Factoring: multiplication in reverse

Expanding turns (x + 2)(x + 3) into x² + 5x + 6. Factoring is the return trip: starting from the sum and recovering the product. It's the single most used skill in the rest of algebra — and the first move is always the same.

Why bother writing it as a product?

Because products have a superpower sums don't: you can tell when they're zero. If A · B = 0 then A = 0 or B = 0 — no third option. A factored expression hands you its roots (that payoff arrives in full in the quadratics module). Factored form also cancels in fractions and reveals structure that a sum hides.

Move one: pull out the GCF

The greatest common factor is everything the terms share. In 6x² + 9x:

GCF = 3x. Divide it out of each term and keep it in front:

$6x^2 + 9x = 3x(2x + 3)$

Check by distributing back: 3x · 2x = 6x² ✓, 3x · 3 = 9x ✓. Factoring is always checkable — multiply it back out.

Always first, whatever comes next

Every factoring problem starts with "is there a GCF?" — because pulling it out shrinks everything that follows. Compare factoring 2x² + 10x + 12 directly with pulling the 2 first: 2(x² + 5x + 6), and now the inside is the friendly x² + 5x + 6 = (x + 2)(x + 3). Full answer: 2(x + 2)(x + 3).

Skipping the GCF isn't wrong, but it leaves you factoring with clumsy numbers — and on a test, out of time.

Completely factored

An expression is factored completely when no factor can be broken further: GCF out, and every polynomial factor irreducible. 4x² − 100 = 4(x² − 25) is a start, but x² − 25 still splits (a preview of the special products lesson): 4(x − 5)(x + 5) is the finish line.

Why this matters

The next two lessons factor trinomials and special patterns; the quadratics module solves equations by factoring. Every one of those begins with the reflex you just built: look for the GCF first.