Factoring Trinomials
Factoring a Trinomial of the Form x2 + bx + c
The product of two binomials can be a trinomial.
For example, let’s multiply
(x + 7) by (x + 3).
| Use FOIL. |
(x + 7)(x + 3) = |
x · x |
+ |
x · 3 |
+ |
7 · x |
+ |
7 · 3 |
| Simplify each term.
Combine like terms. |
= = |
x2 x2 |
+ + |
3x |
+ 10x |
7x |
+ + |
21 21 |
Notice the relationship between the trinomial x2 + 10x + 21 and the
binomials (x + 7) and (x + 3):
| • The first term of the trinomial, x2,
is the product of x and x, the first
term of each binomial. |
(x + 7)(x + 3) |
| • The coefficient of the middle term
of the trinomial is 10, the sum of
3 and 7, the constants in the
binomials. |
= x · x + (7 + 3)x + 7 · 3
= x2 + 10x + 21 |
| • 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 x2 + bx + c:
• c is the product of r and s.
• b is the sum of r and s.
We use this to factor trinomials of the form x2 + bx + c.
This procedure is called the product-sum method of factoring because
we seek two integers whose product is c and whose sum is b.
Procedure —
To Factor x2 + bx + c (Product-Sum 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: x2 + 3x + 2
Solution
This trinomial has the form x2 + 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 · 2
(-1) · (-2) |
Sum 3
-3 |
The first product, 1 · 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:
x2 + 3x + 2 = (x + 1)(x + 2).
We multiply to check the factorization.
Is (x + 1)(x + 2) = x2 + 3x + 2 ?
Is x2 + 2x + 1x + 2 = x2 + 3x + 2 ?
Is x2 + 3x + 2 = x2 3x + 2 ? Yes
Note:
Multiplication is commutative, so the
factorization may also be written: (x + 2)(x + 1).
|
