Trinomial

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

Examples of trinomial expressions

$3 x + 5 y + 8 z$ with $x, y, z$ variables
$3 t + 9 s^{2} + 3 y^{3}$ with $t, s, y$ variables
$3 t s + 9 t + 5 s$ with $t, s$ variables
$a x^{2} + b x + c$ , the quadratic polynomial in standard form with $a, b, c$ variables.^{[note 1]}
$A x^{a} y^{b} z^{c} + B t + C s$ with $x, y, z, t, s$ variables, $a, b, c$ nonnegative integers and $A, B, C$ any constants.
$P x^{a} + Q x^{b} + R x^{c}$ 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]

a x^{2} + b x + c

a^{3} \pm b^{3} = (a \pm b) (a^{2} \mp a b + 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 (Template:Math below). This form is factored as:

x^{2 n} + r x^{n} + s = (x^{n} + a_{1}) (x^{n} + a_{2}),

where

\begin{matrix} a_{1} + a_{2} & = r \\ a_{1} \cdot a_{2} & = s . \end{matrix}

For instance, the polynomial Template:Math is an example of this type of trinomial with Template:Math. The solution Template:Math and Template:Math of the above system gives the trinomial factorization:

The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.

Cite error: <ref> tags exist for a group named "note", but no corresponding <references group="note"/> tag was found