49.6 Nonmonic Trinomial Grouping

Nonmonic Trinomial Grouping is the second stage of the nonmonic trinomial factoring process, taking the four-term polynomial produced by splitting the middle term and applying factoring by grouping to it, ultimately producing the two binomial factors of the original nonmonic trinomial. It picks up exactly where the product-sum search and middle-term split leave off, treating the resulting four-term expression exactly as any other polynomial eligible for grouping would be treated.

This stage completes the connection between nonmonic trinomial factoring and the more general grouping technique, confirming that the specific integer pair chosen during the search was correct by producing a valid shared binomial factor.

Splitting the Four Terms into Groups

First Split-Middle Term Group

The first two terms of the four-term expression, the original squared term and the first of the two split middle terms, are grouped together as the first pair to be factored.

Second Split-Middle Term Group

The remaining two terms, the second of the split middle terms and the original constant term, are grouped together as the second pair to be factored.

Factoring Each Group

First Group Monomial Extraction

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

Second Group Monomial 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.

Extracting the Shared Binomial

Shared Binomial Factor Formation

Comparing the two group extractions confirms that both groups leave behind the identical binomial (2x + 1), forming the shared factor that ties the two groups together into a single factorization.

Shared Binomial Extraction

The matching binomial (2x + 1) is factored out from the combined expression, treated as a single unit exactly the way any common monomial factor would be extracted.

Remaining Binomial Factor Assembly

The two exterior factors left over from each group, x and 3 in this example, are combined into their own binomial, which becomes the second factor of the completed factorization alongside the shared binomial.

Handling Difficult Splits

Split-Term Order Adjustment

If the initial order in which the split middle terms were written does not lead to matching binomial factors after both groups are extracted, the order of the two split terms can be reversed, since either arrangement represents the same underlying four-term expression and one ordering may reveal the shared binomial where the other did not.

Confirming the Final Result

Nonmonic Trinomial Product Completion

The completed factorization, consisting of the shared binomial and the remaining binomial, is confirmed by multiplying the two binomial factors back together and checking that the result matches the original nonmonic trinomial exactly.