WEBVTT

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We're asked to factor
this expression.

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

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

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

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is think about two numbers
whose products are

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going to be equal to.

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You have actually one
coefficient right here, right?

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t squared is the same
thing as 1t squared.

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

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b, a times b.

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the product of these two
numbers needs to be the

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

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the negative 15 right here.

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

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just negative 15.

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

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negative 2.

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

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can use those to factor
by grouping.

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

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this technique works.

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

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

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can somehow get to negative 2.

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

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

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

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

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But let's think about
what happens when

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you take their sum.

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

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

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the negative of that.

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You're going to get 14.

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It does not equal negative 2.

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So what happens if you take
3 and negative 5?

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

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5 is negative 2,
so that works.

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

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that to be positive 2.

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

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would have gotten
a negative 2.

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So these work, 3 and
negative 5 work.

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

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negative 2.

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So what we want to do
here is break this

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middle term up here.

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

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

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

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

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I'll put the minus
15 out here.

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

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

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

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

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say which ones have
common factors?

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

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

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go with this.

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

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negative 2t.

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Positive 3 times negative 5
is equal to negative 15.

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That's where it came from.

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Now, we're ready to factor
by grouping.

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

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terms, right there.

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And what's the common
factor there?

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Well, the common factor
there is t.

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

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divided by t is t.

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3t divided by t is 3.

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So these first two terms
are the same thing as t

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times t plus 3.

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

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What's a common factor?

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

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out a negative 5.

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

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

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

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

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positive 3.

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

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

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

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We're undistributing
the t plus 3.

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

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Times t minus 5.

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

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are really the same thing.

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

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you're just left with that t.

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

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

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

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with the t minus 5, you
have fully factored

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this expression here.

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

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this, but factoring by grouping
is actually the

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

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1, or a non-one coefficient.

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It could also be a negative
coefficient out front here.

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

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actually much easier ways to
factor something like this,

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but it's really the same
thought process.

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