Factoring Trinomials
Factoring a Trinomial of the Form x2 + bx + c
Example 1
Factor: x2 + 2x + 4
Solution
This trinomial has the form x2 + bx + c where b = 2 and c = 4.
Step 1 Find two integers whose product is c and whose sum is b.
• Since the product, c = 4, is positive, both integers must have the same
sign.
• Also the sum, b = 2, is positive. So both integers must be positive.
| Product
1 · 4
2 · 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
x2 + 2x + 4 cannot be factored as (x + r)(x + s) where r and s are
integers.
Note:It is possible to factor
x2 + 2x + 4 using
numbers other than integers, but that is
beyond the scope of this lesson.
Example 2
Factor: x2 - 9
Solution
First we write the binomial x2 - 9 in the form x2 + bx
+ c.
To do this, we insert a middle term, 0x.
The equivalent trinomial is x2 + 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.
• Since the product, c = -9, is negative, one factor must be positive and
the other negative.
• 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:
x2 - 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 x2 - 9.
Example 3
Factor: 3x2 - 6x - 9
Solution
| This trinomial does not have the form
x2 + bx + c because the coefficient of x2 is not 1. However, the terms of this
trinomial have a GCF of 3. |
|
3x2 - 6x - 9 |
| Factor out the common factor, 3. |
= |
3(x2 - 2x - 3) |
Now, the trinomial has the form
x2 + bx + c where b = -2 and c = -3.
Step 1 Find two integers whose product is c and whose sum is b.
• Since the product, c = -3, is negative, one integer must be positive
and the other negative.
• 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 x2 - 2x - 3 is (x + 1)(x - 3).
But don’t forget the factor 3 that we factored out of the original
trinomial.
The factorization of 3x2 - 6x - 9 is:
3x2 - 6x - 9 = 3(x + 1)(x - 3).
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