Factoring by Grouping "Split the Middle" Topic Index | Algebra Index | Regents Exam Prep Center

 for this lesson a = 1.

This process, while generally used for expressions where a is NOT 1,

 First, let's look at a multiplication of binomials -- first step!!!.    Notice that whether you used the distributive method, FOIL, or any other method, you arrived at the point where you see two center terms to combine. In this case,  . So the answer is:
 When we MULTIPLY two binomials, we end up with TWO center terms to combine. When we FACTOR, it makes sense to find these center terms to help us "see" the numbers that factor correctly. Take a look at the method below.

In the method called "factoring by grouping", the goal is to find a way to split the middle term into two appropriate terms.  We will look first at the process and then at a "condensed" statement of the process.

Example:  Factor

1.

Multiply the leading coefficient, 1, and the constant term, c.

1 • (+8) = +8

(Notice:  Since the leading coefficient is 1, you can see that the product is = c.)

2.
Consider all of the possible factors of this new product.
 Factors of +8. (1) • (8) (2) • (4)
3.
From the list of factors, find the one pair that adds to the middle term's coefficient, b.  For this example, we need to find a sum of 6.
2 + 4 = 6

4.

Re-write the middle term, forming two terms, using these two values (order is not important):
x2 + 2x +4x +8

5. Group the first two terms together and group the last two terms together.  Notice the plus sign between the two groups.
(x2 + 2x) + (4
x + 8)

6. Factor the greatest common factor out of each group.  Watch out for those signs in the second group should a negative be involved.
x(x + 2) + 4(x + 2)

7. Notice that the expressions in the parentheses are identical.  By factoring out the parentheses binomial, we have the answer:

Let's see it work on a more difficult problem.

Factor

1.

Multiply the leading coefficient, 1, and the constant term, c.

1 • (-24) = -24

2.
List all of the possible factors of ( - 24 )
 Factors of  - 24. (1) • ( - 24) (2) • ( - 12) (3) • ( - 8) (4) • ( - 6) ( - 1) • (24) ( - 2) • (12) ( - 3) • (8) ( - 4) • (6)
3.
From the list of factors, find the one pair that adds to the middle term's coefficient, b.  For this example, we need to find a sum of  - 10 .
(2) + ( - 12) =  - 10

4.

Re-write the middle term, forming two terms, using these two values
(order is not important):
x2 + 2x - 12x - 24

5. Group the first two terms together and group the last two terms together.  Notice the plus sign between the two groups.
(x2 + 2x) + (-12
x - 24)

6. Factor the greatest common factor out of each group.  Watch out for those signs in the second group.
x(x + 2) - 12(x + 2)

7. Notice that the expressions in the parentheses are identical.  By factoring out the parentheses binomial, we have the answer:
(x + 2)(x - 12 ANSWER:

Factoring by grouping is a very well known procedure used by teachers, students,
textbook authors, and college professors.

 Factoring by Grouping condensed:  1.  Find the product of  1•c. 2.  Find two factors of c that add to up to b. 3.  Split the middle term into two terms using these factors.4.  Group the four terms to form two pairs. 5.  Factor each pair. 6.  Factor out the common (shared) binomial parenthesis. x2 + bx + c

 See how to use your TI-83+ /84+ graphing calculator  with factoring. Click calculator.

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