# Trinomial

Short description: Polynomial that has three terms

In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.[1]

## Examples of trinomial expressions

1. $\displaystyle{ 3x + 5y + 8z }$ with $\displaystyle{ x, y, z }$ variables
2. $\displaystyle{ 3t + 9s^2 + 3y^3 }$ with $\displaystyle{ t, s, y }$ variables
3. $\displaystyle{ 3ts + 9t + 5s }$ with $\displaystyle{ t, s }$ variables
4. $\displaystyle{ A x^a y^b z^c + B t + C s }$ with $\displaystyle{ x, y, z, t, s }$ variables, $\displaystyle{ a, b, c }$ nonnegative integers and $\displaystyle{ A, B, C }$ any constants.
5. $\displaystyle{ Px^a + Qx^b + Rx^c }$ where $\displaystyle{ x }$ is variable and constants $\displaystyle{ a, b, c }$ are nonnegative integers and $\displaystyle{ P, Q, R }$ any constants.

## Trinomial equation

A trinomial equation is a polynomial equation involving three terms. An example is the equation $\displaystyle{ x = q + x^m }$ studied by Johann Heinrich Lambert in the 18th century.[2]

### Some notable trinomials

• sum or difference of two cubes:
$\displaystyle{ (a^3 \pm b^3) = (a \pm b)(a^2 \mp ab + b^2) }$
• A special type of trinomial can be factored in a manner similar to quadratics since it can be viewed as a quadratic in a new variable (xn below). This form is factored as:
$\displaystyle{ x^{2n} + sx^n + p = (x^n + a_1) (x^n + a_2), }$

where

\displaystyle{ \begin{align} a_1+a_2 &= s\\ a_1 \cdot a_2 &= p. \end{align} }

For example, the polynomial (x2 + 3x + 2) is an example of this type of trinomial with n = 1. The solution a1 = 2 and a2 = 1 of the above system gives the trinomial factoring:

(x2 + 3x+ 2) = (x + a1)(x + a2) = (x + 2)(x + 1).

The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.