Trinomial Definition

A trinomial is a polynomial consisting of three terms or monomials which are not like terms. Examples of trinomials include: x² + 4x - 6, 4x⁵ - 3x⁴ + x³, and a²b + 6x + c.

Trinomial Expressions

Examples of trinomial expressions:

3x + 5y + 8z with x,y,z variables.
3t + 9s² + 3y³ with t,s,y variables.
3ts + 9t + 5s with t,s variables.
Ax^ay^bz^c + Bt + Cs with x,y,z,t,s variables, a,b,c nonnegative integers and A,B,C any constants.
Px^a + Qx^b + Rx^c where x is variable and constants a,b,c are nonnegative integers and P,Q,R any constants.

Trinomial Equation

A trinomial equation is a polynomial equation involving three terms. An example is the equation x = q + x^m studied by Johann Heinrich Lambert in the 18th century. Some notable trinomials include:

Sum or difference of two cubes:

(a³ ± b³) = (a ± b)(a² ∓ ab + b²)

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:

x²ⁿ + sxⁿ + p = (xⁿ + a₁)(xⁿ + a₂),
Where:

a₁ + a₂ = s.
a₁ ∙ a₂ = p.

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: