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- 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 squared 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 squared 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.