Weirdness of polynomial long division algorithm

[Cicnar]

"Weirdness" of polynomial long division algorithm

Hello. So, i just started to learn about the polynomial long division. As an introductory example, the book presents the long division of natural numbers, claiming that its basically the same thing.

The example: 8096:23

Solution: 8096:23=352 (23 into 80 goes 3 times)
-69 (3 times 23 is 69, so we subtract)
119 (23 into 119 goes 5 times)
-115 (23 times 5 is 115, subtract)
46 (23 goes into 46 2 times)
-46 (23 times 2 is 46, subtract)
0 (end)

I understand very well (or so i think) the long division of natural numbers, but this is where it gets awkward:

Same problem, different approach:

8096:(20+3)=352 (here i wrote the divisor as a sum of two arbitrary numbers, just to show what is bugging me)

or, if we expand, its like: (8*10³+9*10¹+6):(2*10¹+3)=3*10²+5*10¹+2

Now, as i see it, dividing this by the polynomial long division algorithm is pretty different (at least to me) than the standard one that we learned before, where the divisor was always a mononomial (be that a concrete number like 34, or an algebraic expression, like xy³).
It was NEVER a sum of terms. And it made perfect sense to me dividing that way.

But now, in this problem, for example, I am told that i can:

8000+90+6:(20+3)=300+50+6 1)first divide 20 into 8000 to obtain
400, then i multiply 400 by (20+3)
and subtract that from 8000+90+3

2)and so the procedure repeats

So i find this very confusing, although i can see that it produces the same result as the above mentioned, "standard" approach. I cannot see why are we allowed to divide (8000) by only one member of the sum (20), then multiply that quotient (400) by both summands (20+3) and then subtract it again from the whole dividend. I've tried it on several examples with concrete numbers, and i know it works.But it seems like magic, an arbitrary rule that just happens to works. And the chain of reasoning behind it is not familiar to me, unlike when the divisor is a single term; then i have a very clear reasons to justify the process.So, i think that if i could justify this method on division of natural numbers then it shouldn't be too much of a problem to generalize procedure to any polynomials. Am i right?

 

[HallsofIvy]

I'm a bit confused. What you show is exactly the way most people learn "long division" of numbers. You say "a concrete number like 34" but 34 has two parts: 3x10+ 4. That's not a "single term". To divide 34 into, say, 7854, you would note that 3 divides into 7 twice. Multiplying 2 times 30+ 4 gives 60+ 8 and 78- 68= 10. Bringing the next term, 5, down, we need to divide 34 into 105. 3 divides into 10 three times. 3 times 30+ 4 is 90+ 12= 102. 105- 103= 3 so, bringing down the 3, we have 34. Of course, 34 divides into 34 once so 34 divides into 7954 [itex]2x10^2+ 3x10+ 1= 231[/itex]. It really is exactly the same thing.

 

[Cicnar]

You said "most". That might be true, but not here. I have right in front of me a book which has introduced the division of natural numbers (and that's how i learned it). Not in a single example was there a divisor as a sum of two or more numbers. I know that 34 can be written as a sum, but its pretty different calculation when do not do that. Isnt it? What i really wanted to point out is that i can justify the reasons behind long division when we treat the divisor, say 34, as a single number. And i'll do that with your example to show you how i understand it (or i dont), why it is logical to me, and why this other method, that you say is predominant even in learning the long division of only natural numbers (therefore not only of polynomials) is not logical at all to me.

Here's how my silly brain sees it: we divide 7854 by 34:

first i notice that 7854 is actually 7000+500+80+4, then i see that 34 goes into 7000 at least 200 times (it goes more, but not enough to change the first numeral of the quotient, which is now 2, to, say, 3 or more), so i can write my quotient as (200+ something). Now i multiply 34 by 200 obtaining 6800 and subtract that from 7000 to obtain 200. Now i add 800 to it and get 1000. Next, 34 into 1000 goes 29 times, so my quotient becomes (200+29+something). Again i multiply 34 by 29 obtaining 986. I subtract that from 1000 and get 14. Now i add 14 and 54, getting 68. 68 divided by 34 is 2, so my quotient becomes (200+29+2)=231. And now, what i really like, i can say to myself: "Hey, this works because (200+29+2)*34=6800+986+68=7000+500+80+4." So, this way i can see how it is connected to multiplication and the distributive property. I see it almost as an "unpacking" of a sort, if you know what i mean. I hope that you can see what is going in my mind. I love the fact that i can logically explain it. But that is not the case when the divisor is written as a sum. I can do the division even then, but i cannot see or explain the logic behind it. I cannot explain why it is logical (and not just a "random" rule) to divide, say 7000 with only 30 (of our 30+4=34), which goes 23 times into 7000, and then multiply 23 by both (30+4) and now subtract that product from 7584. Seems so random, and yet it works. Can you see my trouble? And if so, could you give me a valid, logical explanation, or at least correct my corrupted reasoning, if corrupted it is?