Home / All Definitions / Algebra / Trinomial Definition

A trinomial is a polynomial consisting of three terms or monomials which are not like terms. Examples of trinomials include: x^{2} + 4x - 6, 4x^{5} - 3x^{4} + x^{3}, and a^{2}b + 6x + c.

Examples of trinomial expressions:

3x + 5y + 8z with x,y,z variables.

3t + 9s

^{2}+ 3y^{3}with t,s,y variables.3ts + 9t + 5s with t,s variables.

Ax

^{a}y^{b}z^{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.

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:

(a

^{3}± b^{3}) = (a ± b)(a^{2}∓ 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 (x

^{n}below). This form is factored as:x

^{2n}+ sx^{n}+ p = (x^{n}+ a_{1})(x^{n}+ a_{2}),Where:

a

_{1}+ a_{2}= s.a

_{1}∙ a_{2}= p.For example, the polynomial (x

^{2}+ 3x + 2) is an example of this type of trinomial with n = 1. The solution a_{1}= 2 and a_{2}= 1 of the above system gives the trinomial factoring:(x

^{2}+ 3x + 2) = (x + a_{1})(x + a_{2}) = (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.

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