Question

Find the condition that the zeroes of the polynomial $p\left(x\right)=x^3-p{x}^2+qx-r$ may be in arithmetic progression.

Answer 1

$P\left(x\right)=x^3-p{x}^2+qx-r$

sum of roots $\Rightarrow 3a=P\Rightarrow a=P/3$

Product $\Rightarrow (a^2-d^2)a=r\Rightarrow (\frac{P^2}{9}-d^2)=\frac{3r}{P}$

$d^2=\frac{P^2}{9}-\frac{3r}{P}$

$\Rightarrow (a+d)(a-d)+a(a+d)+a(a-d)$

$\Rightarrow a^2-d^2+a^2+ad+a^2-ad\Rightarrow 3a^2-d^2$

$\Rightarrow 3\left(\frac{P^2}{9}\right)+\frac{P^2}{9}+\frac{3r}{P}\Rightarrow \frac{2P^2}{9}+\frac{3r}{P}$