Proving non-existence of real roots

Question

Prove that for any real numbers $a_{85}, a_{84}, a_{83},\dots ,a_3$ the equation $a_{85}x^{85}+ a_{84}x^{84}+ a_{83}x^{83}+\dots +a_3x^3+3x^2+2x+1=0$ has no real roots. (This problem is stated in ALGEBRA book by Arihant Publication).

My doubt is that this is an odd degree equation when $a_{85}\neq 0$ then it will have at least one real root.

Answer 1

And you are right. Unless $a_{85}=0$, that equation must have a real root.