 So what we're doing is we're going to try x is equal to negative one again, which means x plus one We're going to see if x plus one is a factor of that guy and if it is then we've got a double factor because we already have an x plus one and We're just going to take the two so same routine I'm going to take the two multiply by negative one come up zigzag all the way up and find out what the remainder is The remainder is negative two, which means that x plus one is not Another factor of the original polynomial because it wasn't a factor of this guy, right? We already had it once it just doesn't occur again So what happens now is we know that x plus one is not going to work because all these guys are positive if we're Multiplying the terms by positive number then nothing's going to cancel out There's no way for us to be able to get a zero remainder So what we're going to have to do is go into the fractions because the possible factors of this guy This is 2x cube plus x squared plus 2x plus one, right? This is a cubic function So the possible factors of this are possible factors of this which is plus or minus one We've already tried negative one and just looking at it. We know positive ones not going to work so the other two factors that we haven't tried yet are One over two and negative one over two. Okay, what we're going to do is You know try negative x is equal to negative one over two The reason we're going to do that is is because we know that these guys are all positive So there's no way x is equal to a half is going to work out So what we're going to do is try x is equal to negative one over two now I'm going to shift over to camera and we're going to do the synthetic division over here, okay? So what I'm going to do right now is transfer all these numbers up here again and try x is equal to negative one over two So we're trying x is equal to negative one over two the two come straight down right two times negative one over two Is going to be negative one. Okay, so I'm just going to go through the zigzags and find out what our remainder is So what we got is when we go through our zigzags when you go through the synthetic division We've got a remainder of zero So we know that x is equal to negative one over two is a factor of this guy Therefore it's the factor of original polynomial, right? What we have down here. This was x cubed x squared x and the constant So what we have here we just divided an x We just took out an x from an x cube So this guy is now an x squared So this becomes two x squared plus zero x is just a place marker, right? So The next term the x term is missing. So it's going to be two x squared plus two right because that's going to be zero x So I'm not going to bother writing zero x and keep in mind This guy is going to be cross-multiply it up So two goes up here and multiplies the x and we're going to bring the one over So this guy the expression for this guy is actually Two x plus one So the expression for this guy is two x plus one and what we've done because we've got a remainder of zero This guy is a factor of this guy, which means it's a factor of original polynomial So what we started with was a polynomial to a degree of five and we've done three successful synthetic divisions Of an x term. That means we've kicked something to a power of five to something. That's a square. That's a that's a quadratic and That's what we want to do if we get any polynomials, which are higher degree and degree of two We're going to keep on doing our synthetic division until we take them down to A quadratic or an expression where we've got two things subtracted from each other that we can That we can factor using the difference of squares, right? So what we're trying to do with large polynomials is break him down To a level where we can use the other factor in techniques that we've already talked about now keep in mind What I mentioned previously with the warning of whenever you're dividing a fraction What's going to happen when you rewrite it into this form? What's going to happen is you're going to have to take a look at the quotient Okay, and you're going to have to take out a gcf to make sure that This expression this with this fraction rewritten this expression is now Going to give you your original polynomial back So this guy is two x squared plus two But what we're going to do is factor out a two and dump it because once we do that we end up having You know an expression where When multiplied by this it gives us our original term here because we really don't want to bring negative one over two over to have x plus a half That's you know, it's it's bad mathematics. It's bad. It's it's it's not it's not it's not the best way of expressing Factors you don't want fractions and factors. You want to express them like this, right? You want to get rid of your fractions because that's sort of the cleanest way of writing something So this guy here becomes two x squared plus two So we're just going to put brackets around here take out the two and dump it and that's what we're going to end up with Is going to be x squared plus one So what we have here now is x squared plus one that's The polynomial we've gotten down to now you can use the quadratic formula trying to factor this, right? But what you're going to end up having is the discriminant for this is going to be a negative And you can't take the root of a negative number So you know this guy can't be factored any further You can also remember that we can subtract two things We can factor two things subtracted from each other, right? If this was a minus we could still factor these guys But two things added together we can't factor Okay Now what i'm going to do is just show you that the discriminant for this guy is negative that way We can't factor can't use the quadratic formula or we could use the quadratic formula But in the real number world where we're functioning functioning right now this guy doesn't have any roots So the discriminant is b squared minus 4ac now A here this if you write this in the quadratic form a is equal to one B is equal to zero because we're missing the we're missing the middle term We're missing the x term and c is equal to one So i'm just going to do the discriminant over here just to show you that the discriminant is equal to a negative That means we can't factor it So if you set b squared minus 4ac you got zero squared minus four times one times one And that's just going to be negative four, right and negative four is negative It's less than zero. So you got you're going to have a negative sign inside the root symbol in The quadratic formula. So that means that this guy doesn't have any real roots It's got complex roots. It's got imaginary roots But since we're only functioning the real number around right now We're not talking about complex numbers. We're not taking up to that level yet So this is as far as you can factor this. This is as far as you can do go when you're doing your synthetic division What we're going to do right now is Write down our original polynomial over here and write it down in this factored form because we've completely factored it now Which means we found all the x intercepts and uh, you know, that's as far as we can go with it Okay, so let's just do a little Zoom over and rewrite everything and that that's it That's uh, that's as far as it goes and from there We could use all the x intercepts to be able to graph our polynomial Which we will talk about in another series when we get into graphing polynomials more accurately, okay So what we have is our original function, right our original polynomial And these are all its factors that we found using synthetic division x minus one x plus one Two x plus one and x squared plus one and this guy was as far as we could go So this guy up here In his fully factored form in the real number round is this guy right there Okay, so if you foiled all these guys you would end up getting this original expression here One way to check to make sure you did everything correctly Is take the first terms here. That's an x Multiply the first terms together. So x times x is x squared times two x is two x cubed times x squared is two x to the power of five That's our first term, right? So That end of it works out fine And you take the last terms one times one times one times one that just gives you one. That's our last term there So that's a really quick way of Just checking to make sure that you know, you've done everything correctly. Hopefully it all works out And you know, it's it's a good idea to do that once you do, you know if you end up doing a synthetic division or if you end up factoring a large polynomial like this because A question like this would be worth a fair bit of marks All right, because there was a lot of steps involved. So, you know, just take the extra Five seconds ten seconds that it takes to check your answer just to make sure that you ended up getting You know, hopefully you didn't make any mistakes in the middle terms, but you ended up getting the right Right answer, right? And that just gives you some more self-confidence when you find that, you know You got the first term right and you got the last term right Hopefully all the middle terms are correct as well And the odds are they will be if you got the first and the second first and the last term right, okay? Again, this is the way you do Factor large polynomials. You've got to use synthetic division. You start off, you know Possible factors last guy divided by possible factors the first guy always start off with You know, the lowest possible numbers first the lowest possible factors first Make sure everything is in descending order Make sure you put zeros for place markers for any missing x terms, right? And the one important thing to keep in mind is if you end up having fractions Where you have to rewrite across multiply map and move everything over when you get down to your quotient Take a look at it. You may have to take out a gcf and dump it to make sure you're still consistent with your original polynomial So I hope you enjoyed the you know the series that we did in the skate park There was a fair bit of info here as I said initially There was going to be a lot of information coming your way and there was a lot of information coming your way So hopefully you're able to process most of it anyway And whatever you're having problems with hopefully we can clear that up when we go ahead and start doing some examples Solving some equations and graphing them and we're definitely done with the skate park It's a great skate park. We may come back to it later on sometime in the future But we're going to go into the city and See what we can find there and do some work back in the city See you guys in the next videos