Trinomial Factoring Calculator

A quadratic trinomial is an expression with three non-zero terms with two variables in the expression, where the highest power seen in the expression is 2.

A trinomial is typically in the form

This calculator factors quadratic trinomials into two binomials over the integers. It finds binomials of the form

where the factors satisfy:

How to Use This Calculator

1. Enter the coefficients for a, b, and c in the input fields.
2. Once all coefficients are provided, the calculator will attempt to factor the trinomial.

If the trinomial is factorable over the integers, the result will be shown as two binomials. Otherwise, you'll see a message stating that the trinomial is not factorable over the integers.

Understanding the Factorization

To reiterate, the relation between the trinomial (left) and its binomial factors (right) is as follows:

This calculator uses the ac method that utilizes these steps:

1. Find the divisors of a multiplied with c
2. Find pair of divisors that sum to the term b
3. Use the pair to find the factors that make up each binomial factor of the trinomial

You can see a full example later in this article.

Example Problem

Consider the quadratic trinomial

Recall that

and

The steps to factor the trinomial:

1. Plug in the coefficients

   $p\times r=6$$p\times s+q\times r=4$$q\times s=11$

2. Find the divisors of each coefficient.

   • Coefficient a (6) includes 1, 2, 3, 6 (and negatives)
   • Coefficient c (4) includes 1, 2, 4 (and negatives)

3. Check the candidate pairs

   For a candidate such as p = 2,

   $r=a/p$$r=6/2$$r=3$

   For a candidate such as q = 1,

   $s=c/q$$s=4/1$$s=4$

   Repeat the candidate testing process until the following equation is true.

   $p\times s+q\times r=b$

   Testing these candidates with the equation:

   $p=2$$s=4$$q=1$$r=3$$b=11$

   Therefore,

   $p\times s+q\times r=b$$2\times 4+1\times 3=11$$8+3=11$$11=11$

   This equation is true and the binomial factors have been found.

4. Now that the factors have been found, they can be written in the full form

   $6x^2+11x+4=(2x+1)(3x+4)$

   with the binomial factors being

   $(2x+1)(3x+4)$