[tex]5x^3 - 4x + 1 = 0 \\\\5x^3-5x+x+1=0\\\\5x(x^2-1)+1(x+1)=0\\\\5x(x+1)(x-1)+1(x+1)=0\\\\(x+1)(5x(x-1)+1)=0\\\\(x+1)(5x^2-5x+1)=0\\\\x=-1\\\\5x^2-5x+1=0\\\\\Delta=(-5)^2-4\cdot5\cdot1=25-20=5[/tex]
[tex]\Delta>0[/tex] therefore there exist two real solutions.
In total, there are 3 real roots. But, real roots are also complex roots (doesn't work the other way round!), so there are 3 complex roots.
