49 Factoring Trinomials

Factoring Trinomials is the study of rewriting a quadratic trinomial as a product of two binomials, reversing the FOIL multiplication process, using distinct strategies depending on whether the trinomial's leading coefficient is 1 or a value other than 1.

The Scope of Trinomial Factoring

A quadratic trinomial has the form ax² + bx + c, and factoring it means finding two binomials whose product, when expanded by FOIL, reproduces the original trinomial exactly. Not every trinomial factors into binomials with integer coefficients; the techniques in this area apply specifically to trinomials that do factor this way, and part of the skill involved is recognizing when a trinomial does not factor over the integers at all.

Preparing to Factor a Trinomial

Before applying a specific factoring technique, a trinomial should first be checked for a greatest common factor shared by all three terms, which should be factored out first if present, since doing so simplifies the remaining trinomial and its coefficients before the binomial-factoring techniques are applied. The trinomial should also be arranged in standard form, with terms in descending order of degree, to clearly identify a, b, and c before proceeding.

Factoring a Monic Trinomial

A monic trinomial has a leading coefficient of 1, taking the form x² + bx + c. Factoring a monic trinomial means finding two integers whose product equals c and whose sum equals b; those two integers become the constant terms of the two binomial factors.

Here, 3 and 4 are the two integers whose product is 12 and whose sum is 7, and each becomes the constant term of one binomial factor, both paired with x as the variable term.

Sign Analysis for Monic Trinomials

The signs of b and c determine the signs of the two integers being sought, narrowing the search before any pair is tested. If c is positive and b is positive, both integers are positive. If c is positive and b is negative, both integers are negative. If c is negative, the two integers have opposite signs, with the one of larger magnitude taking the same sign as b. Applying this sign analysis first substantially reduces the number of factor pairs of c that need to be checked against the required sum.

Factoring a Nonmonic Trinomial by the Product-Sum Method

A nonmonic trinomial has a leading coefficient other than 1, taking the form ax² + bx + c with a ≠ 1. The product-sum method (sometimes called the "ac method") factors such a trinomial by finding two integers whose product equals a × c and whose sum equals b, then using those two integers to rewrite the middle term bx as two separate terms before factoring by grouping.

Completing Nonmonic Factoring by Grouping

Once the two required integers are found, the middle term is split into two terms using those integers as coefficients, converting the trinomial into a four-term polynomial, which is then factored by grouping exactly as in general four-term factoring.

The order in which the split middle term's two parts are written does not affect the final factored result, though a different order may require pairing the four terms differently to reveal the shared binomial factor.

Trinomials with a Negative Leading Coefficient

When a trinomial's leading coefficient is negative, it is often easiest to first factor out -1 (or another negative common factor) from every term, converting the remaining trinomial inside the parentheses into one with a positive leading coefficient, which is then factored using the monic or nonmonic techniques as appropriate, with the extracted -1 carried through as part of the final factored form.

Trinomials That Do Not Factor

Some trinomials have no pair of integers satisfying the required product-and-sum conditions, meaning the trinomial does not factor into binomials with integer coefficients and is described as prime or irreducible over the integers. Confirming that no valid integer pair exists, after a reasonably thorough search of the factor pairs of c (or ac in the nonmonic case), is itself a valid and complete conclusion for such a trinomial within elementary algebra's integer-based factoring techniques.

Verifying a Factored Trinomial

A factored trinomial is verified by expanding the two binomial factors using FOIL and confirming the result matches the original trinomial exactly, term for term, including the sign of every term.

Diagnosing Errors in Trinomial Factoring

Common errors in this area include searching for a factor pair of c alone when the trinomial is actually nonmonic and the product a × c should have been used instead, misapplying the sign analysis and testing factor pairs of the wrong sign combination, making an arithmetic error when splitting the middle term in the product-sum method, and declaring a trinomial prime without having checked every relevant factor pair of the required product.