Factoring a Trinomial of the Form x2 + bx + c
Example 1
Factor: x2 - 7x + 12
Solution
This trinomial has the form x2 + 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 · 12
2 · 6
3 · 4
-1 · (-12)
-2 · (-6)
-3 · (-4) |
Sum 13
8
7
-13
-8
-7 |
The last possibility, -3 · (-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:
x2 - 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: x2 + x - 30
Solution
This trinomial has the form x2 + 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.
Since the product, c = -30, is negative, one factor must be positive
and the other negative.
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
· 30
-2
· 15
-3
· 10
-5
· 6 |
Sum
29
13
7
1 |
The last possibility, -5
· 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:
x2 + 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.
|
