# Trinomial

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Polynomial that has three terms

This article is about mathematics. For the use in taxonomy, see [Trinomial name](/source/Trinomial_name). For the use identifying archaeological sites in the United States, see [Smithsonian trinomial](/source/Smithsonian_trinomial).

Layers of [Pascal's pyramid](/source/Pascal's_pyramid) derived from [coefficients](/source/Coefficient) in an upside-down [ternary plot](/source/Ternary_plot) of the terms in the [expansions of the powers of a trinomial](/source/Trinomial_expansion)

In [elementary algebra](/source/Elementary_algebra), a **trinomial** is a [polynomial](/source/Polynomial) consisting of three terms or [monomials](/source/Monomial).[1]

## Examples of trinomial expressions

1. 3 x + 5 y + 8 z {\displaystyle 3x+5y+8z} with x , y , z {\displaystyle x,y,z} variables

1. 3 t + 9 s 2 + 3 y 3 {\displaystyle 3t+9s^{2}+3y^{3}} with t , s , y {\displaystyle t,s,y} variables

1. 3 t s + 9 t + 5 s {\displaystyle 3ts+9t+5s} with t , s {\displaystyle t,s} variables

1. a x 2 + b x + c {\displaystyle ax^{2}+bx+c} , the [quadratic polynomial](/source/Quadratic_polynomial) in standard form with a , b , c {\displaystyle a,b,c} variables.[note 1]

1. A x a y b z c + B t + C s {\displaystyle Ax^{a}y^{b}z^{c}+Bt+Cs} with x , y , z , t , s {\displaystyle x,y,z,t,s} variables, a , b , c {\displaystyle a,b,c} nonnegative [integers](/source/Integer) and A , B , C {\displaystyle A,B,C} any constants.

1. P x a + Q x b + R x c {\displaystyle Px^{a}+Qx^{b}+Rx^{c}} where x {\displaystyle x} is variable and constants a , b , c {\displaystyle a,b,c} are nonnegative integers and P , Q , R {\displaystyle P,Q,R} any constants.

## Trinomial equation

A trinomial equation is a [polynomial equation](/source/Polynomial_equation) involving three terms. An example is the equation x = q + x m {\displaystyle x=q+x^{m}} studied by [Johann Heinrich Lambert](/source/Johann_Heinrich_Lambert) in the 18th century.[2]

### Some notable trinomials

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

- - a x 2 + b x + c {\displaystyle ax^{2}+bx+c}

- [sum or difference of two cubes](/source/Factorization#Sum/difference_of_two_cubes):

- - a 3 ± b 3 = ( a ± b ) ( a 2 ∓ a b + b 2 ) {\displaystyle a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})}

- A special type of trinomial can be [factored](/source/Factorization) 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 2 n + r x n + s = ( x n + a 1 ) ( x n + a 2 ) , {\displaystyle x^{2n}+rx^{n}+s=(x^{n}+a_{1})(x^{n}+a_{2}),}

- where - a 1 + a 2 = r a 1 ⋅ a 2 = s . {\displaystyle {\begin{aligned}a_{1}+a_{2}&=r\\a_{1}\cdot a_{2}&=s.\end{aligned}}}

- For instance, the polynomial *x*2 + 3*x* + 2 is an example of this type of trinomial with *n* = 1. The solution *a*1 = −2 and *a*2 = −1 of the above system gives the trinomial factorization: - *x*2 + 3*x* + 2 = (*x* + *a*1)(*x* + *a*2) = (*x* + 2)(*x* + 1).

- The same result can be provided by [Ruffini's rule](/source/Ruffini's_rule#Example), but with a more complex and time-consuming process.

## See also

- [Trinomial expansion](/source/Trinomial_expansion)

- [Monomial](/source/Monomial)

- [Binomial](/source/Binomial_(polynomial))

- [Multinomial](/source/Multinomial_(disambiguation))

- [Simple expression](/source/Simple_expression)

- [Compound expression](/source/Compound_expression)

- [Sparse polynomial](/source/Sparse_polynomial)

## Notes

1. **[^](#cite_ref-2)** Quadratic expressions are not always trinomials, the expressions' appearance can vary.

## References

1. **[^](#cite_ref-1)** ["Definition of Trinomial"](https://www.mathsisfun.com/definitions/trinomial.html). *Math Is Fun*. Retrieved 16 April 2016.

1. **[^](#cite_ref-3)** Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jerey, D. J.; Knuth, D. E. (1996). ["On the Lambert *W* Function"](http://www.cs.uwaterloo.ca/research/tr/1993/03/W.pdf) (PDF). *Advances in Computational Mathematics*. **5** (1): 329–359. [doi](/source/Doi_(identifier)):[10.1007/BF02124750](https://doi.org/10.1007%2FBF02124750).

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