48.7 Factoring by Grouping

Factoring by Grouping is a technique for factoring polynomials with four or more terms in which no single monomial factor is shared by every term, but the terms can be split into smaller groups, each of which does share its own common factor, revealing a further shared binomial factor once those individual group factors have been extracted. It extends common-factor factoring to polynomials that would otherwise resist the basic technique, by applying that same basic technique twice in sequence, once to each group, before combining the results.

The method succeeds specifically when the two extracted group factors leave behind identical binomial expressions, since that shared binomial is what ultimately becomes the second factor of the completed factorization.

Splitting the Polynomial into Groups

Polynomial Term Group Formation

The polynomial's terms are divided into two groups, typically the first two terms and the last two terms in the order the polynomial is written, though other groupings are possible if the initial arrangement does not lead to a successful factorization.

Factoring Each Group Independently

First Group GCF Extraction

The greatest common factor of the first group is identified and extracted, exactly as in ordinary common-factor factoring applied to just those two terms.

Second Group GCF Extraction

The greatest common factor of the second group is identified and extracted independently, using the same procedure applied to that separate pair of terms.

Recognizing and Extracting the Shared Binomial

Matching Binomial Factor Recognition

After both groups have been factored individually, the resulting expression is examined to check whether the binomial factor left inside each group's parentheses is identical; this matching binomial is the signal that the grouping was chosen successfully.

Common Binomial Factor Extraction

Once the matching binomial is confirmed, it is factored out from the combined two-term expression exactly as any common monomial factor would be, treating the entire binomial (c + d) as if it were a single unit being extracted.

Remaining Factor Assembly

The two remaining exterior factors from each group, a and b in this example, are combined into their own binomial, which becomes the second factor of the fully factored polynomial alongside the shared binomial.

When the First Attempt Does Not Work

Alternative Grouping Arrangement

If the initial grouping of terms fails to produce matching binomial factors, the terms can be rearranged into a different grouping, since factoring by grouping does not require the terms to remain in their original order, and a different pairing may reveal the shared binomial that the first pairing did not.

Unsuccessful Grouping Recognition

If no rearrangement of the terms into two groups produces matching binomial factors after extracting each group's GCF, the polynomial does not factor by grouping using this four-term technique, and a different factoring approach must be considered instead.

Preparing Before Grouping

Initial Polynomial GCF before Grouping

Before splitting a polynomial into groups, the entire expression should first be checked for an overall common factor shared by all four or more terms together; extracting that overall factor first, if one exists, simplifies the coefficients and variable parts that grouping will then need to handle.

Worked Example

Full Grouping Procedure

Applying this to x³ + 3x² + 2x + 6, the groups (x³ + 3x²) and (2x + 6) each yield a GCF extraction of x²(x + 3) and 2(x + 3), revealing the shared binomial (x + 3), which combines with x² and 2 to give the final factorization.