Question

The number of real roots of the equation $x^3-8x^2+4x+2=0$ is

Answer 1

$f\left(0\right)=2\&f\left(1\right)=-1$
$f\left(0\right)\cdot f\left(1\right)<0$
$\therefore$ by Intermediate value theorem there exists at least one real root in $(0,1)$
Also $f\left(0\right)=2\&f(-1)=-11$
$f\left(0\right)\cdot f(-1)<0$
by Intermediate value theorem there exists at least one real root in $(-1,0)$
So, remaining one root must be real as complex roots always exist in pairs.
Hence, $3$ real roots are there.