$a)$

$x^3-9x^2+2x-18=0$

$x^2\left(x-9\right)+2\left(x-9\right)=0$

$\left(x-9\right)\stackrel{\Delta =0-4\cdot 2<0}{\left(x^2+2\right)}=0$

$x-9=0$

$x=9$

 

 

 

$b)$

$2x^3+x^2-8x-4=0$

$x^2\left(2x+1\right)-4\left(2x+1\right)=0$

$\left(2x+1\right)\left(x^2-4\right)=0$

$\left(2x+1\right)\left(x-2\right)\left(x+2\right)=0$

$2x+1=0\vee x-2=0\vee x+2=0$

$x=-\frac{1}{2}\vee x=2\vee x=-2$

 

 

 

$c)$

$x^5+5x^4+x^3+5x^2=0$

$x^5+x^3+5x^4+5x^2=0$

$x^3\left(x^2+1\right)+5x^2\left(x^2+1\right)=0$

$\left(x^2+1\right)\left(x^3+5x^2\right)=0$

$\stackrel{\Delta =0-4<0}{\left(x^2+1\right)}\left(x+5\right)x^2=0$

$x=-5\vee x=0$

 

 

 

$d)$

$3x^4-5x^3-6x^2+10x=0$

$3x^4-6x^2-5x^3+10x=0$

$3x^2\left(x^2-2\right)-5x\left(x^2-2\right)=0$

$\left(x^2-2\right)\left(3x^2-5x\right)=0$

$\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)\left(3x-5\right)x=0$

$x=\sqrt{2}\vee x=-\sqrt{2}\vee x=\frac{5}{3}\vee x=0$

 

 

 

$e)$

$x^5-4x^3-8x^2+32=0$

$x^3\left(x^2-4\right)-8\left(x^2-4\right)=0$

$\left(x^2-4\right)\left(x^3-8\right)=0$

$\left(x-2\right)\left(x+2\right)\left(x-2\right)\stackrel{\Delta =4-4\cdot 4<0}{\left(x^2+2x+4\right)}=0$

$x=2\vee x=-2$

 

 

 

 

$f)$

$x^3-3\sqrt{2}{x}^2+\sqrt{2}x-6=0$

$x^3+\sqrt{2}x-3\sqrt{2}{x}^2-6=0$

$x\left(x^2+\sqrt{2}\right)-3\sqrt{2}\left(x^2+\frac{6}{3\sqrt{2}}\right)=0$

$x\left(x^2+\sqrt{2}\right)-3\sqrt{2}\left(x^2+\sqrt{2}\right)=0$

$\stackrel{\Delta =0-4\sqrt{2}<0}{\left(x^2+\sqrt{2}\right)}\left(x-3\sqrt{2}\right)=0$

$x=3\sqrt{2}$