How many real roots has the equation? How many real roots (depends on the parametr $a$)  has the equation $x^{13}=a(x^{14}+1)?$
I ploted left and right sides and observed that for all $a \neq 0$ there is only one root. Is it true? 
Edit.  Correct typos.

The equation is equivalent to
$P_a(x)=ax^{14}-x^{13} +a=0$
Check out Descartes' Rule of Signs:
https://en.wikipedia.org/wiki/Descartes'_rule_of_signs
It is very useful for this kind of problems. In this specific case, it says that $P_a(x)$ has exactly one zero, because 


*

*$P_a(x)$ has 1 change of signs for $a>0$ and 0 change of signs for $a<0$.

*$P_a(-x)$ has 0 change of signs for $a<0$ and 1 change of signs for $a>0$.

