 So let's do something that's a little bit more complicated in the previous little four plus six thing that we wrote up, right? Let's say you had something so what you have here is two terms and what you're trying to do is find a pattern between each one and Take that out find something similar between each other and take that factor that out to the front That's called the GCF greatest common factor So what you would do here is you have a 24 and a 16 now I'm going to break one more of these things down to their prime prime numbers. Actually, we're not even going to bother 24 16 what's similar to me these two numbers, right? If you look at this eight goes into this that's the that's the highest number that goes into both of those terms You can take out a 12 out of this but you can't take out a 12 out of this you can take a 24 out of this We can't do 24 out of that so the biggest number that you can take out of both these numbers 24 is 16 is 8 so you can take an 8 out of both of these So what you can do is I usually end up writing it like this that way I could put whatever similar between them in here And then we figure out what's left over so From 24 and 16 you can take out an 8 so an 8 comes out Right if you want you can write down what's left over after you take the 8 out Right so if you took an 8 out of 24 you have three left here Put your plus sign here leave enough room for these guys if you're gonna have any of these guys left in there But let's put this in the middle, right? 8 out of 16 is going to be 2 Right so this is basically the principle right if you brought the 8 in multiplied it here This would go in and multiply this guy and this guy and you would get back your original 24 and 16 Right you're taking out the GCF Over here now. You're done with the numbers now. Let's go to the next letter you have an a here You have an a here so you can take out an a from both of them so a comes out You don't need anything here because you only need an a there and you have an a here already So you've already taken out the greatest color factor from these guys nothing left over You have a b squared here and a b cubed here b squared means there's two b's b cubed means There's three b's so what you can do is take out two b's from both of them now You couldn't take out a b cube because this doesn't have three b's to be taken out So what you do is you go b squared comes out Right now you don't have any more b's here You'll you needed a b squared and you've got a b squared here because when this guy goes in and multiplies this guy You're gonna get 8 times 3 is 24 a b squared, right? Over here you had a b cube you have a b squared here, so all you need is another b one more Now obviously we left too much space between these guys, so what you would do is just go eight a b squared Three plus and this is this guy Factor you just took out the GCF from this guy, and you ended up with this DCF is something that you're gonna use all the time throughout your you know math experience if you want to call it So it's something that you really need to learn so when it comes to quadratic equations. Let's say you had something like If they ask you to factor this Looking to see what's similar between these two terms here to take it out to the front and turn it into two things Two terms multiplied together now. What's similar between, you know two and eight is a two right, so you can take out a two What's similar between x squared and x, you know, there's an x there and two x is here So you can take out a maximum of one x right so x comes in Right now what you have left here is the way you should think about it is what do you multiply two x five? To give you two x squared, and that's just an x Right because two x times x gives you two x squared What do you multiply two x five to give you eight x? Well, you multiply it by four So that becomes four and this is this guy factored and where this becomes useful is If they give you look for example if you had it like this all they would be saying is you know factor this Expression and this is really an expression Because there is no equal sign right what they would do is just say factor this But what they what they could do and what you know the way you work towards functions Where we're gonna go is working towards functions is you would have an f of x here if it was a function When it comes to solving equations what they would do is just say add equals to zero here and say solve this equation and To solve this equation what we're gonna do is use the property of zero where if you you know Right now we can't solve this equation because x squared and x you know you can't combine them You can't add them if you remember your exponents or radicals So we need a new technique to be able to solve this and that's called factor gcf right and what we're gonna Do is use the property of zero where it says The only way you can have something multiple things you know things multiply together to give you zero is at least One of them is equal to zero. So if they said solve for this Right, they will say solve for this and this would be zero what we're gonna do is break this into you know Things multiply together to give you zero that way you can set each one equal to zero and that's exactly what we have here Two things multiply together to give you zero. So what you can do is set each one of these equal to zero So what you would do you go? Just solve for this for this you just divide by two. So this would be x is equal to zero And for this you just move the four over and with the x is Negative four and those are your solutions and as we talked about before those are your x intercepts for a parabola So we've got 9x squared over 60 plus 3x cubed over 28 and again If there was no equal sign here, they would just say factor this expression But what we're gonna do is continuously add the equal sign because you know, it's one step more complicated than this So what we're gonna do is just try to solve the equations right away So what you would do is I mean this doesn't change anything, right? You have a fraction compared to where we didn't have fractions which has had straight up, you know Numerator problems. We have something like this right with fractions works the same way if you're taking out a GCF Whatever is in the denominator stays in the denominator. So what you're gonna do here is Take out the GCF for each one of these terms You got a nine and you have a three so What's the GCF? What's the common greatest common factor between nine and three? Well, that's just straight up three, right? So let's put our brackets So you take out a three from here you took out three from here you have an x squared here You have an x cubed so you can take out an x squared Now what you're gonna do is just go into the denominator You took out whatever you could from the numerators, right? What you're gonna do is look at 16 16 and 28 now Two goes into 16 you got an eight left two goes into 28 you got 14 left But that's not the complete GCF because you can take out another two so the GCF 16 is gonna be 16 and 28 is gonna be four so you're gonna take out a four from this and you're gonna take out a four Is the GCF from both of those terms? So what you're gonna do now is figure out what's left over after you take this out Now again, the way I think about it is I always go back to this and I go What do I need to put here and what what am I gonna multiply this guy by to give me back this? Okay, so We have three x squared here. We've got nine x squared here What do you multiply three x squared by to give you nine x squared? That's just straight up three because we already got all the x's that we need so there's gonna be three going here With the denominator, what do you multiply four by to give you 16? That's just full Plus and What do you multiply three x squared by to give you three x cubed? Well, you need another x because this is x squared you need x cube And what do you multiply four by to give you 28? That's just seven And that is this guy fact now what we had was this guy equaling zero, right? Unfortunately zero is not coming up, but there's a zero there equals to zero So all you do is you set each one of these terms equal to zero and solve for the x So what you're gonna have here is you're gonna have x squared over four Is equal to zero Right and over here you're gonna have three quarters plus x over seven is equal to zero and these two You know the solutions for these is going to be the solution for this now a brand out of room This one's just gonna be straightforward you cross multiply to four up four times zero is zero