trinomials

Trinomials are polynomials that consist of exactly three nonzero terms. They can have any degree, for example x^3 + 2x^2 − 5, 3x^5 − x^2 + 4, or x^2 + 4x + 4. The term trinomial distinguishes them from binomials (two terms) and monomials (one term). When the highest degree is two, the trinomial is usually called a quadratic trinomial and has the form ax^2 + bx + c with a ≠ 0.

Quadratic trinomials and factoring. Factoring a quadratic trinomial involves finding two numbers that multiply to ac and add to b. If possible, it factors as (dx + e)(fx + g) with df = a, eg = c, and dg + ef = b. For example, 2x^2 + 7x + 3 factors as (2x + 1)(x + 3). The discriminant D = b^2 − 4ac determines real factorability: if D is a perfect square, the trinomial factors over the integers; if D < 0, there are no real roots.

Higher-degree trinomials. A trinomial of degree greater than two can sometimes factor into products of binomials or quadratics, but not always. For instance, x^4 + 5x^2 + 6 = (x^2 + 2)(x^2 + 3), a three-term polynomial that factors into two quadratics. Substitution, such as t = x^k, can aid factoring in some cases by reducing to a quadratic in t. However, many high-degree trinomials do not have a simple factorization over the integers.

Solutions and methods. Quadratic trinomials yield roots via the quadratic formula. For higher-degree trinomials, roots may be real or complex and are often found by numerical methods or by applying specific factoring patterns when present.