Trinomial

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

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
$ax^{2}+bx+c$ , the quadratic polynomial in standard form with $a,b,c$ variables.^{[note 1]}
$Ax^{a}y^{b}z^{c}+Bt+Cs$ 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.^[2]

ax^{2}+bx+c

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 ( $x n$ below). This form is factored as:

x^{2n}+rx^{n}+s=(x^{n}+a_{1})(x^{n}+a_{2}),

where

{\begin{aligned}a_{1}+a_{2}&=r\\a_{1}\cdot a_{2}&=s.\end{aligned}}

For instance, the polynomial

x 2 + 3 x + 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 factorization:

x 2 + 3 x + 2 = (x + a 1)(x + a 2) = (x + 2)(x + 1)

.

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

^ Quadratic expressions are not always trinomials, the expressions' appearance can vary.

^ "Definition of Trinomial". Math Is Fun. Retrieved 16 April 2016.
^ Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jerey, D. J.; Knuth, D. E. (1996). "On the Lambert W Function" (PDF). Advances in Computational Mathematics. 5 (1): 329–359. doi:10.1007/BF02124750.

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