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Cicnar

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**"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?