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^2-12x+32
In this case, we have 32. 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 32
32
-1 and -32
32
2 and 16
32
-2 and -16
32
4 and 8
32
-4 and -8
32
Next, let's consider the coefficient of the linear term.
x^2-12x+32
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, -12.
Factors
Sum of Factors
1 and 32
33
-1 and -32
-33
2 and 16
18
-2 and -16
- 18
4 and 8
12
- 4 and - 8
-12
We found the factors whose product is 32 and whose sum is -12.
x^2-12x+32 ⇔ (x-4)(x-8)
