{{Short description|Polynomial that has three terms}} {{About|mathematics|the use in taxonomy|Trinomial name|the use identifying archaeological sites in the United States|Smithsonian trinomial}} [[File:Pascal_pyramid_trinomial.svg|thumb|upright=2|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.<ref>{{cite web |url=https://www.mathsisfun.com/definitions/trinomial.html | title=Definition of Trinomial | work=Math Is Fun | accessdate=16 April 2016}}</ref>
==Examples of trinomial expressions== # <math>3x + 5y + 8z</math> with <math>x, y, z</math> variables # <math>3t + 9s^2 + 3y^3</math> with <math>t, s, y</math> variables # <math>3ts + 9t + 5s</math> with <math>t, s</math> variables # <math>ax^2+bx+c</math>, the quadratic polynomial in standard form with <math>a,b,c</math> variables.<ref group=note>Quadratic expressions are not always trinomials, the expressions' appearance can vary.</ref> # <math>A x^a y^b z^c + B t + C s</math> with <math>x, y, z, t, s</math> variables, <math>a, b, c</math> nonnegative integers and <math>A, B, C</math> any constants. # <math>Px^a + Qx^b + Rx^c</math> where <math>x</math> is variable and constants <math>a, b, c</math> are nonnegative integers and <math>P, Q, R</math> any constants.
==Trinomial equation== A trinomial equation is a polynomial equation involving three terms. An example is the equation <math>x = q + x^m</math> studied by Johann Heinrich Lambert in the 18th century.<ref>{{cite journal |first=R. M. |last=Corless |first2=G. H. |last2=Gonnet |first3=D. E. G. |last3=Hare |first4=D. J. |last4=Jerey |first5=D. E. |last5=Knuth |year=1996 |url=http://www.cs.uwaterloo.ca/research/tr/1993/03/W.pdf |title=On the Lambert ''W'' Function |journal=Advances in Computational Mathematics |volume=5 |issue=1 |pages=329–359 |doi=10.1007/BF02124750 }}</ref>
===Some notable trinomials ===
* The quadratic trinomial in standard form (as from above): :: <math>ax^2+bx+c</math> * sum or difference of two cubes: :: <math>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 ({{math|''x''<sup>''n''</sup>}} below). This form is factored as: :: <math>x^{2n} + rx^n + s = (x^n + a_1)(x^n + a_2),</math> :where :: <math>\begin{align} a_1+a_2 &= r\\ a_1 \cdot a_2 &= s. \end{align}</math> :For instance, the polynomial {{math|''x''<sup>2</sup> + 3''x'' + 2}} is an example of this type of trinomial with {{math|1=''n'' = 1}}. The solution {{math|1=''a''<sub>1</sub> = −2}} and {{math|1=''a''<sub>2</sub> = −1}} of the above system gives the trinomial factorization: ::{{math|1=''x''<sup>2</sup> + 3''x'' + 2 = (''x'' + ''a''<sub>1</sub>)(''x'' + ''a''<sub>2</sub>) = (''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== {{div col}} *Trinomial expansion *Monomial *Binomial *Multinomial *Simple expression *Compound expression *Sparse polynomial {{div col end}}
==Notes== {{reflist|group=note}}
==References== {{reflist}}
{{polynomials}}
Category:Elementary algebra Category:Polynomials
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