Trinomial Tree, Trinomial

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

Examples of trinomial expressions

# $3x + 5y + 8z$ with $x, y, z$ variables
# $3t + 9s^2 + 3y^3$ with $t, s, y$ variables
# $3ts + 9t + 5s$ with $t, s$ variables
# $ax^2+bx+c$, the quadratic polynomial in standard form with $a,b,c$ variables.Quadratic expressions are not always trinomials, the expressions' appearance can vary.
# $A x^a y^b z^c + B t + C s$ with $x, y, z, t, s$ variables, $a, b, c$ nonnegative integers and $A, B, C$ any constants.
# $Px^a + Qx^b + Rx^c$ where $x$ is variable and constants $a, b, c$ are nonnegative integers and $P, Q, R$ any constants.

Trinomial equation

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.

Some notable trinomials

* The quadratic trinomial in standard form (as from above):
:: $ax^2+bx+c$
* sum or difference of two cubes:
:: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$
* 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 ( below). This form is factored as:
:: $x^ + rx^n + s = (x^n + a_1)(x^n + a_2),$
:where
:: $\begin a_1+a_2 &= r\\ a_1 \cdot a_2 &= s. \end$
:For instance, the polynomial is an example of this type of trinomial with . The solution and of the above system gives the trinomial factorization:
::.
: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

References

Elementary algebra
Polynomials

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