Factoring Trinomials by Grouping 2

U09_L1_T2_we2 Factoring Trinomials by Grouping 2

We're asked to factor
this expression.

And there's going to be simpler
ways to factor it, but

in this video I'm going to
factor it by grouping.

And when you factor by grouping,
what you need to do

is think about two numbers
whose products are

going to be equal to.

You have actually one
coefficient right here, right?

t squared is the same
thing as 1t squared.

So we're looking for two
numbers, let's call them a and

b, a times b.

the product of these two
numbers needs to be the

product of the coefficient on
the t squared, which is 1, and

the negative 15 right here.

So a times b has to be equal
to 1 times negative 15, or

just negative 15.

And the sum of a and b, and a
plus b, needs to be equal to

negative 2.

And once we have these two
numbers, I can show you how we

can use those to factor
by grouping.

And in other videos I've
actually broken down to why

this technique works.

Now, let's think of the
different factors of negative

15 when we take their product,
and if we take the sum, if we

can somehow get to negative 2.

So let's look at the different
factors of negative 15.

So we could do-- let me do it in
this other, let me do it in

pink-- see if 1 and negative 15,
these are-- so everything

I list here, their product is
going to be negative 15.

But let's think about
what happens when

you take their sum.

So 1 and negative 15, the
sum is negative 14.

And if you did negative 1 and
15, you're just going to get

the negative of that.

You're going to get 14.

It does not equal negative 2.

So what happens if you take
3 and negative 5?

So their product is definitely
negative 15, 3, plus negative

5 is negative 2,
so that works.

And if we tried negative 3 and
5 first, we would have gotten

that to be positive 2.

It's just like, oh, we just have
to swap the signs and we

would have gotten
a negative 2.

So these work, 3 and
negative 5 work.

3 times negative 5 is negative
15, 3 plus negative 5 is

negative 2.

So what we want to do
here is break this

middle term up here.

We know that 3 plus negative
5 is equal to negative 2.

So we can break up this middle
term here as a sum of-- and

I'll do it right here; I'll
actually do it in the same

color-- so this thing here, we
can rewrite as t squared.

I'll put the minus
15 out here.

But the negative 2t we can
rewrite as the sum of 3t.

We could write it here
as plus 3t, minus 5t.

And when you're trying to figure
out which one to put

first or second, you should look
at these other terms, and

say which ones have
common factors?

The 3 and 5 both have a common
factor with 15, so it's not as

obvious which one to put first,
so we're just going to

go with this.

3t minus 5t, I got that from
3t minus 5t is equal to

negative 2t.

Positive 3 times negative 5
is equal to negative 15.

That's where it came from.

Now, we're ready to factor
by grouping.

So let's take the first group,
let's take these first two

terms, right there.

And what's the common
factor there?

Well, the common factor
there is t.

So if I factor a t out, that
becomes t times-- t squared

divided by t is t.

3t divided by t is 3.

So these first two terms
are the same thing as t

times t plus 3.

Now, let's look at the
second two terms.

What's a common factor?

Well, they're both divisible by
negative 5, so let's factor

out a negative 5.

And negative 5t divided by
negative 5, if you factor out

the negative 5, you're just
going to have a t there.

And then negative 15, if you
factor out a negative 5, you

divide negative 15 by negative
5, you're just going to have a

positive 3.

And then notice, you now have
two terms here, two products,

and they both have the common
factor of t plus 3.

So we can rewrite this right
here as a product of t plus 3.

We're undistributing
the t plus 3.

We're factoring out the t plus
3. t plus 3 times t, right?

Times t minus 5.

And I want you to really make
sure you feel good that these

are really the same thing.

If you take t times t plus 3 and
factor out the t plus 3,

you're just left with that t.

If you take negative 5 times t
plus 3 and you factor out the

t plus 3, you're just left
with that negative 5.

But once you factor out the t
plus 3, and you're just left

with the t minus 5, you
have fully factored

this expression here.

And in the future we're going
to see easier ways of doing

this, but factoring by grouping
is actually the

easiest way to do it if you have
a coefficient higher than

1, or a non-one coefficient.

It could also be a negative
coefficient out front here.

When you have 1 as your
coefficient here, there's

actually much easier ways to
factor something like this,

but it's really the same
thought process.