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

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 polynomial 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.

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]

• The quadratic trinomial in standard form (as from above):

${\displaystyle ax^{2}+bx+c}$

• sum or difference of two cubes:

${\displaystyle a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})}$

• 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.

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