1. organize polynomial criteria?

I'm dividing polynomials, operations like: $\displaystyle a^{2}-b^{2}+2bc-c^{2}\div a+b-c$
I know that I can organize first the polynomial in decreasing order for one letter by taking the biggest exponent of that letter and reorganizing form biggest exponent for that letter to lowest. But how can I do with the polynomial I just wrote? What's the criteria? For example for this case, I can organize decreasingly for the variable $\displaystyle b$, and I get up to this point in the reorganizing process: $\displaystyle -b^{2}+2bc$... but now the exponent for $\displaystyle b$ is zero for both the terms, $\displaystyle a^{2}$ and $\displaystyle -c^{2}$
Which one should be put first in these situations?

2. Re: organize polynomial criteria?

Are you sure that you are meant to deal with it that way ? I suspect that you are meant to factorise the top line.

3. Re: organize polynomial criteria?

Let me put it in a different way, since unfortunately some words that I use are being translated and perhaps incorrectly used in English. If I have for example $\displaystyle 2a^{3}+9a^{4}-14a^{2}$ and it's the dividend in a division problem, I can put the polynomial in order: here I'd get: $\displaystyle 9a^{4}+2a^{3}-14a^{2}$ The exponents go downwards from the term with the highest degree to the term with the lowest one. How can I do the same for the polynomial that I wrote: $\displaystyle a^{2}-b^{2}+2bc-c^{2}$

4. Re: organize polynomial criteria?

This is a strange operation to want to carry out, but if that's what's required then you have three options.

Write the dividend as $\displaystyle a^{2} - b^{2} + 2bc - c^{2}$ and divide by $\displaystyle a+b-c.$

Write the dividend as $\displaystyle -b^{2}+2bc-c^{2}+a^{2}$ and divide by $\displaystyle b-c+a.$

Or, write the dividend as $\displaystyle -c^{2}+2bc - b^{2} +a^{2}$ and divide by $\displaystyle -c+b+a.$

The easiest way of dealing with your particular example though is to factorise the top line.

5. Re: organize polynomial criteria?

I tried to understand what you meant by factorizing the top line but I still can't see how I could do that...
When I factorize I'm converting the dividend polynomial into a product. But then I don't see any simplification with the divisor.
I add the following two images to the question, where I show in orange rectangles, both the same problem. How does one decides to put those spaces to separate the terms in the dividend?

6. Re: organize polynomial criteria?

$\displaystyle a^{2}-b^{2}+2bc-c^{2}=a^{2}-(b^{2}-2bc+c^{2})=a^{2}-(b-c)^{2}$

$\displaystyle =(a-(b-c))(a+(b-c))=(a-b+c)(a+b-c),$

and the $\displaystyle a+b-c$ cancels.

If the numerator did not contain $\displaystyle a+b-c$ as a factor, division would lead to a remainder.

As to division, you have to decide on which variable you are dividing wrt and arrange for terms in that variable to occur in descending order. The order of the constant terms will not matter.

7. Re: organize polynomial criteria?

Thanks BobP for the help, I was struggling by doing $\displaystyle (a+b)(a-b)+c(2b-c)$ with the dividend but now I see it wasn't the right way of dealing with it!