Question

If $\alpha $ is the only real root of $x^{3}+bx^{2}+cx+1=0 \, \left(b < c\right),$ then the value of $\left|\left[\alpha \right]\right|$ is (where, $\left[.\right]$ represents the greatest integer function)

Answer 1

Let, $f\left(x\right)=x^{3}+bx^{2}+cx+1$
$f\left(x\right)=0$ has only one real root so it is an increasing function
Now, $f\left(0\right)=1,f\left(- 1\right)=-1+b-c+1$
$f\left(- 1\right)=b-c < 0$
using mean value theorem, real root must lie between $\left(- 1,0\right)$
so, $\left[\alpha \right]=-1$
$\left|\left[\alpha \right]\right|=1$