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.
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.
Axaybzc + Bt + Cs with x,y,z,t,s variables, a,b,c nonnegative integers and A,B,C any constants.
Pxa + Qxb + Rxc where x is variable and constants a,b,c are nonnegative integers and P,Q,R any constants.
A trinomial equation is a polynomial equation involving three terms. An example is the equation x = q + xm 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),
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:
(x2 + 3x + 2) = (x + a1)(x + a2) = (x + 2)(x + 1).
The same result can be provided by the Ruffini's rule, but with a more complex and time-consuming process.
“Trinomial.” Wikipedia, Wikimedia Foundation, 29 Oct. 2019, en.wikipedia.org/wiki/Trinomial.