Can You Find Real Roots in an Integer Polynomial Using Calculus?

[Dragonfall]

I'll try my luck here:

How do you determine whether an integer coefficient single variable polynomial has at least one real root?

 

[trambolin]

if the order is odd and coefficients are real?

 

[mathman]

I'll try my luck here:

How do you determine whether an integer coefficient single variable polynomial has at least one real root?

Integer coefficients don't make any difference. If the coefficients are real, then for odd degree, there has to be (as trambolin notes) at least one real root. For quadratic, the discriminant has to be non-negative. For higher even degree it is more complicated. Try looking at Sturm sequences.

 

[Dragonfall]

Of course if the degree is odd or is 2 then it's trivial. I have to work out the general case, which "only requires elementary calculus, but is difficult."