h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Were you trying to explain factoring?? I'm really sorry, and it made sense. It just looks so hard. I like FOIL better, because it sounds much simpler.

Anna

It's a technique of factoring. And I don't think that it is really that hard, and I do think that it saves time... *grin* But certainly you are entitled to your opinion! The problem with FOIL is that it goes the other direction - you can expand the equations with First/Inside/Outside/Last, not condense (factor) them...


Shee

Put that way, it is true. Your way you could go both directions easily. With FOIL it's only one way. I ought to write that down somewhere....

Anna

*grin* Which part?

Hello Sheer, I am a bit confused. I can't seem to find the "first factor of term #1" neither of "term #3:. How do I find it in this example ?

6a^2 + 13a + 6

How should I factor 6a^ and 6 ??

YOUR QUESTION:
The first term (6a^2) can only be factored into:
1a -- 6a
2a -- 3a
(notice that the a^2 is factored out into two individual a's)


The third term (6) can only be factored into:
1 -- 6
2 -- 3


You should be able to find these just from thinking about the number 6 and what whole numbers it can be divided by and into. Don't worry about fractions; there's whole number answers.


SOLVING THE EQUATION (and testing some of these sets of numbers):
Now just choose a pair of those numbers and plug them in. Let's take both first sets, so we have a table that looks like:

6a^2 -- 6 -- 36a^2
1a ---- 1
6a ---- 6

But that order won't work, because in multiplying across the second and third rows, we get 1a and 36a. These add up to 37a, not to the 13a that was the second term in the original expression! So let's try again, this time with both second pairs:

6a^2 -- 6 -- 36a^2
2a ---- 2
3a ---- 3

Multiplying across the second row: 4a.
Multiplying across the third row: 9a.
4a * 9a = 36a^2, which is the same number as in box C.
4a + 9a = 13a, which is the same number as the third term in the original expression.

THIS IS OUR SOLUTION!:
6a^2 -- 6 -- 36a^2
2a ---- 2 -- 4a
3a ---- 3 -- 9a

So reading it from the table, the factored form is (2a+3)(3a+2).



In factoring the numbers, some of it is guesswork. Usually, it is very very little and just by looking at the numbers, one can see which sets will multiply together or add together resulting in a too-high number or too-low number for the goal, and thus quickly eliminate those. The first example I did in this post is an example of a test box that one probably wouldn't try in real life, but I needed an example.

Oh, and of course the factored pairs can be paired with any other factored pair -- it isn't necessarily first group to first group, second to second.