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 center 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 find the coefficients of these two center terms.
The process is 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
(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.
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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 factors of 24 (that add to 10) as the coefficients. |
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 out the shared binomial (x + 4).
You now have the factors of x2 + 10x + 24. |
(x + 4)(x + 6) ANSWER |
