To factor a trinomial with a leading coefficient of 1, think of the process as multiplying two binomials in reverse. Let's start by taking a look at the constant term.
m^2+18m+81
In this case, we have 81. 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 81
81
-1 and -81
81
3 and 27
81
-3 and -27
81
9 and 9
81
-9 and -9
81
Next, let's consider the coefficient of the linear term.
m^2+18m+81
For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, 18.
Factors
Sum of Factors
1 and 81
82
-1 and -81
-82
3 and 27
30
-3 and -27
-30
9 and 9
18
-9 and -9
- 18
We found the factors whose product is 81 and whose sum is 18.
m^2+18m+81 ⇔ (m+9)(m+9)
Both factors are the same, so we can write the expression as (m+9)^2.
