# Trinomial

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

## Trinomial expressions

1. ${\displaystyle 3x+5y+8z}$ with ${\displaystyle x,y,z}$ variables
2. ${\displaystyle 3t+9s^{2}+3y^{3}}$ with ${\displaystyle t,s,y}$ variables
3. ${\displaystyle 3ts+9t+5s}$ with ${\displaystyle t,s}$ variables
4. ${\displaystyle Ax^{a}y^{b}z^{c}+Bt+Cs}$ with ${\displaystyle x,y,z,t,s}$ variables, ${\displaystyle a,b,c}$ nonnegative integers and ${\displaystyle A,B,C}$ any constants.
5. ${\displaystyle Px^{a}+Qx^{b}+Rx^{c}}$ where ${\displaystyle x}$ is variable and constants ${\displaystyle a,b,c}$ are nonnegative integers and ${\displaystyle P,Q,R}$ any constants.

## 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. [2]

### Some notable trinomials

${\displaystyle (a^{3}\pm b^{3})=(a\pm b)(a^{2}\mp ab+b^{2})}$
• 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:
${\displaystyle x^{2n}+sx^{n}+p=(x^{n}+a_{1})(x^{n}+a_{2}),}$

where

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

For example, 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 factoring:

(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.

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## References

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.