Question

Let a and b be real numbers such that $a\ne 0$. Prove that not all the roots of $a{x}^4+b{x}^3+x^2+x+1=0$ can be real.

Answer 1

Let ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ be the roots of $a{x}^4+b{x}^3+x^2+x+1=0$ .
Observe none of these is zero since their product is 1/a.
Then the roots of $x^4+x^3+x^2+bx+a=0$ are
${\beta }_1=\frac{1}{{\alpha }_1},{\beta }_2=\frac{1}{{\alpha }_2},{\beta }_3=\frac{1}{{\alpha }_3},{\beta }_4=\frac{1}{{\alpha }_4}$
We have
$\sum _{j=1}^4{\beta }_j=-1,\sum 1\le j<k\le 4{\beta }_j{\beta }_k=1$
Hence
$\sum _{j=1}^4{\beta }_j^2=\left(\sum _{j=1}^4{\beta }_j\right)2-2\left(\sum _{1\le j<k\le 4}{\beta }_j{\beta }_k\right)=1-2=-1$
This shows that not all ${\beta }_j$ can be real. Hence not all ${\alpha }_j$'s can be real.