Factoring Trinomials
Factoring a Trinomial of the Form x^{2} + bx + c
Example 1
Factor: x^{2} + 2x + 4
Solution
This trinomial has the form x^{2} + bx + c where b = 2 and c = 4.
Step 1 Find two integers whose product is c and whose sum is b.
â€¢ Since the product, c = 4, is positive, both integers must have the same
sign.
â€¢ Also the sum, b = 2, is positive. So both integers must be positive.
Product
1 Â· 4
2 Â· 2 
Sum 5
4 
These are the only possibilities, and neither gives the required sum, 2.
Since there are no two integers with product 4 and sum 2, the trinomial
x^{2} + 2x + 4 cannot be factored as (x + r)(x + s) where r and s are
integers.
Note:It is possible to factor
x^{2} + 2x + 4 using
numbers other than integers, but that is
beyond the scope of this lesson.
Example 2
Factor: x^{2}  9
Solution
First we write the binomial x^{2}  9 in the form x^{2} + bx
+ c.
To do this, we insert a middle term, 0x.
The equivalent trinomial is x^{2} + 0x  9.
Now, we see that b = 0 and c = 9.
Step 1 Find two integers whose product is c and whose sum is b.
â€¢ Since the product, c = 9, is negative, one factor must be positive and
the other negative.
â€¢ Since the sum is b = 0, the integers must be opposites.
There is only one possibility.
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}  9 = (x + 3)(x  3).
You can multiply to check the factorization. We leave the check to you.
Note:
These are the three integer pairs with
product 9:
1, 9
1, 9
3, 3
Only one pair, 3 and 3, gives the
required sum, 0.
Recall that the product of (x  3) and
(x + 3) is the difference of two squares,
this case x^{2}  9.
Example 3
Factor: 3x^{2}  6x  9
Solution
This trinomial does not have the form
x^{2} + bx + c because the coefficient of x^{2} is not 1. However, the terms of this
trinomial have a GCF of 3. 

3x^{2}  6x  9 
Factor out the common factor, 3. 
= 
3(x^{2}  2x  3) 
Now, the trinomial has the form
x^{2} + bx + c where b = 2 and c = 3.
Step 1 Find two integers whose product is c and whose sum is b.
â€¢ Since the product, c = 3, is negative, one integer must be positive
and the other negative.
â€¢ Also, the sum, b = 2, is negative. So the integer with the greater
absolute value must be negative.
There is only one possibility.
Step 2 Use the integers from Step 1 as the constants, r and s, in the
binomial factors (x + r) and (x + s).
The factorization of x^{2}  2x  3 is (x + 1)(x  3).
But donâ€™t forget the factor 3 that we factored out of the original
trinomial.
The factorization of 3x^{2}  6x  9 is:
3x^{2}  6x  9 = 3(x + 1)(x  3).
