Factoring Trinomials
Factoring a Trinomial of the Form x^{2} + bx + c
Example 1
Factor: x^{2}  7x + 12
Solution
This trinomial has the form x^{2} + bx + c where b = 7 and c = 12.
Step 1 Find two integers whose product is c and whose sum is b.
Since c is 12, list pairs of integers whose product is 12. Then, find the
sum of each pair of integers.
Product
1 Â· 12
2 Â· 6
3 Â· 4
1 Â· (12)
2 Â· (6)
3 Â· (4) 
Sum 13
8
7
13
8
7 
The last possibility, 3 Â· (4), gives the required sum,
7.
Step 2 Use the integers from Step 1 as the constants, r and s, in the
binomial factors (x + r) and (x + s).
The result is:
x^{2}  7x + 12 = (x  3)(x  4).
You can multiply to check the factorization. We leave the check to you.
Note:
The product, c = 12, is positive, so both
integers are positive or both are negative.
Since we also know the sum, b = 7, is
negative, we can conclude that both
integers are negative.
So we did not have to try the positive
integers.
Example 2
Factor: x^{2} + x  30
Solution
This trinomial has the form x^{2} + bx
+ c where b = 1 and c
= 30.
Step 1 Find two integers whose product is c and whose sum is b.
There are eight possible integer pairs whose product is 30.
To reduce the list, think about the signs of 1 and 30.
â€¢ Since the product, c = 30, is negative, one factor must be positive
and the other negative.
â€¢ Also, the sum, b = 1, is positive. So the integer with the greater
absolute value must be positive. We need only list pairs of integers
whose sum is positive.
Product
1
Â· 30
2
Â· 15
3
Â· 10
5
Â· 6 
Sum
29
13
7
1 
The last possibility, 5
Â· 6, gives the required sum, 1.
Step 2 Use the integers from Step 1 as the constants, r and s, in the
binomial factors (x + r) and (x + s).
The result is:
x^{2} + x  30
= (x  5)(x + 6).
You can multiply to check the factorization. We leave the check to you.
Note:
These are the eight integer pairs with
product 30:
1, 30
2, 15
3, 10
5, 6
1, 30
2, 15
3, 10
5, 6
Only one pair, 5 and 6, gives the
required sum, 1.
