Review question

# Can we show $x^4 - px^3 - 6x^2 + px +1 =0$ always has four real roots? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9189

## Suggestion

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?