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In this video, I want to focus on a few more techniques for

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factoring polynomials.

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And in particular, I want to focus on quadratics that don't

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have a 1 as the leading coefficient.

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For example, if I wanted to factor 4x squared

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plus 25x minus 21.

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Everything we've factored so far, or all of the quadratics

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we've factored so far, had either a 1 or negative 1 where

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this 4 is sitting.

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All of a sudden now, we have this 4 here.

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So what I'm going to teach you is a technique called,

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factoring by grouping.

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And it's a little bit more involved than what we've

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learned before, but it's a neat trick.

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To some degree, it'll become obsolete once you learn the

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quadratic formula, because, frankly, the quadratic formula

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is a lot easier.

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But this is how it goes.

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I'll show you the technique.

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And then at the end of this video, I'll actually show you

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why it works.

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So what we need to do here, is we need to think of two

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numbers, a and b, where a times b is equal 4 times

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negative 21.

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So a times b is going to be equal to 4 times negative 21,

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which is equal to negative 84.

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And those same two numbers, a plus b, need

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to be equal to 25.

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Let me be very clear.

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This is the 25, so they need to be equal to 25.

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This is where the 4 is.

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So we go, 4 times negative 21.

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That's a negative 21.

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So what two numbers are there that would do this?

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Well, we have to look at the factors of negative 84.

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And once again, one of these are going

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to have to be positive.

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The other ones are going to have to be negative, because

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their product is negative.

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So let's think about the different

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factors that might work.

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4 and negative 21 look tantalizing, but when you add

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them, you get negative 17.

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Or, if you had negative 4 and 21, you'd get positive 17.

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Doesn't work.

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Let's try some other combinations.

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1 and 84, too far apart when you take their difference.

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Because that's essentially what you're going to do, if

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one is negative and one is positive.

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Too far apart.

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Let's see you could do 3-- I'm jumping the gun.

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2 and 42.

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Once again, too far apart.

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Negative 2 plus 42 is 40.

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2 plus negative 42 is negative 40-- too far apart.

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3 and-- Let's see, 3 goes into 84-- 3 goes into 8 2 times.

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2 times 3 is 6.

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8 minus 6 is 2.

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Bring down the 4.

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Goes exactly 8 times.

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So 3 and 28.

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This seems interesting.

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And remember, one of these has to be negative.

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So if we have negative 3 plus 28, that is equal to 25.

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Now, we've found our two numbers.

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But it's not going to be quite as simple of an operation as

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what we did when this was a 1 or negative 1.

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What we're going to do now is split up this term right here.

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We're going to split it up into positive 28x minus 3x.

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We're just going to split that term.

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That term is that term right there.

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And of course, you have your minus 21 there, and you have

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your 4x squared over here.

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Now, you might say, how did you pick the 28 to go here,

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and the negative 3 to go there?

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And it actually does matter.

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The way I thought about it is 3 or negative 3, and 21 or

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negative 21 , they have some common factors.

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In particular, they have the factor 3 in common.

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And 28 and 4 have some common factors.

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So I grouped the 28 on the side of the 4.

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And you're going to see what I mean in a second.

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If we, literally, group these so that term becomes 4x

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squared plus 28x.

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And then, this side, over here in pink, it's plus

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negative 3x minus 21.

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Once again, I picked these.

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I grouped the negative 3 with the 21, or the negative 21,

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because they're both divisible by 3.

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And I grouped the 28 with the 4, because they're both

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divisible by 4.

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And now, in each of these groups, we factor as

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much out as we can.

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So both of these terms are divisible by 4x.

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So this orange term is equal to 4x times x-- 4x squared

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divided by 4x is just x-- plus 28x divided by 4x is just 7.

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Now, this second term.

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Remember, you factor out everything that

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you can factor out.

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Well, both of these terms are divisible by 3 or negative 3.

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So let's factor out a negative 3.

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And this becomes x plus 7.

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And now, something might pop out at you.

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We have x plus 7 times 4x plus, x plus 7

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times negative 3.

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So we can factor out an x plus 7.

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This might not be completely obvious.

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You're probably not used to factoring

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out an entire binomial.

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But you could view this could be like a.

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Or if you have 4xa minus 3a, you would be able to

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factor out an a.

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And I can just leave this as a minus sign.

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Let me delete this plus right here.

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Because it's just minus 3, right?

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Plus negative 3, same thing as minus 3.

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So what can we do here?

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We have an x plus 7, times 4x.

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We have an x plus 7, times negative 3.

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Let's factor out the x plus 7.

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We get x plus 7, times 4x minus 3.

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Minus that 3 right there.

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And we've factored our binomial.

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Sorry, we've factored our quadratic by grouping.

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And we factored it into two binomials.

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Let's do another example of that, because it's a little

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bit involved.

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But once you get the hang of it's kind of fun.

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So let's say we want to factor 6x squared plus 7x plus 1.

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Same drill.

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We want to find a times b that is equal to 1 times 6, which

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is equal to 6.

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And we want to find an a plus b needs to be equal to 7.

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This is a little bit more straightforward.

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What are the-- well, the obvious one is 1 and 6, right?

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1 times 6 is 6.

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1 plus 6 is 7.

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So we have a is equal to 1.

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Or let me not even assign them.

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The numbers here are 1 and 6.

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Now, we want to split this into a 1x and a 6x.

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But we want to group it so it's on the side of something

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that it shares a factor with.

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So we're going to have a 6x squar ed here, plus-- and so

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I'm going to put the 6x first because 6

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and 6 share a factor.

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And then, we're going to have plus 1x, right?

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6x plus 1x equals 7x .

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That was the whole point.

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They had to add up to 7 .

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And then we have the final plus 1 there.

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Now, in each of these groups, we can factor out

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as much as we like.

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So in this first group, let's factor out a 6x.

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So this first group becomes 6x times-- 6x squar ed divided by

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6x is just an x.

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6x divided by 6x is just a 1.

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And then, the second group-- we're going

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to have a plus here.

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But this second group, we just literally have a x plus 1.

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Or we could even write a 1 times an x plus 1.

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You could imagine I just factored out of 1 so to speak.

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Now, I have 6x times x plus 1, plus 1 times x plus 1.

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Well, I can factor out the x plus 1.

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If I factor out an x plus 1, that's equal to x plus 1 times

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6x plus that 1.

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I'm just doing the

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distributive property in reverse.

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So hopefully you didn't find that too bad.

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And now, I'm going to actually explain why this little

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magical system actually works.

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Let me take an example.

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I'll do it in very general terms.

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Let's say I had ax plus b, times cx-- actually, I'm

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afraid to use the a's and b's.

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I think that'll confuse you, because I

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use a's and b's here.

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They won't be the same thing.

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So let me use completely different letters.

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Let's say I have fx plus g, times hx plus, I'll use j

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instead of i.

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You'll learn in the future why don't like

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using i as a variable.

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So what is this going to be equal to?

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Well, it's going to be fx times hx which is fhx.

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And then, fx times j.

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So plus fjx.

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And then, we're going to have g times hx.

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So plus ghx.

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And then g times j.

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Plus gj.

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Or, if we add these two middle terms, you have fh times x,

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plus-- add these two terms-- fj plus gh x.

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Plus gj.

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Now, what did I do here?

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Well, remember, in all of these problems where you have

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a non-1 or non-negative 1 coefficient here, we look for

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two numbers that add up to this, whose product is equal

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to the product of that times that.

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Well, here we have two numbers that add up-- let's say that a

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is equal to fj.

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That is a.

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And b is equal to gh.

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So a plus b is going to be equal to that middle

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coefficient.

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And then what is a times b? a times b is going to be equal

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to fj times gh.

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We could just reorder these terms. We're just multiplying

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a bunch of terms. So that could be rewritten as f times

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h times g times j.

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These are all the same things.

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Well, what is fh times gj?

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This is equal to fh times gj.

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Well, this is equal to the first coefficient times the

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constant term.

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So a plus b will be equal to the middle coefficient.

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And a times b will equal the first coefficient times the

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constant term.

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So that's why this whole factoring by grouping even

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works, or how we're able to figure out what

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a and b even are.

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Now, I'm going to close up with something slightly

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different, but just to make sure that you have a

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well-rounded education in factoring things.

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What I want to do is to teach you to factor things a little

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bit more completely.

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And this is a little bit of a add-on.

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I was going to make a whole video on this.

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But I think, on some level, it might be a

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little obvious for you.

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So let's say we had-- let me get a good one here.

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Let's say we had negative x to the third, plus 17x

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squared, minus 70x.

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Immediately, you say, gee, this isn't even a quadratic.

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I don't know how to solve something like this.

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It has an x to third power.

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And the first thing you should realize is that every term

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here is divisible by x.

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So let's factor out an x.

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Or even better, let's factor out a negative x.

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So if you factor out a negative x, this is equal to

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negative x times-- negative x to the third divided by

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negative x is x squared.

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17x squared divided by negative x is negative 17x.

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Negative 70x divided by negative x is positive 70.

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The x's cancel out.

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And now, you have something that might look

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a little bit familiar.

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We have just a standard quadratic where the leading

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coefficient is a 1.

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We just have to find two numbers whose product is 70,

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and that add up to negative 17.

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And the numbers that immediately jumped into my

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head are negative 10 and negative 7.

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You take their product, you get 70.

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You add them up, you get negative 17.

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So this part right here is going to be x minus 10,

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times x minus 7.

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And of course, you have that leading negative x.

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The general idea here is just see if there's anything you

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can factor out.

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And that'll get it into a form that you might recognize.

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Hopefully, you found this helpful.

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I want to reiterate what I showed you at the beginning of

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this video.

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I think it's a really cool trick, so to speak, to be able

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to factor things that have a non-1 or non-negative 1

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leading coefficient.

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But to some degree, you're going to find out easier ways

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to do this, especially with the quadratic

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formula, in not too long.