For a polynomial $ax^3+bx^2+cx+d=0$ can we show that it has atleast one real root where $a,b,c\in\mathbb{R}$. Can we generalise and conclude the same thing about any polynomial of odd degree?
Note:- For a cubic polynomial we can say that it can have two complex roots which are conjugate and hence the third root can be real(I am unable to prove), but for any higher odd degree polynomial how can we assume that the conjugate complex roots exist like the case of cubic polynomial. Thank you.