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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Polynomials and polynomial functions
By degree	Zero polynomial (degree undefined or −1 or −∞) Constant function (0) Linear function (1) Linear equation Quadratic function (2) Quadratic equation Cubic function (3) Cubic equation Quartic function (4) Quartic equation Quintic function (5) Sextic equation (6) Septic equation (7)
By properties	Univariate Bivariate Multivariate Monomial Binomial Trinomial Irreducible Square-free Homogeneous Quasi-homogeneous
Tools and algorithms	Factorization Greatest common divisor Division Horner's method of evaluation Resultant Discriminant Gröbner basis