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.