polynomial long division to understand how to divide polynomials let's remember how to just divide so we have 297 divided by 14. do you remember how to do this we say 14 goes into 29 how many times well 14 times 2 is 28 yeah let's do that put the 2 above that 9 and then 2 times 14 is 28 we write it under the 29 and then we subtract there you go the big thing subtract 28 from 29 is 1. bring down our 7 and then 14 goes into 17 one time multiply 1 times 14 subtract and we have 3 left over now we're not in elementary school so we're not going to say remainder 3 no we're going to say 3 divided by 14 because we still haven't divided 3 by 14. so we have 21 and 3 14. let's try example b notice that 7 goes into 4508 evenly so it's a factor of 4508 next let's apply this to long division with polynomials so we have the the 2x cubed polynomial being divided by the 2x plus 5. so we still set it up the same the 2x plus 5 goes outside and then we put that big polynomial inside the box let's go we want to look at just the first term of each polynomial so i have this 2x and i need to go into 2x cubed for our purposes we're really going to think about like 2x times what gives me 2x cubed or you can think 2x cubed divided by 2x x squared so that means we want to multiply 2x plus 5 by x squared we just have to distribute that x squared into the binomial so we're going to go ahead x squared times 2x i get 2x cubed if those aren't the same then i did something wrong start all over right but then i also have to multiply the x squared into the 5. so plus 5 x squared all right do you remember the process now the next step was to subtract so the most common error here is to forget to subtract to subtract that second term to distribute the minus sign so let's make sure that we're very obvious with those subtraction signs alrighty so 2x cubed minus 2x cubed gone that's what we wanted negative 7x squared minus 5x squared so negative 12x squared then what's the next step in the process bring down so we're going to bring down that 12x okay now we repeat the process so we're still looking at that 2x the first term of the binomial going into the first term of what's in that next part of the polynomial that we're dividing into so negative 12 x squared so over here on my think section i can think negative 12 x squared divided by that 2x because i really want to say 2x times what equals negative 12x squared well that's going to be a negative 6x so that's the next term that i'll multiply into the binomial so minus 6x goes above that negative 12x and then we distribute into the 2x plus 5. i'm going to continue to make a big deal about that subtraction sign so we're going to subtract both terms notice that when i get to that minus a negative 30x that means i'm actually going to be adding the 30x so negative 12x squared plus 12x squared negative 12x plus 30x and now we're almost done 2x plus 5 into 18x plus 45. do you remember what we think we think how many times does 2x go into 18x well that's going to be 9. [Music] woohoo remainder 0 there's nothing left over what does that tell us about 2x plus 5 it tells us that 2x plus 5 is a factor of that larger cubic polynomial as a way of checking our division we could now multiply the 2x plus 5 into the x squared minus 6x plus 9. also if we were wanting to get fully factored form we could take this and now factor the trinomial and have fully factored form let's try this next one same idea x squared times what gives me 3x to the fourth so think 3x to the fourth divided by x squared 3x squared now i'm going to distribute multiplying to the binomial 3x squared times x squared 3x to the fourth 3x squared times 1 is 3x squared wait a second i have an x cubed term in this polynomial what's going on oh sometimes we need to go back and save a space so notice that i went from x squared no x term and then a 1. so i need to go ahead let's rewrite this really quick actually when i was rewriting it i noticed that i was also missing the x term in the other polynomial so keep in mind that you may have to put in zeros for any time you are missing one of the terms nice thing about division long division is you'll notice it because it won't match up so let's go ahead back to where we were we're going to multiply by 3x squared distribute into now the trinomial [Music] do yourself a favor really do something distinct to show that subtraction change colors show that you're subtracting each term go ahead and finish this problem come back and check in with me [Music] here we have a binomial left over so x squared plus 1 is not a factor of that larger polynomial how do we write the remainder we write it showing that it still needs to be divided so three x squared minus four x plus nine plus four x minus four divided by x squared plus one whoo-hoo i think we have the hang of this number three let's go ahead i think you got this do it check back with me [Music] [Music] 4x plus 3 plus negative 31 divided by x plus five whoa wait a second i think i have a quicker way to do that problem okay okay i got this [Music] look at that i mean we could have clocked it i beat you there hey um they haven't learned that yet oh let's learn synthetic division so since i was dividing by a linear factor of one x plus five i take the zero for that factor so x plus five equal to zero subtract five i get x equals negative five so i set that up right here then i take my coefficients of the polynomial i'm dividing into so that's the 4 the 23 the negative 16. and i go and i start doing some really cool synthetic division i bring down that 4 multiply the 4 by the negative 5 i get negative 20 i add down multiply add down until i run out of terms the reason we like synthetic division is because it's adding now instead of subtracting so i don't have any of that little mistake then if we look when i'm done i have this 4 which is the 4x from my answer then i have this 3 which is that plus 3 from my answer and then that negative 31 left is my remainder let's try it so let me break this down a little bit more in example 5. okay so i've got my polynomial that's a cubic and i'm dividing it by a linear factor where i have that one x minus three okay so to set it up i kind of set up a little box here i have a line a little box here okay in this box right here i'm gonna put my zero so x minus three equal to zero is three if i add the three to the other side so that's what i'm gonna set up there now parallel to that is my coefficients to the polynomial i'm dividing into so i've got two x cubed right then i got my negative 3 x squared then i got my negative 11 x and then my positive 6. now if i was missing any of those exponents it has to go x cubed to x to the zero so sometimes what i do is i'll set them up at the top so that i don't accidentally forget one all right so now i'm going to bring down that 2 like i talked about before and i start to multiply 2 times 3 is 6. then i get to add down that's the beauty of synthetic division adding so negative 3 plus 6 is 3 and then i just repeat the same process multiply 3 times 3 is 9 then i add down negative 11 plus 9 is 2 negative 2. and i do it again repeat the process negative two times three is negative six i add down and i get a zero now i'm out of terms i've completed my synthetic division well that zero is in the remainder spot so what does it mean if i have a remainder of zero it means that x minus 3 is a factor of this polynomial that's awesome but i don't even have a a leftover polynomial a quotient so what i need to do is i need to take these this 2 this 3 and this negative 2 and those are my coefficients for my quotient so 2 x cubed was being divided by x so that means the leading term is going to be x squared because if i divide x cubed by x i get x squared so i have 2x squared and then i'm just going to decrease by one exponent each time so plus 3 x because x to the 1 power and then minus 2 x to the 0 power then i don't have a remainder so i can just write it that way but does this factor out any further i think it does let's use crisscross so that polynomial factors out into 2x minus 1 x plus 2 and then the original factor was x minus 3. so this big polynomial up here can be factored into x minus three two x minus one and x plus two whoa this is cool okay why don't you set example six up synthetic division okay let's check your setup does yours look like mine that's a problem we definitely can't have that happen right what did i forget this is an x to the fourth power so that means i need a place for x to the fourth x to the third x squared x and x to the zero well awesome i have a place for my x to the fourth my x squared and my x to the zero but i don't have my x to the third or my x so since i didn't have those terms i need to put zero placeholders in so that's one bummer part about synthetic division we can think we're doing it totally right and get an answer and be totally wrong because we forgot our zeros we're in long division we can kind of catch that because it's part of the process okay so go ahead and finish it from there once you've put in those zeros [Music] all right let's check your final answer now make sure you've actually written out five x to the third right not just a five then i have my plus ten x squared i have my minus four x and then i have my minus eight with my remainder minus six divided by x minus 2.