How Do You Expand Polynomials with Three Terms Using the FOIL Method?
Multiplying Trinomials: Extending the FOIL Metaphor
The Linguistic and Pedagogical Metaphor
In mathematics education, FOIL is a popular mnemonic device standing for First, Outer, Inner, Last. Linguistically, it functions as a cognitive chunking mechanism—a mnemonic that helps students map the distributive property of multiplication over addition when multiplying two binomials (e.g., $(a + b)(c + d)$). However, strictly speaking, FOIL is linguistically and mathematically bound to two-term expressions.
When students ask about using FOIL with "three terms" (trinomials, such as $(a + b + c)(d + e + f)$), they are stretching a pedagogical acronym beyond its native definition. Etymologically and structurally, applying FOIL to three terms requires shifting from a simple 4-step linear mnemonic to a broader application of the distributive property.
Step-by-Step Mathematical Mechanics
To multiply a trinomial by a trinomial (or a binomial by a trinomial), you cannot rely on the literal "First-Outer-Inner-Last" sequence. Instead, you generalize the underlying operation:
- Substitution / Grouping: Treat one of the trinomials as a single complex term using parentheses. For example, in $(x + y + 2)(x + y - 3)$, let $u = (x + y)$.
- Distribution: Rewrite the expression as $(u + 2)(u - 3)$.
- Apply FOIL to the Binomial Form: Multiply out to get $u^2 - 3u + 2u - 6$, which simplifies to $u^2 - u - 6$.
- Back-Substitution: Substitute $(x + y)$ back in for $u$: $(x + y)^2 - (x + y) - 6$.
- Final Expansion: Expand $(x + y)^2$ to $x^2 + 2xy + y^2 - x - y - 6$.
Alternatively, you can use the extended distributive method (sometimes called the grid or tabular method), multiplying every single term in the first polynomial by every single term in the second polynomial. For two trinomials, this yields $3 \times 3 = 9$ distinct products before combining like terms.