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6.4, number 7--
here we have a trinomial that we want to factor.
And instead of factoring by grouping like the last
section, we're going to factor it by trial and error.
So we're just going to basically list a bunch of
different binomials and just see which one works.
So in order to narrow down some of our choices, we're
going to look at what are the factors of the first term, 4.
So we have a 2 times 2 or 4 times 1 are the possibilities.
So we're either going to start off our binomial with 2x times
2x or 4x times x.
And I'm just going to start with the 2x times 2x and try
out different factors of the 9 to see if any of those work
and give us the right middle term.
If they don't, then we'll switch over to 4x times x.
And so you see that's why it's called the trial by error.
You just kind of try out a bunch of stuff, hopefully try
all possibilities, and see which one gives you the right
middle term.
So let's see if we can get the signs to help
us out a bit first.
The last term is a 9, positive 9.
That means that the 3 times 3 or 9 times 1 is going to be
the same signs.
So we either have plus plus or minus minus.
They have to add up to the middle term of a positive
number, so that means I must use the plus plus combination.
So I'll do plus plus, and let's try the 3s first and see
what we get.
And now we just look at the middle terms, because we
already know that we picked the first terms and the last
terms to work.
2x times 2x is 4x squared.
3 times 3 is 9.
They're all set.
We don't have to think about them.
Middle terms--
3 times 2x is 6x plus 2x times 3 is 6x.
So to figure middle terms, always do inner, outer, then
add them up.
And that gives us 12x.
That's not what we want, so we keep going.
And you just keep going until you find what you want,
because we want 15x to be our sum.
So then I stick with the 2x, because I still have another
option to try, the 9 times 1.
And I still know they're going to be plus and plus.
And now I look at the middle--
18x plus 2x, 20x.
That didn't work either.
All right, well, we're done with the 2x's.
And then we go over to 4x x.
And again, I'll just start with the 3
and see what happens--
plus 3 plus 3.
Middle terms, 3x plus 12x--
and that gives us 15x.
That's what we're looking for.
If that didn't work, I would've kept going.
So I'm going to circle that, so we know
that's our correct answer.
But let's say that I hadn't gotten to that one yet.
I would keep going.
And I want to show you one other thing you have to watch
for when you're using trial and error.
If I use the 9 and 1 first, so 4x x and 9 and 1, like that.
Then I would say, OK, well, let's see what the
middle terms are.
9x and 4x is 13x didn't work.
So then I would keep going with the 4x and x and then,
because the 4x and x are two different factors, unlike up
here where we had 2x times 2x, the same thing.
I also need to think about, well, what if the 1 and the 9
had switched positions?
Right, because if they switch positions--
plus 1 plus 9--
now instead of the 9 hitting the x, I have the
9 hitting the 4x.
And can you see how this combination is going to give
us a completely different middle term?
Whereas if I switched things around up here, no matter
where I put the 9 and the 1, 9 always hits the 2.
And the 1 always hits the 2 also.
So it doesn't matter.
We're going to get the same middle term anyways.
All right, so here I would say I get 1x plus 36x, 37x.
And that still doesn't work, so we keep going and
eventually we would try this guy right here.
And that one is our answer.
And the way I usually work these is if you try out a few
of these and you like this method, it's a really good
method use.
Because after a while, you're going to start seeing patterns
that help you more quickly arrive at the correct numbers.
If you don't really like this method too much, you can
always use the factor by grouping method in 6.3 and
just do it out that way.