To factor a trinomial with a leading coefficient of one, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
x^2+13x+40
In this case, we have 40. This is a positive number, so for the product of the constant terms in the factors to be positive, these constants must have the same sign (both positive or both negative.)
Factor Constants
Product of Constants
1 and 40
40
-1 and - 40
40
2 and 20
40
- 2 and - 20
40
4 and 10
40
- 4 and - 10
40
5 and 8
40
- 5 and - 8
40
Next, let's consider the coefficient of the linear term.
x^2+13x+40
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 13.
Factors
Sum of Factors
1 and 40
41
-1 and - 40
- 41
2 and 20
22
- 2 and - 20
- 22
4 and 10
14
- 4 and - 10
- 14
5 and 8
13
- 5 and - 8
- 13
We found the factors whose product is 40 and whose sum is 13.
x^2+13x+40 ⇔ (x+5)(x+8)
