Question

The number of distinct real roots of the equation
$x ^5\left( x ^3- x ^2- x +1\right)+ x \left(3 x ^3-4 x ^2-2 x +4\right)-1=0 \text { is }$

Answer 1

$x^5\left(x^3-x^2-x+1\right)+x\left(3 x^3-4 x^2-2 x+4\right)-1 =0$
$\Rightarrow(x-1)^2(x+1)\left(x^5+3 x-1\right)=0$
Let $f(x)=x^5+3 x-1$
$f^{\prime}(x)>0 \forall x \in R$
Hence $3$ real distinct roots.