Trinomial

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Short description: Polynomial that has three terms

In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.[1]

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{ 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.
  5. [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.[2]

Some notable trinomials

  • 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} + sx^n + p = (x^n + a_1) (x^n + a_2), }[/math]

where

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

For example, 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 factoring:

(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

References

  1. "Definition of Trinomial". Math Is Fun. https://www.mathsisfun.com/definitions/trinomial.html. 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". Advances in Computational Mathematics 5 (1): 329–359. doi:10.1007/BF02124750. http://www.cs.uwaterloo.ca/research/tr/1993/03/W.pdf.