In elementary algebra, a trinomial is a polynomial consisting of three terms or monomials.
Examples of trinomial expressions
3 x + 5 y + 8 z {\displaystyle 3x+5y+8z} with x , y , z {\displaystyle x,y,z} variables
3 t + 9 s 2 + 3 y 3 {\displaystyle 3t+9s^{2}+3y^{3}} with t , s , y {\displaystyle t,s,y} variables
3 t s + 9 t + 5 s {\displaystyle 3ts+9t+5s} with t , s {\displaystyle t,s} variables
a x 2 + b x + c {\displaystyle ax^{2}+bx+c} , the quadratic polynomial in standard form with a , b , c {\displaystyle a,b,c} variables.
A x a y b z c + B t + C s {\displaystyle Ax^{a}y^{b}z^{c}+Bt+Cs} with x , y , z , t , s {\displaystyle x,y,z,t,s} variables, a , b , c {\displaystyle a,b,c} nonnegative integers and A , B , C {\displaystyle A,B,C} any constants.
P x a + Q x b + R x c {\displaystyle Px^{a}+Qx^{b}+Rx^{c}} where x {\displaystyle x} is variable and constants a , b , c {\displaystyle a,b,c} are nonnegative integers and P , Q , R {\displaystyle 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 {\displaystyle 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):
a x 2 + b x + c {\displaystyle ax^{2}+bx+c}
sum or difference of two cubes:
a 3 ± b 3 = ( a ± b ) ( a 2 ∓ a b + b 2 ) {\displaystyle 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 2 n + r x n + s = ( x n + a 1 ) ( x n + a 2 ) , {\displaystyle x^{2n}+rx^{n}+s=(x^{n}+a_{1})(x^{n}+a_{2}),}
where
a 1 + a 2 = r a 1 ⋅ a 2 = s . {\displaystyle {\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.
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