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Factoring Trinomials by Grouping

Factoring a Trinomial of the Form ax2 + bx + c by Grouping

Example 1

Factor: 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.

 

 
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