Layers of Pascal's pyramid derived from coefficients in an upside-down ternary plot of the terms in the expansions of the powers of a trinomial

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

## Examples of 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. ${\displaystyle ax^{2}+bx+c}$, the quadratic form in standard form with ${\displaystyle a,b,c}$ variables.[note 1]
5. ${\displaystyle Ax^{a}y^{b}z^{c}+Bt+Cs}$ with ${\displaystyle x,y,z,t,s}$ variables, ${\displaystyle a,b,c}$ nonnegative integers and ${\displaystyle A,B,C}$ any constants.
6. ${\displaystyle Px^{a}+Qx^{b}+Rx^{c))$ 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

• The quadratic trinomial in standard form (as from above):
${\displaystyle ax^{2}+bx+c}$
${\displaystyle a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})}$
• 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:
${\displaystyle x^{2n}+rx^{n}+s=(x^{n}+a_{1})(x^{n}+a_{2}),}$
where
{\displaystyle {\begin{aligned}a_{1}+a_{2}&=r\\a_{1}\cdot a_{2}&=s.\end{aligned))}
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.