Trinomial

Short description: Polynomial that has three terms

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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.

Examples of trinomial expressions

1. [math]\displaystyle{ 3x + 5y + 8z }[/math] with [math]\displaystyle{ x, y, z }[/math] variables
2. [math]\displaystyle{ 3t + 9s^2 + 3y^3 }[/math] with [math]\displaystyle{ t, s, y }[/math] variables
3. [math]\displaystyle{ 3ts + 9t + 5s }[/math] with [math]\displaystyle{ t, s }[/math] variables
4. [math]\displaystyle{ ax^2+bx+c }[/math], the quadratic polynomial in standard form with [math]\displaystyle{ a,b,c }[/math] variables.
5. [math]\displaystyle{ A x^a y^b z^c + B t + C s }[/math] with [math]\displaystyle{ x, y, z, t, s }[/math] variables, [math]\displaystyle{ a, b, c }[/math] nonnegative integers and [math]\displaystyle{ A, B, C }[/math] any constants.
6. [math]\displaystyle{ Px^a + Qx^b + Rx^c }[/math] where [math]\displaystyle{ x }[/math] is variable and constants [math]\displaystyle{ a, b, c }[/math] are nonnegative integers and [math]\displaystyle{ P, Q, R }[/math] any constants.

Trinomial equation

A trinomial equation is a polynomial equation involving three terms. An example is the equation [math]\displaystyle{ x = q + x^m }[/math] studied by Johann Heinrich Lambert in the 18th century.

Some notable trinomials

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

[math]\displaystyle{ ax^2+bx+c }[/math]

• sum or difference of two cubes:

[math]\displaystyle{ a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) }[/math]

• 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:

[math]\displaystyle{ x^{2n} + rx^n + s = (x^n + a_1)(x^n + a_2), }[/math]

where

[math]\displaystyle{ \begin{align} a_1+a_2 &= r\\ a_1 \cdot a_2 &= s. \end{align} }[/math]

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.

See also

• Trinomial expansion
• Monomial
• Binomial
• Multinomial
• Simple expression
• Compound expression
• Sparse polynomial