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  4. The number of distinct real roots of the equation x 5( x 3- x 2- x +1)+ x (3 x 3-4 x 2-2 x +4)-1=0 text is

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

JEE MainJEE Main 2022Complex Numbers and Quadratic Equations

Solution:

$ 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.