 Today we're going to review how to do long division with whole numbers and then we'll use those same properties in order to learn how to do long division with polynomials. So if you think back a long time ago when you first learned how to divide with whole numbers, what you would do if you look at this example here is you first would look to see if three fits into just this number one. In this case three doesn't fit into one so then we just extend it and see just three fit into seventeen. It does, it fits into seventeen five times. So we'll put a five up here and then we do five times three is fifteen and that's what will subtract from seventeen. So now seventeen minus fifteen is two and if you thought of the biggest number that fit into seventeen you should get something smaller than three which we did. So since three doesn't fit into two we'll bring this five down and now we're looking at twenty five. So does three fit into twenty five? Yes it fits into twenty five eight times and so we write the eight at the top and three times eight is twenty four and then we subtract again and we get one. Now since one is less than three we could keep going and get a decimal but in this case we're just going to call this the remainder. With this long division our answer could be fifty eight remainder one or if you wanted to write it as a fraction what we're left with is one for what we're dividing and we're still dividing it by three so we keep that the same as this number up here. Now the reason I like to review that is because we're going to be using those same ideas when oops went one slide too many we do long division with polynomials. So if you look at your handout this is the first example under long division with polynomials. So here we're taking six x squared plus five x minus ten and we'll divide the entire expression by two x plus three. Now you can't just cancel something from each part individually. Instead we have to look at each of them as an entire expression. So I recommend rewriting the expression using our division sign because that can give you a little bit better organization. So just like we did before we first look at what we are dividing by the two x plus three but we just have to look at the term with the highest degree or your leading term. In this case that would be two x and then for what we are dividing we just again look at the leading term. So two x and six x squared is all we're looking at for the first one. The question is does two x fit into six x squared or two x times what gives you six x squared and two x multiplied by three x will give us six x squared so that is the first term that will be in our answer. So once we get that first term what we'll do is we'll multiply it by each part here both the two x and the three and we'll subtract just like we did before. So three x times two x is six x squared three x multiplied by three is nine x and notice I'm writing them below their like terms. So here six x squared minus six x squared will cancel five x minus nine x is a negative four x. Okay now this negative ten we didn't do anything with so I just like to bring it down each time. You don't need to bring it down until the end but I just bring it down each term. Okay now we're going to repeat that same process again we're just looking at the two x and this time we only will look at the negative four x. So two x times what gives us negative four x well in this case it would be a negative two and so that's our next term in our quotient up here at the top. Once we get the negative two we multiply by each term over here negative two multiplied by two x is negative four x negative two multiplied by three is a negative six and then remember each time we subtract that entire expression. So this leading term should cancel each time as long as you did it correctly and then negative ten minus a negative six will be a negative four and the way that we know we're done is if you compare the degree of what we're dividing by here the degree is one with what we have down here here the degree of what we have left is zero because we have no x's anymore in our answer. Once the degree down here is less than the degree of what we were dividing by then we're done so this negative four would be the remainder. So the answer that we can write it's two ways three x minus two from the top remainder negative four or with that remainder we also can make it a fraction. So make this a full expression. The remainder goes on the numerator of the fraction and then the denominator is just what we were dividing by the two x plus three. Okay I want us to do one more example using long division just to notice a few more of the little intricacies with it. So this example here we have a little bit longer expression three x to the fourth plus two x cubed minus x minus one. Now what I notice right away when I look at this is here when we look at the terms we're missing a term we have a fourth power a third power but right here there's no squared then we go to just the x and then the constant. So if I were doing this problem I would add in a section for the squared terms. Again you don't have to do this but I find that it helps to stay a little bit better organized because when we're subtracting we always like to write things under their like terms and if we don't have any x squareds then we kind of mess up our columns if we get an x squared while doing the division. If you don't end up getting an x squared then that just can go away at the end but this way it can help us stay organized to start. So what we'll look at here is we'll look at our leading term the x and our leading term here the three x to the fourth. So the question is x multiplied by what gives you three x to the fourth and here our answer is three x cubed. So just like before we have to multiply this three x cubed by both pieces over to the left. Three x cubed multiplied by three x to the fourth is a negative three x to the fourth. Oops I think I said that wrong. Three x cubed multiplied by x is a negative three x to the fourth and three x cubed multiplied by four is twelve x cubed. And remember we are subtracting both of these so that subtraction sign needs to go in front. When we subtract our x to the fourth cancel and then two x cubed minus twelve x cubed is negative ten x cubed. And here we're just going to bring these down because nothing changed with any of these last three terms. Now we repeat the process but with the new leading term. So x multiplied by what gives us negative ten x cubed it would in this case be negative ten x squared. And so we multiply negative ten x squared by both pieces of the x plus four negative ten x squared times x is negative ten x cubed. Negative ten x squared multiplied by four is negative forty x squared. So when we subtract the x cubed terms cancel but this is where that zero comes in handy. If we hadn't had a zero that written you could have just assumed it was a zero and gone on. But this way the like terms could stay in their columns. So zero minus a negative forty x squared will be a positive forty x squared. And then the negative x and negative one will stay. So down here our degree is two up here the degree is one so we have to keep going. X multiplied by what gives us forty x squared it is forty x. And so we'll multiply that by x plus four and we end up with forty x squared plus one hundred sixty x. Okay now we're subtracting so the forty x squareds cancel and then you just have to be careful here you're going to do a negative x that's like a negative one as the coefficient minus one hundred sixty. And it gives you negative one hundred sixty one x and that one the negative one will stay. Now notice these still have the same degree so we have to do it yet another time. X multiplied by what gives us negative one hundred sixty one x it's negative one sixty one. I'm running out of room there but hopefully you could see that. So we'll multiply each of these by negative one sixty one and then subtract. So we get negative one hundred sixty one x and then negative one hundred sixty one multiplied by four is a negative six hundred forty four. When we subtract the x's they cancel and then negative one minus a negative six forty four is a positive six hundred forty three. So that will be our remainder because the degree is less than the degree up here. Okay so the answer this one's a long one two options for writing this will rewrite this entire quotient here at the top and then you can just write remainder six forty three or to write it as a full function instead we can take the remainder and put that in the numerator of the fraction and then take our x plus four and put that in the denominator. So there is a quick explanation of how to do long division with polynomials.