Trinomial

This article is about mathematics. For the use in taxonomy, see Trinomial name. For the use identifying archaeological sites in the United States, see Smithsonian trinomial.

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

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

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. $Ax^{a}y^{b}z^{c}+Bt+Cs$ 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 (xⁿ below). This form is factored as:

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

where

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

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.

See also

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