Descartes Rule of Signs: Why must leading coeff. be pos?

[axrw]

Sorry, I've got another one. This is a little silly perhaps, but in my book it says:

"To use the rule, we must have the polynomial arranged in descending or ascending order, with no zero terms written in, the leading coefficient positive, and the constant term not 0."

My question is why does the leading coefficient need to be positive? I'm probably missing something, but how would that change anything? I can see how the order would be important, but yeah. I've looked around on the internet and I can't find that mentioned anywhere else.

Thanks.

 

[tkhunny]

No difference. f(x) and -f(x) have the same zeros. However, convention is important. Hanging negative signs are easy to overlook.

I'm a little surprised you didn't ask about the constant term. Why does that have to be non-zero?

 

[axrw]

Ok, thank you for clearing that up for me.

As for the constant term, I just figured it was because it had something to do with 0 being a real zero of the polynomial that was neither negative or positive, so you had to factor the polynomial to get a constant term, and then figure out how many of the remaining zeros could be +- real. Otherwise you would be missing a real zero (or possibly more) in your count.

...or something.

I think my head is going to explode. *sigh*

 

[tkhunny]

Close enough, on the zero stuff. Stay away from the head explosions.