In algebra, specifically in the factorization of polynomials, a special trinomial is a polynomial that can be expressed in the form:p.419
and for which there is a known method for factoring it as the product of two first-degree binomials.
Factoring Method We can distinguish between the cases in which the coefficient of the quadratic term is equal to or different from .
a = 1 If the coefficient of the quadratic term is equal to , the trinomial takes the form:
in this case, it can be factored into the product of two first-degree binomials, in the form:
,
where and are two terms with the following two properties:
.
In fact, performing the calculations yields:
A practical method to find and is to solve for the two roots of the polynomial. In fact, if
,
then:
To find the roots of the quadratic trinomial, simply use the quadratic formula:
a ≠ 1 If the coefficient of the quadratic term is not , the trinomial can be factored as follows:
,
where and have the following properties:p.872
.
In this case as well, the factorization can be demonstrated as follows:p.277
As in the previous case, and can be found by solving for the roots of the polynomial using the quadratic formula.
Trinomials of degree greater than 2
Factoring by substitution More generally, if we consider the trinomial:p.99
This can be factored by making the substitution , which gives the trinomial:
,
that can be factored using the methods described above and then by reapplying the substitution in reverse.
Direct factoring Considering the previous properties:
.
The trinomial can also be directly factored as follows:
The factorization can be demonstrated as follows:
Also in this case and can be found by solving for the roots of the polynomial using the quadratic formula.
Notes
Bibliography