Trinomial

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

Examples of trinomial expressions

 * 1) $$3x + 5y + 8z$$ with $$x, y, z$$ variables
 * 2) $$3t + 9s^2 + 3y^3$$ with $$t, s, y$$ variables
 * 3) $$3ts + 9t + 5s$$ with $$t, s$$ variables
 * 4) $$ax^2+bx+c$$, the quadratic polynomial in standard form with $$a,b,c$$ variables.
 * 5) $$A x^a y^b z^c + B t + C s$$ with $$x, y, z, t, s$$ variables, $$a, b, c$$ nonnegative integers and $$A, B, C$$ any constants.
 * 6) $$Px^a + Qx^b + Rx^c$$ 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.

Some notable trinomials

 * The quadratic trinomial in standard form (as from above):
 * $$ax^2+bx+c$$


 * 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:
 * $$x^{2n} + rx^n + s = (x^n + a_1)(x^n + a_2),$$
 * where
 * $$\begin{align}

a_1+a_2 &= r\\ a_1 \cdot a_2 &= s. \end{align}$$
 * For instance, the polynomial $x^{2} + 3x + 2$ is an example of this type of trinomial with $n =&thinsp;1$. The solution $a_{1} = −2$ and $a_{2} = −1$ 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.
 * The same result can be provided by Ruffini's rule, but with a more complex and time-consuming process.