² − 16 = (x + 4)(x − 4)
4x² − 25 = (2x + 5)(2x − 5)

Note: a² + b² does not factor over the real numbers. (Sum of squares only factors using imaginary numbers.)

3. Perfect square trinomials

a² ± 2ab + b² = (a ± b)². The middle term must be exactly 2ab for the pattern to apply.

x² + 6x + 9 = (x + 3)²    (because 2 · 1 · 3 = 6)
x² − 10x + 25 = (x − 5)²

4. Sum and difference of cubes

a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

Memorize the signs. The first factor matches the sign of the original; the middle term of the trinomial factor has the opposite sign.

5. Factoring by grouping

When you have four terms, try pairing them so each pair has a common factor that yields the same binomial.

x³ + 2x² + 3x + 6
= x²(x + 2) + 3(x + 2)
= (x + 2)(x² + 3)

If the binomials don’t match after pulling out factors, try a different grouping or check if you missed a sign.

Why factor at all?

Roots solve p(x) = 0. If p(x) = (x − r)(x − s), then setting each factor to zero gives the roots immediately: x = r or x = s. Without factoring, you’d need numerical methods.

ML connection

Finding eigenvalues of a matrix means factoring its characteristic polynomial. PCA’s principal components come from solving det(A − λI) = 0, which for small matrices is a hand-factorable polynomial.