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Factoring by Grouping

When four terms have no factor common to all of them, split them into two pairs, factor each pair, and watch a shared binomial appear. That shared binomial is the key to grouping.

When there is no common factor for all four

Some polynomials have four terms and no single GCF shared by all of them — yet they still factor. The trick is [[factoring-by-grouping|factoring by grouping]]: split the four terms into two pairs, pull a common factor out of each pair separately, and hope the two pairs are left holding the same binomial. If they are, that binomial factors out and you are done.

Factor x^3 + 3x^2 + 2x + 6

Group:        (x^3 + 3x^2) + (2x + 6)
GCF of pair 1: x^2(x + 3)
GCF of pair 2:   2(x + 3)

            = x^2(x + 3) + 2(x + 3)
Shared binomial (x + 3) factors out:
            = (x + 3)(x^2 + 2)

Check: (x + 3)(x^2 + 2) = x^3 + 2x + 3x^2 + 6  ✓
Two pairs, each leaving (x + 3) — so (x + 3) comes out.

The steps, and a sign trap

  1. First pull out any overall GCF from all four terms (often there is none — that is fine).
  2. Group into two pairs and factor the GCF out of each pair.
  3. Confirm the two parentheses are identical, then factor that binomial out front.
  4. Multiply back to check.

When the pairs disagree

If the two parentheses come out different, do not give up — try a different pairing of the four terms. Grouping often only works for one arrangement. If no pairing produces a matching binomial after a couple of honest tries, the polynomial may not factor by grouping at all. Grouping is also the engine behind the AC method for trinomials, which the next guide builds on directly.