Polynomials and sign conventions

Question

I can never make up my mind when dealing with polynomials how to "format" them. Are there any general guidelines to follow? Which of the following is the best practice?

1. $-a^2 + b$
2. $b - a^2$
3. $-(a^2 − b)$

Answer 1

There aren't any strict rules or conventions obviously, so it comes down to personal preference. Even if there were some widely used convention for anything, its technically not wrong to not follow it. So for anything, format it the way that you like best.

For me, I don't like negative signs at the front because you can avoid it and its one less sign to write down. But I also don't want to ruin a cyclical or symmetric expression or anything which is better organized in some specific order.

So in your example, I would write $b - a^2$ unless there was a significance in naming the variables in alphabetical order.

Another example is if I have the polynomial $-ax^3 + bx^2 + cx + d$ then in that case I would not rearrange it to avoid the negative sign since I almost always order the terms from highest power of $x$ to lowest.

Answer 2

All the above are correct and equivalent it is the beauty of mathematics. Cherish it. There are no guidelines to follow on the above.But sometimes these formatting play there role as when you want to take a common factor in an expression like

$(a+b)(a-b)+(b-a)=c$ which is equivalent to $(a+b)(a-b)-(a-b)=c$ or $(a+b-1)(a-b)=c$