trinomial

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.Quadratic expressions are not always trinomials, the expressions' appearance can vary.
5. $A\; x^a\; y^b\; z^c\; +\; B\; t\; +\; C\; s$ 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.

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.{{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 }}

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

:: $ax^2+bx+c$

• sum or difference of two cubes:

:: $a^3\; \backslash pm\; b^3\; =\; (a\; \backslash pm\; b)(a^2\; \backslash mp\; ab\; +\; b^2)$

• 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 ({{math|xⁿ}} below). This form is factored as:

:: $x^\{2n\}\; +\; rx^n\; +\; s\; =\; (x^n\; +\; a\_1)(x^n\; +\; a\_2),$

:where

:: $\backslash begin\{align\}$

a_1+a_2 &= r\\

a_1 \cdot a_2 &= s.

\end{align}

:For instance, the polynomial {{math|x² + 3x + 2}} is an example of this type of trinomial with {{math|1=n = 1}}. The solution {{math|1=a₁ = −2}} and {{math|1=a₂ = −1}} of the above system gives the trinomial factorization:

::{{math|1=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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• Trinomial expansion
• Monomial
• Binomial
• Multinomial
• Simple expression
• Compound expression
• Sparse polynomial

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Category:Elementary algebra

Category:Polynomials

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