Factoring Trinomials

Reverse FOIL: split a trinomial back into two binomials.

The idea

Factoring a trinomial $x^{2} + bx + c$ means writing it as a product of two binomials. Expanding such a product shows what to look for: $(x + p)(x + q) = x^{2} + (p + q)x + pq.$ The constant term is the product $pq$ and the middle coefficient is the sum $p + q$. Seen as areas, a rectangle with sides $x + p$ and $x + q$ splits into the four pieces $x^{2}$, $px$, $qx$ and $pq$, the two middle pieces together making the $(p + q)x$ term. So we seek two numbers that multiply to $c$ and add to $b$; once found, they are the constants in the two factors. The method is called reverse FOIL, after the FOIL expansion it undoes.

In practice, list the integer factor pairs of $c$ and test each pair's sum against $b$. The signs narrow the search: when $c$ is positive, the two numbers share the sign of $b$; when $c$ is negative, they have opposite signs, and the one larger in size carries the sign of $b$.

When no pair of integers works, the trinomial does not factor this way, and finding its roots requires a different method.

Ways to work on it

• Walkthrough. Find the number pair, then factor.
• Practice. Factor a monic trinomial with integer roots.
• Hardest. Factor a trinomial whose leading coefficient is not 1.

Not sure where to start? Take the ten-question placement test.