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## Summary

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

## Examples of trinomial expressions

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.[note 1]
5. $Ax^{a}y^{b}z^{c}+Bt+Cs$  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.

## 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):
$ax^{2}+bx+c$
$a^{3}\pm b^{3}=(a\pm b)(a^{2}\mp ab+b^{2})$
• 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:
$x^{2n}+rx^{n}+s=(x^{n}+a_{1})(x^{n}+a_{2}),$
where
{\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.