Factoring Trinomials
AlgebraFactoring trinomials means expressing a three-term polynomial, usually in the form ax^2 + bx + c, as a product of two binomials.
Formula
ax^2 + bx + c = (px + q)(rx + s)
Definition
Factoring a trinomial means finding two binomials that multiply to give that trinomial. For $x^2 + bx + c$, find two numbers with product $c$ and sum $b$.
Example
Factor $x^2 + 7x + 12$. Find two numbers: product $= 12$, sum $= 7$. That is $3$ and $4$. Answer: $(x + 3)(x + 4)$. Check by FOIL.
Key Insight
The "product and sum" trick: the two numbers you need must multiply to $c$ and add to $b$. List factor pairs of $c$ and find the one that adds to $b$.
Definition
For trinomials of the form $x^2 + bx + c$ (leading coefficient $1$), find two numbers $p$ and $q$ such that $pq = c$ and $p+q = b$. For $ax^2 + bx + c$ ($a \neq 1$), use the "ac method": find two numbers with product $ac$ and sum $b$, then split the middle term and factor by grouping.
Example
Factor $2x^2 + 7x + 3$. Product: $2 \cdot 3 = 6$. Need sum $7$: $6$ and $1$. Split: $2x^2 + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3)$.
Key Insight
If you cannot find two integers with the required product and sum, the trinomial does not factor over the integers. Use the quadratic formula to find irrational roots in that case.
Definition
A trinomial $ax^2 + bx + c$ factors over $\mathbb{Q}$ if and only if the discriminant $b^2 - 4ac$ is a perfect square. Factoring corresponds to finding rational roots via the Rational Root Theorem. If the trinomial factors as $a(x - r_1)(x - r_2)$, the roots $r_1$ and $r_2$ satisfy $r_1 + r_2 = -b/a$ (Vieta's formula for sum) and $r_1r_2 = c/a$ (Vieta's formula for product).
Example
$6x^2 - 5x - 6$: discriminant $= 25 + 144 = 169 = 13^2$. Roots: $(5 \pm 13)/12 = 3/2$ or $-2/3$. Factored: $6(x - 3/2)(x + 2/3) = (2x-3)(3x+2)$.
Key Insight
Vieta's formulas generalize: for a degree-$n$ polynomial, the elementary symmetric polynomials of the roots equal the coefficients (up to sign and leading coefficient). These formulas connect roots to coefficients without explicitly solving.