If one root of the equation \[ x^4 - px^3 - 6x^2 + px +1 =0\] is \(\alpha\), prove that the others are \[-\frac{1}{\alpha},\frac{\alpha-1}{\alpha+1},\frac{1+\alpha}{1-\alpha}.\]
Could we use the fact that a quartic equation with roots \(\alpha, \beta, \gamma\) and \(\delta\) can we written as \((x-\alpha)(x-\beta)(x-\gamma)(x-\delta)=0?\)
Hence or otherwise show that for all real values of \(p\) the equation has four real roots.
Can we show the equation must have one real root?
