Let's remember what happens, in the very first step, when we multiply two binomials.
(x + 4)(x + 6) we get x2 + 4x + 6x + 24
Regardless of the method you used to multiply,
you always arrive at the point where you see two middle terms,
that usually add up as combined like terms.
In this case, + 4x + 6x = 10x
So, the answer is: x2 + 10x + 24
Let's "think about this" for a minute:
The middle term that you see in a trinomial is ACTUALLY the combination of TWO other terms.
In this case, 10 is actually the sum of two other numbers (6 and 4).
Those other numbers are seen in the factors (x + 4) and (x + 6). |
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If we can find these TWO center terms in a trinomial,
we could easily see what the binomials, that created the trinomial, look like.
That is, we could easily "factor" the trinomial.
The goal of
"factoring by grouping" is
to use the coefficients of these two middle terms to form factors.
For this to happen, we need to find a way to split the middle term
into two appropriate terms.
Thus the nickname for this process as "split the middle".
Method: Split the Middle
(also referred to as the "ac" method of factoring)
(Remember that this process is designed for harder problems where a ≠1.
We are taking a look when a = 1, so that we will better understand how the
terms and coefficients relate to one another when factoring.)
Factor: x2 +10x + 24 |
Step 1: The first step is to multiply the leading coefficient times the constant term.
The leading coefficient is 1.
The constant term is 24.
This is the product of a • c.
|
1 • 24 = 24 |
Step 2:
Find two factors of a • c that will add up to b.
Since all signs in this example are positive, we will only consider positive factors of 24. |
sum of possible factors of 24
24 + 1 = 25
12 + 2 = 14
8 + 3 = 11
6 + 4 = 10
stop! we found them (6 and 4) |
Step 3: Split the middle term into two terms using the newly found factors of 24 (that add to 10) as the coefficients. The order in which you list the two middle terms if of no importance, You will end up with the correct answer either way. |
x2 + 4x + 6x + 24
Order of middle terms does not matter. |
Step 4: Group the four terms to form two binomials. |
(x2 + 4x) + (6x + 24) |
Step 5: Factor each binomial. |
x(x + 4) + 6(x + 4) |
Step 6: "Factor by Grouping": factor out the shared binomial (x + 4). Distributive property in reverse.
You now have the factors of x2 + 10x + 24. |
(x + 4)(x + 6) ANSWER |
