The number of distinct real roots of the equation x 7 - 7 x - 2 = 0 is

Given,

$x^7-7x-2=0$

$x^7-7x=2$

Now taking $f\left(x\right)=x^7-7x$ (odd) & $y=2$

Now differentiating $f\left(x\right)$ w.r.t $x$ we get,

$f^{\text{'}}\left(x\right)=7\left(x^6-1\right)=7\left(x^2-1\right)\left(x^4+x^2+1\right)$

$f^{\text{'}}\left(x\right)=0\Rightarrow x=\pm 1$

So points of extrema are $-1 \&\hspace{0.17em}1$, Now plotting the graph to check number of solution we get,

$f\left(x\right)=2$ has $3$ real distinct solution.