Trinomial
AlgebraA 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 terms. "Tri" means three.
Example
$x^2 + 5x + 6$, $3y^2 - 2y + 1$, and $a^2 + ab + b^2$ are trinomials. Each has three unlike terms.
Key Insight
"Tri" means three. A triangle has three sides. A trinomial has three terms.
Definition
A trinomial has exactly three unlike terms. The standard quadratic trinomial $ax^2 + bx + c$ is the most common form in Algebra I. It results from multiplying two binomials: $(x + p)(x + q) = x^2 + (p+q)x + pq$, where the middle coefficient is $p+q$ and the constant is $pq$.
Example
$x^2 + 7x + 12$: find two numbers with product $12$ and sum $7$: that is $3$ and $4$. So $x^2 + 7x + 12 = (x + 3)(x + 4)$.
Key Insight
Most trinomial factoring problems are reverse FOIL: you are finding what two binomials multiply to give the trinomial.
Definition
A quadratic trinomial $ax^2 + bx + c$ factors over $\mathbb{R}$ if and only if the discriminant $b^2 - 4ac$ is greater than or equal to $0$. If the discriminant equals $0$, the trinomial is a perfect square. If the discriminant is negative, the trinomial is irreducible over $\mathbb{R}$ but factors over $\mathbb{C}$. The factored form is $a(x - r_1)(x - r_2)$ where $r_1, r_2$ are the roots.
Example
$2x^2 + 5x + 3$: discriminant $= 25 - 24 = 1$. Roots: $x = (-5 \pm 1)/4 = -1$ or $-3/2$. Factored: $2(x+1)(x+3/2) = (x+1)(2x+3)$.
Key Insight
The discriminant test for factorability over $\mathbb{Q}$ requires not just $b^2 - 4ac \ge 0$ but also that $b^2 - 4ac$ is a perfect square (if $a$, $b$, $c$ are integers). Otherwise, the roots are irrational and the trinomial does not factor over $\mathbb{Q}$.