Example 5: Factoring by grouping - Engineering Mathematics Video Lecture

We're asked to factor 35k
squared plus 100k, minus 15.
And because we have a non-1
coefficient out here, the best
thing to do is probably to
factor this by grouping.
But before we even do that,
let's see if there's a common
factor across all of these
terms, and maybe we can get a
1 coefficient, out there.
If we can't get a 1 coefficient,
we'll at least
have a lower coefficient here.
And if we look at all of these
numbers, they all look
divisible by 5.
In fact their greatest
common factor is 5.
So let's at least
factor out a 5.
So this is equal to 5
times-- 35k squared
divided by 5 is 7k squared.
100k divided by 5 is 20k.
And then negative 15 divided
by 5 is negative 3.
So we were able to factor out a
5, but we still don't have a
1 coefficient here, so we're
still going to have to factor
by grouping.
But at least the numbers here
are smaller so it'll be easier
to think about it in terms of
finding numbers whose product
is equal to 7 times negative
3, and whose
sum is equal to 20.
So let's think about that.
Let's figure out two numbers
that if I were to add them, or
even better if I were to take
their product, I get 7 times
negative 3, which is equal
to negative 21.
And if I were to take their
sum, if I add those two
numbers, it needs to
be equal to 20.
Now, once again, because their
product is a negative number,
that means they have to be of
different signs, so when you
add numbers of different signs,
you could view it as
you're taking the difference
of the positive versions.
So the difference between the
positive versions of the
number has to be 20.
So the number that immediately
jumps out is we're probably
going to be dealing with 20
and 21, and 1 will be the
negative, because we want
to get to a positive 20.
So let's think about it.
So if we think of 20 and
negative 1, their product is
negative 21.
Sorry.
If we take 21 and negative 1,
their product is negative 21.
21 times negative 1
is negative 21.
and if you take their sum,
21 plus negative 1,
that is equal to 20.
So these two numbers right
there fit the bill.
Now, let's break up this 20k
right here into a 21k and a
negative 1k.
So let's do that.
So let's rewrite the
whole thing.
We have 5 times 7k squared, and
I'm going to break this
20k into a-- let me do it in
this color right here-- I'm
going to break that 20k into
a plus 21k, minus k.
Or you could say minus
1k if you want.
I'm using those two factors
to break it up.
And then we finally have the
minus 3 right there.
Now, the whole point of doing
that is so that we can now
factor each of the two groups.
This could be our first
group right here.
And so what can we factor out
of that group right there?
Well, both of these are
divisible by 7k, so we can
write this as 7k times-- 7k
squared divided by 7k, you're
just going to have
a k left over.
And then plus 21k divided by
7k is just going to be a 3.
So that factors into that.
And then we can look at
this group right here.
They have a common factor.
Well, we can factor out a
negative 1 if we like, so this
is equal to negative 1 times--
k divided by negative 1 is k.
Negative 3 divided by negative
1 is positive 3.
And, of course, we have this
5 sitting out there.
Now, ignoring that 5 for a
second, you see that both of
these inside terms have
k plus 3 as a factor.
So we can factor that out.
So let's ignore this
5 for a second.
This inside part right here,
the stuff that's inside the
parentheses, we can factor k
plus 3 out, and it becomes k
plus 3, times k plus 3,
times 7k minus 1.
And if this seems a little
bizarre to you, distribute the
k plus 3 on to this.
K plus 3 times 7k is that term,
k plus 3 times negative
1 is that term.
And, of course, the whole
time you have
that 5 sitting outside.
You have that 5.
We don't even have to put
parentheses there.
5 times k plus 3, times
7k minus 1.
And we factored it,
we're done.