Trinomial

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

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.

Trinomial equation

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]

Some notable trinomials

sum or difference of two cubes:

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

where

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

For example, 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 factoring:

(x 2 + 3 x + 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.

References

"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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[1] "Definition of Trinomial". Math Is Fun. Retrieved 16 April 2016.

[2] 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.

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

Trinomial

Trinomial expressions

Trinomial equation

Some notable trinomials

See also

References