Factoring Trinomials
Factoring a Trinomial of the Form x^{2} + bx + c
The product of two binomials can be a trinomial.
For example, letâ€™s multiply
(x + 7) by (x + 3).
Use FOIL. 
(x + 7)(x + 3) = 
x Â· x 
+ 
x Â· 3 
+ 
7 Â· x 
+ 
7 Â· 3 
Simplify each term.
Combine like terms. 
= = 
x^{2} x^{2} 
+ + 
3x 
+ 10x 
7x 
+ + 
21 21 
Notice the relationship between the trinomial x^{2} + 10x + 21 and the
binomials (x + 7) and (x + 3):
â€¢ The first term of the trinomial, x2,
is the product of x and x, the first
term of each binomial. 
(x + 7)(x + 3) 
â€¢ The coefficient of the middle term
of the trinomial is 10, the sum of
3 and 7, the constants in the
binomials. 
= x Â· x + (7 + 3)x + 7 Â· 3
= x^{2} + 10x + 21 
â€¢ The last term of the trinomial is 21, is the product of 3 and 7, the
constants in the binomials. 
This relationship holds in general.
That is, if the product of two binomials (x + r) and (x + s) is a trinomial
of the form x^{2} + bx + c:
â€¢ c is the product of r and s.
â€¢ b is the sum of r and s.
We use this to factor trinomials of the form x^{2} + bx + c.
This procedure is called the productsum method of factoring because
we seek two integers whose product is c and whose sum is b.
Procedure â€”
To Factor x^{2} + bx + c (ProductSum Method)
Step 1 Find two integers whose product is c and whose sum is b.
Step 2 Use the integers from Step 1 as the constants, r and s, in the
binomial factors (x + r) and (x + s).
To check the factorization, multiply the binomial factors.
Example
Factor: x^{2} + 3x + 2
Solution
This trinomial has the form x^{2} + bx + c where b = 3 and c = 2.
Step 1 Find two integers whose product is c and whose sum is b.
Since c is 2, list pairs of integers whose product is 2.
Then, find the sum of each pair of integers.
Product
1 Â· 2
(1) Â· (2) 
Sum 3
3 
The first product, 1 Â· 2, gives the required sum, 3.
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} + 3x + 2 = (x + 1)(x + 2).
We multiply to check the factorization.
Is (x + 1)(x + 2) = x^{2} + 3x + 2 ?
Is x^{2} + 2x + 1x + 2 = x^{2} + 3x + 2 ?
Is x^{2} + 3x + 2 = x^{2} 3x + 2 ? Yes
Note:
Multiplication is commutative, so the
factorization may also be written: (x + 2)(x + 1).
