Lesson 5: Factoring x² + bx + c

Property

To factor a trinomial of the form $x^2 + bx + c$, we need two factors $(x + m)$ and $(x + n)$ where the two numbers $m$ and $n$ multiply to $c$ and add to $b$.

How to factor trinomials of the form $x^2 + bx + c$

1. Write the factors as two binomials with first terms $x$: $(x \quad)(x \quad)$.
2. Find two numbers $m$ and $n$ that multiply to $c$, $m \cdot n = c$, and add to $b$, $m + n = b$.
3. Use $m$ and $n$ as the last terms of the factors.
4. Check by multiplying the factors.

Strategy for Determining Signs
When $c$ is positive, $m$ and $n$ have the same sign as $b$.

• If $b$ is positive, $m$ and $n$ are positive. Example: $x^2 + 5x + 6 = (x+2)(x+3)$.
• If $b$ is negative, $m$ and $n$ are negative. Example: $x^2 - 6x + 8 = (x-4)(x-2)$.

Property

To factor $x^2 + bx + c$, we look for two numbers $p$ and $q$ so that

$pq = c$ and $p + q = b$

When we expand the factored form $(x + p)(x + q)$, we get $x^2 + (p + q)x + pq$.