True or False: Some odd powered functions have no real roots.?

thanks! :)True or False: Some odd powered functions have no real roots.?
False.





Think about this. Recall that an equation with real coefficients (this is actually true if you are talking about complex coefficients) will have an even number of complex roots.





Let's say that there is a function with odd-degree such that there are no real roots. If that is the case, then it will have an odd number of complex roots. But since a complex solution must also have it's conjugate as a solution when it comes to real coefficients, there MUST be an even number of solutions. Thus, a contradiction.





I hope this helps!