Factoring trinomials of the form where a = 1.

Factoring and expanding are reverse processes, so let's start by imagining multiplying two factors and then looking to see how you could work backwards from the product. Suppose a trinomial could be factored into (x + p) and (x + q).

Multiplying these factors gives

Notice that p + q is the b-value and pq is the c-value of the trinomial . This helpful result means that if there are two numbers, p and q, that add to give b and multiply to give c, then the trinomial can be factored as (x + p)(x + q).

Example 3

Factor .

The b value is −1 and the c value is −6. This means you need p + q = −1 and pq = −6. Sometimes you can determine these values quickly by inspection and sometimes a table is more helpful. Start with p and q values that will multiply to −6 and then check to see which pairing gives p + q = −1.

p
q
pq
p + q
−2
3
−2 × 3 = −6
−2 + 3 = 1
−1
6
−1 × 6 = −6
−1 + 6 = 5
1
−6
1 × −6 = −6
1 + −6 = −5
2
−3
2 × −3 = −6
2 + −3 = −1

In the last line of the table, p + q = −1 and pq = −6. Now you know a p-value and a q-value that satisfy the requirements for factoring the trinomial . Use these to write the trinomial in factored form.

You can check by expanding.