Factoring Trinomials by Grouping
Factoring a Trinomial of the Form ax2 + bx + c by Grouping
Example 1Factor: 32x2 - 20x - 3
Solution
Step 1 Factor out common factors (other than 1 or -1).
There are no common factors other than 1 and -1.
Step 2 List the values of a, b, and c. Then find two integers whose
product is ac and whose sum is b.
32x2 - 20x - 3 has the form ax2 + bx + c where a = 32, b
= -20, and c = -3.
The product ac is 32 · (-3) = -96. Thus, find two integers whose
product, ac, is -96 and whose sum, b, is -20.
• Since their product is negative, the integers must have different signs.
• Also, their sum is negative, so the integer with the greater absolute
value must be negative.
Here are some of the possibilities:
| Product
1 · (-96)
2 · (-48)
3 · (-32)
4 · (-24) |
Sum -95
-46
-29
-20 |
Since 4 and -24 have product -96 and sum -20, we do not need to
consider any other pairs of integers.
| Step 3 Replace the middle term, bx,
with a sum or difference using the
two integers found in Step 2. |
32x2 - 20x - 3
|
|
Replace -20x with 4x - 24x.
Step 4 Factor by grouping.
Group the first pair of terms and group
the second pair of terms.
Factor 4x out of the first group; factor
-3 out of the second group.
Factor out the common factor, (8x + 1). |
= 32x2 + 4x - 24x - 3
= (32x2 + 4x) + (-24x - 3)
= 4x(8x + 1) + (-3)(8x + 1)
= (8x + 1)(4x - 3) |
The result is: 32x2 - 20x - 3
= (8x + 1)(4x - 3)
Note:
We replaced -20x with 4x - 24x. If we
switch 4x and -24x, we can still group
and factor:
= 32x2 - 24x + 4x - 3
= (32x2 - 24x) + (4x - 3)
= 8x(4x - 3) + 1(4x - 3)
= (4x - 3)(8x + 1)
Example 2
Factor: 2x2 + 4x + 3.
Solution
Step 1 Factor out common factors (other than 1 or -1).
There are no common factors other than 1 and -1.
Step 2 List the values of a, b, and c. Then find two integers whose
product is ac and whose sum is b.
2x2 + 4x + 3 has the form ax2 + bx + c where a = 2, b = 4, and c
= 3.
The product ac is 2 · 3 = 6.
Thus, find two integers whose product, ac, is 6 and whose sum, b, is 4.
• Since their product is positive, the integers must have the same sign.
• Since their sum is also positive, the integers must both be positive.
Here are the possibilities:
| Product
1 · (6)
2 · (3) |
Sum 7
5 |
Neither possibility has the required sum, 4.
Since there are no two integers whose product is 6 and whose sum is 4, we
conclude that 2x2 + 4x + 3 is not factorable over the integers.
Note:
This approach tells us directly when the
trinomial is not factorable.
That’s a major advantage of this method.
Example 3
Factor: 2x2 - 8x - 10
Solution
Step 1 Factor out common factors (other than 1 or -1).
| Factor out the common factor of 2.
The trinomial has the form x2 + bx + c. |
2x2 - 8x - 10
= 2(x2 - 4x - 5) |
| Since the coefficient of the x2-term is 1, we can factor the trinomial by
the product-sum method. |
| That is, we find two integers whose
product is -5 and whose sum is -4.
The integers are -5 and 1. |
= 2(x - 5)(x + 1) |
The result is: 2x2 - 8x - 10 = 2(x - 5)(x + 1).
You can multiply to check the factorization. We leave the check to you.
