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## Factoring polynomials with quadratic forms

Current time:0:00Total duration:5:01

# Factoring quadratics with common factor (old)

## Video transcript

Factor 8k squared
minus 24k minus 144. Now the first thing
we can do here, just eyeballing
each of these terms, if we want to simplify it a
good bit is all of these terms are divisible by 8. Clearly, 8k squared is divisible
by 8, 24 is divisible by 8, and 144-- it might not be as
obvious is divisible by 8-- but it looks like it is. 8 goes into one and 144,
8 goes into 14 one time. 1 times 8 is 8. Subtract, you get a 6. 14 minus 8 is 6. Bring down the 4. 8 goes into 64 eight times. So it goes into 144 18 times. So let's just factor
out an 8 of this. And then that will
simplify our expression. It will actually give us
a leading 1 coefficient. So this will become 8 times k
squared minus 24 divided by 8 is 3k minus 18. Now we have to factor
this business in here. And remember if anything has the
form x squared plus bx plus c, where you have a leading
1 coefficient-- this is implicitly a one--
we have that here in this expression
in parentheses. Then we literally
just need to-- and we can do this multiple
ways-- but we need to find two
numbers whose sum is equal to the coefficient on x. So two numbers whose sum
is equal to negative 3 and whose product is equal
to the constant term. And whose product is
equal to negative 18. So let's just think about the
factors of negative 18 here. Let's see if we can do
something interesting. So it could be 1. And since it's negative, one of
the numbers has to be positive, one has to be negative 1 and
18 is if it was positive. And then one of these could be
positive and then one of these could be negative. But no matter what
if this is negative and this is positive
then they add up to 17. If you switch them, then
they add up to negative 17. So those won't work. So either we could
write it this way, positive or negative 1, and
then negative or positive 18 to show that they have
to be different signs. So those don't work. Then you have positive
or negative 3. And then negative
or positive 6, just to know that they
are different signs. So if you have positive
3 and negative 6, they add up to
negative 3 which is what we need them to add up to. And clearly, positive 3 and
negative 6, their product is negative 18. So it works. So we're going to
go with positive 3 and negative 6 as
our two numbers. Now, for this example-- just
for the sake of this example-- We'll do this by grouping. So what we can do is we can
separate this middle term right here as the sum
of 3k negative 6k. So I could write the negative
3k as plus 3k minus 6k. And then let me
write the rest of it. So we have k squared up
here, plus 3k minus 6k, which is the same thing
as this over here. And then we have minus 18. And then all of that's
being multiplied by 8. Now we're ready to
group this thing. We can group these first two
terms, they're both divisible by k. And then we can group-- let
me put a positive sign-- let's group these
second two terms. So then we have 8 times--
I'll write brackets here instead of drawing
double parentheses. Brackets are really
just parentheses that look a little
bit more serious. Now let's factor out a k
from this term right here. I'm going to do this
in a different color. Let's factor out the k here. So this is k times k plus 3. And then we have plus. And then over here
it looks like we could factor out a negative 6. So let's factor
out-- I'm going to do this in a different color--
let's factor out a negative 6 over here. So plus negative
6 times k plus 3. So now it looks like we
can factor out a k plus 3. There's a k plus 3 times k, and
then we have a k plus 3 times the negative 6. So let's factor that out. So we have this 8 out
front, that's not changing. So let me write that
in the brackets. We're factoring out a k plus 3. So then we have the k plus
3 that we factored out. And then inside of that
we just have left this k. Instead of writing
plus negative 6, I could just write k minus 6. We factor out the k
plus 3 and we're done. And then we can rewrite this. The way we wrote it here
it's 8 times the product of k plus 3 times k minus 6. But we know from the
properties of multiplication, this is the exact same
thing as 8 times k plus 3 times k minus 6. And we are done.