Math101 · Independent Ontario learning library
Math101Factoring by Grouping
Factoring by grouping rewrites a polynomial as groups with a shared binomial or other common factor. For four terms, pair terms, factor the GCF from each pair, then factor the repeated bracket.
Grouping factors polynomials that do not have a single GCF and reveals the structure behind many cubic and trinomial factorizations.
Intuition and core definition
Factoring by grouping rewrites a polynomial as groups with a shared binomial or other common factor. For four terms, pair terms, factor the GCF from each pair, then factor the repeated bracket. The method is valid because it applies the distributive property twice in reverse.
Notation, language, and conditions
$ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)$. A negative GCF may be chosen from one group so the bracket expressions match exactly. Grouping also supports splitting the middle term of a trinomial after finding numbers with a required product and sum.
Why this idea matters
Factoring by grouping exposes a shared binomial by first creating common factors within carefully chosen pairs of terms.
A dependable method
- Remove a GCF common to all terms.
- Arrange and group terms so each group has a useful GCF.
- Factor the GCF from each group, taking a negative factor if needed.
- Confirm the remaining group factors are identical.
- Factor that shared expression and expand to verify.
