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Trinomial

Polynomial that has three terms

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated August 23, 2026
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In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.

01Examples 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.

02Trinomial 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
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.
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Sources and credits

This article is adapted from the Wikipedia article “Trinomial”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Pascal's pyramid

In mathematics, Pascal's pyramid is a three-dimensional arrangement of the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial coefficients that appear in the binomial expansion and the binomial distribution.

Trinomial expansion

In mathematics, a trinomial expansion is the expansion of a power of a sum of three terms into monomials. The expansion is given by n = ∑ i , j , k i + j + k = n ( n i , j , k ) a i b j c k , {\displaystyle (a+b+c)^{n}=\sum _{{i,j,k} \atop {i+j+k=n}}{n \choose i,j,k}\,a^{i}\,b^{\;\!j}\;\!c^{k},} where n is a nonnegative integer and the sum is taken over all combinations of nonnegative indices i, j, and k such that i + j + k = n.

Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions.