okay uh i might have added this um this was a canvas quiz problem that uh people usually blow up so um i thought i'd go over this so we're gonna take um we're gonna take this problem um and i gave you a zero so when i give you a factor when it's like x uh minus square root of two you change the sign but if given a zero you don't change the sign remember the zero is what goes outside of synthetic division and i'm actually going to use synthetic division for this this is a this is a linear factor so i'm going to take the square root of 2 outside and i'm going to divide it into this and we can see if we can find the other two linear factors so this is a one a two a negative two and a negative four we're gonna we're gonna do that one right there everybody know where i got the one two negative two and four yes are we good on that okay and again the zero goes outside here so i don't change the sign because i gave you the zero okay drop it one multiply it square root of add it 2 plus the square root of 2. multiply it 2 root 2 that times that is two root two what's root two times root two it's two plus two a lot of room in my blue box add it um root 2 times root 2 is 2 times 2 is 4. and remainder constant x x squared here we go the answer here is x squared plus two plus radical two x plus 2 radical 2. so i wrote that okay i'm not done yeah this is a quadratic factor quadratic because it's x squared they want two linear factors this quadratic is actually factorable guesses on what the front will be okay x and x okay here's the rule whenever you do factoring what numbers multiply to that and add to that here's your two linear factors this is a quadratic factor quadratic factor the directions on your canvas quiz will say linear factors well those are linear factors they're linear because it's x to the first it's x to the first that's what a linear factor means it's a canvas quiz problem all right last thing we have to do today this is awesome this is why we do this chapter this is why we just do what i i just did here we go this is the this is the this is the pinnacle right here it's number 60 in the book um i created a canvas quiz problem like this it's great okay before i go on um show me the show me the picture of that um [Music] maybe maybe got really wobbly look at 71 14 8 14. well those are big numbers i wonder if it really wobbles around zero but it's u-shaped it gets wobbly in the middle if you missed dan's party yesterday right now all right here we go these are factors of this they told you by telling you their factors that you're going to get a zero so watch this this is kind of cool if x minus 2 is a factor what will divide negative 2. okay now this is tricky watch this 8 negative 14 negative 7 1 negative 10 okay so 8 negative 14 negative 71 negative 10 and 24. it's the zero of that so you have to do that twice just wait i'll show you yeah but wait it gets trickier just do that and get your zero so do the zero you're gonna get zero here if you don't get zero here you did something wrong okay all right drop it multiply ads multiply put it there add do the do the synthetic division algorithm [Music] [Music] yeah i heard you [Music] negative 60 or positive 60 because those two make positive 60 negative 11. multiply 22 add 12 multiply negative 24 add zero now here's the cool part said verify the factors well you can actually use what's left over and do it again with uh negative four except it's going to be what positive 4. now what we're doing here when we do it one after another is what we're doing is we're actually reducing the polynomial i'll show you what i mean here in a second so let's do this together eight drops four times eight thirty-two thirty-two add two two two multiply eight eight add negative three negative three multiply negative 12 and zero so here's an essence what you just did you took something had x to the fourth in it and you reduced it down so it was only x squared because remainder constant x x squared stop so here we go we have these excuses eight x squared plus two x minus three and you could have done it in any order you would have gotten the same bottom one for real for real so it doesn't matter which one you could have put four there and negative two there and you would have gotten the same bottom yes okay why didn't you take like zero times four and then put the other zero and then have like your first well that's the you mean this one yeah that's the remainder it's just gone there's no remainder it's not it's not does that existing when it's only a zero yep yep nope stop stop they told you it was factors oh yes it had to be zeros there was no other option yeah these are factors of this you were going to get zeros so it says find the remaining factors how am i going to find the remaining two linear factors [Music] wow me too positive 3 and negative 1 yes i think negative 4 yes 6 is 2. so the remaining two factors are that four x plus three and that two x minus one one i simply just okay now all your work is done all you have to do now is compile all of your answers the way they want it on the bottom so watch right the complete factorization what was the first one was it x plus two x minus two uh minus minus no plus plus plus k one x plus two x minus four x plus 2 x minus 4. and the other two factors are 4x plus 3 and 2x minus 1. guess what would happen if i foiled all of this out it would it would equal the original polynomial what would be a great way of checking to make sure i've done this right plug the first one in plug the second one into y one and y two and the two graphs should lie right on top of each other you can check this done that list all real zeros x equals negative two four positive four negative three four negative three fourths confirm the results on a graphing calculator okay take your original function here and plug it in to your graphing calculator in y1 take the time to do this you need to see it decimals works pretty well on this if you want to do this on decimals i'm fine with that but you'll have to use your calculator on your test so you might as well do it on the calculator now 8x to the fourth minus 14 x to the third minus 71 x to the second minus 10 x plus 24. and i graphed it and mine's funky i got a zoom standard oh cool okay when i graphed it i got that picture now the bump does go down and does go up and we know it's in shape but here's what i got they're not lines if you zoom out they go up and they go down so i got a dot here i got an error i got a dot here i got a dot here and i got a dot here and the gist of the picture i'm going to draw it like i would want to even though i know it goes up it's u-shaped it comes down here somewhere down there it turns around it goes like that somehow and it comes down like that that's what it looks like with this spot being negative two this spot here being negative three-fourths this spot here being one-half and this spot here being four and i just verified those answers on that graph awesome that's the end of it um hopefully you took good notes you should be ready for the synthetic division quiz or the 2.3 quiz good luck