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for this lesson
a
= 1.
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This process, while generally
used for expressions where a is NOT 1,
may help you to see how factoring develops.
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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:
 |
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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. |
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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
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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.) |
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2. |
| Consider all
of the possible factors of this new product.
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Factors of +8.
(1) • (8)
(2) • (4)
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| 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
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| 4.
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Re-write the middle term, forming two terms,
using these two values (order is not important):
x2 + 2x
+4x
+8
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| 5. |
Group
the first two terms together and group the
last two terms together.
Notice the plus sign between the two groups.
(x2 + 2x) + (4x + 8)
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| 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)
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| 7. |
Notice that the expressions in the
parentheses are identical. By factoring out the
parentheses binomial, we have the answer:
(x + 2)(x
+4)
ANSWER:
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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 )
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Factors of
- 24.
(1) • ( - 24)
(2) • ( - 12)
(3) • ( - 8)
(4) • ( - 6)
( - 1) • (24)
( - 2) • (12)
( - 3) • (8)
( - 4) • (6) |
|
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| 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
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| 4.
|
Re-write the middle term, forming two terms,
using these two values
(order is not important):
x2 + 2x
- 12x
- 24
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| 5. |
Group
the first two terms together and group the
last two terms together.
Notice the plus sign between the two groups.
(x2 + 2x) + (-12x - 24)
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| 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)
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| 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:
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Factoring by grouping is a very well
known procedure used by teachers, students,
textbook authors, and
college professors.
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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
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|
 |
See how to use
your
TI-83+ /84+ graphing calculator with
factoring.
Click calculator. |
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