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.

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. ^[1]

Examples of trinomial expressions

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

Trinomial equation

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

Some notable trinomials

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

${\displaystyle ax^{2}+bx+c}$

• sum or difference of two cubes:

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

• 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:

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

where

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

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

Notes

1. Quadratic expressions are not always trinomials, the expressions' appearance can vary.

References

1. "Definition of Trinomial". Math Is Fun. Retrieved 16 April 2016.
2. Corless, R. M.; Gonnet, G. H.; Hare, D. E. G.; Jerey, D. J.; Knuth, D. E. (1996). "On the Lambert W Function" (PDF). Advances in Computational Mathematics. 5 (1): 329–359. doi:10.1007/BF02124750.