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CCSS.Math: , ,

In this video, I want to focus
on a few more techniques for factoring polynomials. And in particular, I want to
focus on quadratics that don't have a 1 as the leading
coefficient. For example, if I wanted
to factor 4x squared plus 25x minus 21. Everything we've factored so
far, or all of the quadratics we've factored so far, had
either a 1 or negative 1 where this 4 is sitting. All of a sudden now, we
have this 4 here. So what I'm going to teach you
is a technique called, factoring by grouping. And it's a little bit more
involved than what we've learned before, but
it's a neat trick. To some degree, it'll become
obsolete once you learn the quadratic formula, because,
frankly, the quadratic formula is a lot easier. But this is how it goes. I'll show you the technique. And then at the end of this
video, I'll actually show you why it works. So what we need to do here,
is we need to think of two numbers, a and b, where a
times b is equal 4 times negative 21. So a times b is going to be
equal to 4 times negative 21, which is equal to negative 84. And those same two numbers,
a plus b, need to be equal to 25. Let me be very clear. This is the 25, so they need
to be equal to 25. This is where the 4 is. So we go, 4 times negative 21. That's a negative 21. So what two numbers are there
that would do this? Well, we have to look at the
factors of negative 84. And once again, one
of these are going to have to be positive. The other ones are going to have
to be negative, because their product is negative. So let's think about
the different factors that might work. 4 and negative 21 look
tantalizing, but when you add them, you get negative 17. Or, if you had negative 4 and
21, you'd get positive 17. Doesn't work. Let's try some other
combinations. 1 and 84, too far apart when
you take their difference. Because that's essentially what
you're going to do, if one is negative and
one is positive. Too far apart. Let's see you could do 3--
I'm jumping the gun. 2 and 42. Once again, too far apart. Negative 2 plus 42 is 40. 2 plus negative 42 is negative
40-- too far apart. 3 and-- Let's see, 3 goes into
84-- 3 goes into 8 2 times. 2 times 3 is 6. 8 minus 6 is 2. Bring down the 4. Goes exactly 8 times. So 3 and 28. This seems interesting. And remember, one of these
has to be negative. So if we have negative 3 plus
28, that is equal to 25. Now, we've found our
two numbers. But it's not going to be quite
as simple of an operation as what we did when this was
a 1 or negative 1. What we're going to do now is
split up this term right here. We're going to split it up into
positive 28x minus 3x. We're just going to
split that term. That term is that term
right there. And of course, you have your
minus 21 there, and you have your 4x squared over here. Now, you might say, how did you
pick the 28 to go here, and the negative
3 to go there? And it actually does matter. The way I thought about it is
3 or negative 3, and 21 or negative 21 , they have
some common factors. In particular, they have
the factor 3 in common. And 28 and 4 have some
common factors. So I grouped the 28 on
the side of the 4. And you're going to see what
I mean in a second. If we, literally, group these
so that term becomes 4x squared plus 28x. And then, this side, over
here in pink, it's plus negative 3x minus 21. Once again, I picked these. I grouped the negative 3 with
the 21, or the negative 21, because they're both
divisible by 3. And I grouped the 28 with the
4, because they're both divisible by 4. And now, in each of these
groups, we factor as much out as we can. So both of these terms
are divisible by 4x. So this orange term is equal
to 4x times x-- 4x squared divided by 4x is just x-- plus
28x divided by 4x is just 7. Now, this second term. Remember, you factor
out everything that you can factor out. Well, both of these terms are
divisible by 3 or negative 3. So let's factor out
a negative 3. And this becomes x plus 7. And now, something might
pop out at you. We have x plus 7 times
4x plus, x plus 7 times negative 3. So we can factor out
an x plus 7. This might not be completely
obvious. You're probably not
used to factoring out an entire binomial. But you could view this
could be like a. Or if you have 4xa minus 3a,
you would be able to factor out an a. And I can just leave this
as a minus sign. Let me delete this
plus right here. Because it's just
minus 3, right? Plus negative 3, same
thing as minus 3. So what can we do here? We have an x plus 7, times 4x. We have an x plus 7,
times negative 3. Let's factor out the x plus 7. We get x plus 7, times
4x minus 3. Minus that 3 right there. And we've factored
our binomial. Sorry, we've factored our
quadratic by grouping. And we factored it into
two binomials. Let's do another example of
that, because it's a little bit involved. But once you get the hang
of it's kind of fun. So let's say we want to factor
6x squared plus 7x plus 1. Same drill. We want to find a times b that
is equal to 1 times 6, which is equal to 6. And we want to find an a plus
b needs to be equal to 7. This is a little bit more
straightforward. What are the-- well, the obvious
one is 1 and 6, right? 1 times 6 is 6. 1 plus 6 is 7. So we have a is equal to 1. Or let me not even
assign them. The numbers here are 1 and 6. Now, we want to split this
into a 1x and a 6x. But we want to group it so it's
on the side of something that it shares a factor with. So we're going to have a 6x
squar ed here, plus-- and so I'm going to put the
6x first because 6 and 6 share a factor. And then, we're going to
have plus 1x, right? 6x plus 1x equals 7x . That was the whole point. They had to add up to 7 . And then we have the
final plus 1 there. Now, in each of these groups,
we can factor out as much as we like. So in this first group,
let's factor out a 6x. So this first group becomes 6x
times-- 6x squar ed divided by 6x is just an x. 6x divided by 6x is just a 1. And then, the second
group-- we're going to have a plus here. But this second group, we just
literally have a x plus 1. Or we could even write a
1 times an x plus 1. You could imagine I just
factored out of 1 so to speak. Now, I have 6x times x plus
1, plus 1 times x plus 1. Well, I can factor
out the x plus 1. If I factor out an x plus 1,
that's equal to x plus 1 times 6x plus that 1. I'm just doing the distributive property in reverse. So hopefully you didn't
find that too bad. And now, I'm going to actually
explain why this little magical system actually works. Let me take an example. I'll do it in very
general terms. Let's say I had ax plus b,
times cx-- actually, I'm afraid to use the a's and b's. I think that'll confuse
you, because I use a's and b's here. They won't be the same thing. So let me use completely
different letters. Let's say I have fx plus g,
times hx plus, I'll use j instead of i. You'll learn in the future
why don't like using i as a variable. So what is this going
to be equal to? Well, it's going to be fx
times hx which is fhx. And then, fx times j. So plus fjx. And then, we're going
to have g times hx. So plus ghx. And then g times j. Plus gj. Or, if we add these two middle
terms, you have fh times x, plus-- add these two terms--
fj plus gh x. Plus gj. Now, what did I do here? Well, remember, in all of these
problems where you have a non-1 or non-negative 1
coefficient here, we look for two numbers that add up to this,
whose product is equal to the product of
that times that. Well, here we have two numbers
that add up-- let's say that a is equal to fj. That is a. And b is equal to gh. So a plus b is going to be
equal to that middle coefficient. And then what is a times b? a
times b is going to be equal to fj times gh. We could just reorder these
terms. We're just multiplying a bunch of terms. So that could
be rewritten as f times h times g times j. These are all the same things. Well, what is fh times gj? This is equal to fh times gj. Well, this is equal to the first
coefficient times the constant term. So a plus b will be equal to
the middle coefficient. And a times b will equal the
first coefficient times the constant term. So that's why this whole
factoring by grouping even works, or how we're able
to figure out what a and b even are. Now, I'm going to close up
with something slightly different, but just to make
sure that you have a well-rounded education
in factoring things. What I want to do is to teach
you to factor things a little bit more completely. And this is a little
bit of a add-on. I was going to make a
whole video on this. But I think, on some level,
it might be a little obvious for you. So let's say we had-- let
me get a good one here. Let's say we had negative
x to the third, plus 17x squared, minus 70x. Immediately, you say, gee, this
isn't even a quadratic. I don't know how to solve
something like this. It has an x to third power. And the first thing you should
realize is that every term here is divisible by x. So let's factor out an x. Or even better, let's factor
out a negative x. So if you factor out a negative
x, this is equal to negative x times-- negative
x to the third divided by negative x is x squared. 17x squared divided by negative
x is negative 17x. Negative 70x divided by negative
x is positive 70. The x's cancel out. And now, you have something
that might look a little bit familiar. We have just a standard
quadratic where the leading coefficient is a 1. We just have to find two numbers
whose product is 70, and that add up to
negative 17. And the numbers that immediately
jumped into my head are negative 10
and negative 7. You take their product,
you get 70. You add them up, you
get negative 17. So this part right here is
going to be x minus 10, times x minus 7. And of course, you have that
leading negative x. The general idea here is just
see if there's anything you can factor out. And that'll get it into a form
that you might recognize. Hopefully, you found
this helpful. I want to reiterate what I
showed you at the beginning of this video. I think it's a really cool
trick, so to speak, to be able to factor things that have
a non-1 or non-negative 1 leading coefficient. But to some degree, you're going
to find out easier ways to do this, especially
with the quadratic formula, in not too long.