Math Stream – Why does the “AC” (“
Many algebra books now cover
the “AC” or “
First, let’s review the steps in the method, using the
example 4x² + 5x – 21.
I. Multiply
A and C – in this case, 4 and -21 4(-21)
= -84
II. Find
factors of this product whose sum is B
In
this case, factors of –84 whose sum is 5.
Factors
of –84 Sum (is it 5)?
42,
- 2 40
21,
-4 17
28,
-3 25
14,
-6 8
12,
-7 5
Step
II says the number to use are 12 and –7.
III. Write
the original middle term (5x) as a sum of terms using step II’s
numbers.
Since
5x = 12x – 7x
4x²
+ 5x – 21 is written as 4x² + 12x – 7x – 21
IV. Factor this 4-term polynomial
by
(4x²
+ 12x) + (-7x –
21)
4x(x
+ 3) – 7(x + 3)
(x
+ 3)(4x – 7)
Answer: (x + 3)(4x – 7)
WHY DOES THIS WORK??
This gets a little formal (lots of symbols), but I’ll try to
say it in words as well.
First, think about any trinomial. We can represent it by Ax² + Bx + C
(In other words, the A, B and C represent the coefficients.)
If this trinomial factors (besides a GCF), it is likely to
be two binomial factors. The two
binomial factors can be represented by (ax + b)(cx
+ d).
[In other words, we don’t know the coefficients in the two
factors).]
Third, multiply these two factor. (I’ll use “FOIL”.) (ax + b)(cx + d)
F O
I L
acx² + adx + bcx + bd
You may have noticed that the two middle terms in
FOIL are often like terms. We combine them. acx² + (ad + bc)x + bd
(In other words, both are x-terms so we add
the coefficients ad and bc.)
Fifth, use the AC step on this expression. AC = (ac)(bd) = abcd
In other words, AC is the product of all 4
coefficients in the two factors.
Notice that the middle term has to be the sum (ad + bc)x = Bx
Which means that the products ad and bc have
to add up to B.
Those last two steps show what we need. The numbers ad and bc are factors of AC, and they add up to B.
This is an informal proof; like any proof, it may not make
sense right away – you may need to read all steps several times before it makes
sense.
Added benefit: This
WHY also shows why we can use the factors in any order. Look –
Factoring by the AC (
and writes it as 4 terms acx² + adx + bcx + bd
We can then factor ax from the first pair and b
from the last pair.
If we switch the order of the two middle terms acx² + bcx + adx + bd
We can then factor cx from the
first pair and d
from the last pair.
The choice of order may produce a different order of the
factors (ax + b) and
(cx + d),
but we know that the order of factors is not important.
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