Class notes from West Texas A&M University state that the primary trick used to multiply special case polynomials is the FOIL method. Mathematicians also multiply two monomial terms together or multiply a monomial by a polynomial, depending on the specific equation. The FOIL method is usually the fastest when multiplying binomials by binomials.
The FOIL method states which terms come first in a binomial multiplication problem. The method's letters stand for "first terms," "outside terms," "inside terms" and "last terms." Mathematicians use the distributive property on these four terms in each binomial. The distributive property combines like terms to make the binomial easier to read and solve for the variable.
Consider the example of the problem (3x + 5)(2x - 7). To solve using the FOIL method, the first terms need to be multiplied, then the outside terms, followed by the inside terms and then the last terms. This is written out as (3x)(2x) + (3x)(-7) + (5)(2x) + (5)(-7). Using the distributive property, this simplifies to 6x^2 - 21x + 10x - 35, which further simplifies to the solution 6x^2 - 11x - 35.
The FOIL method only works for binomial problems. A monomial has one term, so FOIL cannot apply. Mathematicians working with trinomials and larger terms distribute the first term for every term in the second polynomial.