By Maria O'Brien, Updated Aug 30, 2022
A trinomial is a polynomial with three terms, such as x^2 + 5x + 6. If it can be expressed as the product of two or more lower‑degree polynomials, it is considered factored. A prime trinomial (or irreducible trinomial) cannot be factored over the integers; it remains in its simplest form.
For a monic quadratic of the form x^2 + bx + c, write down all factor pairs of the constant term c. Include both positive and negative pairs when appropriate.
If any pair of factors of c sums to the coefficient b, the trinomial is factorable and therefore not prime.
When none of the factor pair sums equal b, the trinomial is prime. For example, in x^2 – 11x – 10, the factor pairs of –10 are –1×10, –2×5, –5×2, and –10×1. Their sums are –9, 3, –3, and –9, none of which equals –11. Hence, x^2 – 11x – 10 is a prime trinomial.
Sometimes assignments present a “prime” trinomial as a trick question or due to a typo. If you encounter a trinomial that appears factorable but your method shows it is prime, double‑check the coefficients or consult your instructor.
Prime trinomials cannot be factored over the integers. Verify by testing the sums of factor pairs of the constant term against the middle coefficient.
