trinomial

A trinomial is a polynomial with exactly three terms. The prefix tri- means three, as in triangle (three sides). A classic example is x^2 + 5x + 6 — three terms joined by plus and minus signs, none of which combine into one another.

The most famous trinomials are the quadratic ones, written ax^2 + bx + c with a not equal to 0. Factoring these — turning x^2 + 5x + 6 back into (x + 2)(x + 3) — is a cornerstone skill, because it lets you solve quadratic equations and simplify rational expressions.

Not every three-term expression is a trinomial in the strict sense, and not every trinomial factors over the integers. Some, like x^2 + x + 1, are prime over the rationals — they simply cannot be broken into nicer factors with whole-number coefficients, and that is a perfectly normal outcome.