Trinomial

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.

Some notable trinomials

• The quadratic trinomial in standard form (as from above):

${\displaystyle a{x}^2+bx+c}$

• 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:

${\displaystyle x^{2n}+r{x}^n+s=(x^n+a_1)(x^n+a_2),}$

where

${\displaystyle \begin{array}{cc}{a}_1+a_2& =r\\ a_1\cdot a_2& =s.\end{array}}$

For instance, 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 factorization:

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.