Trinomial
A trinomial is a polynomial with exactly three unlike terms connected by addition or subtraction.
Formula
ax^2 + bx + c
Definition
A trinomial is a polynomial with exactly three unlike terms ("tri" means three), and the standard quadratic trinomial $ax^2 + bx + c$ is the most common form encountered in Algebra I. It results from multiplying two binomials, $(x + p)(x + q) = x^2 + (p+q)x + pq$, so the middle coefficient is $p+q$ and the constant is $pq$, making most trinomial factoring problems a kind of reverse FOIL. A quadratic trinomial factors over $\mathbb{R}$ if and only if the discriminant $b^2 - 4ac$ is greater than or equal to $0$; if it equals $0$ the trinomial is a perfect square, and if it is negative the trinomial is irreducible over $\mathbb{R}$ but factors over $\mathbb{C}$, with factored form $a(x - r_1)(x - r_2)$ for roots $r_1, r_2$.
Example
$x^2 + 5x + 6$, $3y^2 - 2y + 1$, and $a^2 + ab + b^2$ are trinomials. Since $2 \cdot 3 = 6$ and $2 + 3 = 5$, $x^2 + 5x + 6 = (x+2)(x+3)$. For $2x^2 + 5x + 3$, the discriminant is $25 - 24 = 1$, giving roots $x = (-5 \pm 1)/4 = -1$ or $-3/2$, so it factors as $(x+1)(2x+3)$.
Key Insight
"Tri" means three, like a triangle's three sides. The discriminant test for factorability over $\mathbb{Q}$ requires not just $b^2 - 4ac \ge 0$ but also that it be a perfect square (when $a$, $b$, $c$ are integers); otherwise the roots are irrational and the trinomial does not factor over $\mathbb{Q}$, even though it still factors over $\mathbb{R}$.