Skill: Factor simple trinomials with leading coefficient one

To factor a trinomial of the form

$$x^2+bx+c,$$

look for two integers, $m$ and $n$, such that:

• $m+n=b$ (their sum is the coefficient of $x$)
• $mn=c$ (their product is the constant)

Then write:

$$x^2+bx+c=(x+m)(x+n).$$

Example

Factor:

$$x^2+5x+6$$

Step 1: Identify $b$ and $c$.

In $x^2+5x+6$:

$$b=5,\qquad c=6$$

Step 2: Find two numbers whose product is $6$ and whose sum is $5$.

The factor pairs of $6$ are:

$$1\text{ and }6,\qquad 2\text{ and }3$$

Since $2+3=5$ and $2\cdot3=6$, the numbers are $2$ and $3$.

Step 3: Write the factors.

Place the numbers in two binomials:

$$x^2+5x+6=(x+2)(x+3)$$

Step 4: Check by multiplying.

$$(x+2)(x+3) =x^2+3x+2x+6 =x^2+5x+6$$

Therefore,

$$\boxed{x^2+5x+6=(x+2)(x+3)}$$

When the constant $c$ is negative, the two numbers must have opposite signs. When $c$ is positive, they have the same sign; choose the signs that make their sum equal $b$.