Question

Find the condition that the zeros of the polynomial $f\left(x\right)=x^3+3p{x}^2+3qx+r$ may in A. P.

Answer 1

Consider the zeros of the polynomial as $a-d,a,a+d$.

$a-d+a+a+d=-\frac{3p}{1}$

$p=-a$

And,

$a\left(a-d\right)+a\left(a+d\right)+\left(a-d\right)\left(a+d\right)=\frac{3q}{1}$

$3q=3a^2-d^2$

$d^2=3p^2-3q$

And,

$\left(a-d\right)a\left(a+d\right)=-\frac{r}{1}$

$r=a{d}^2-a^3$

$r=\left(-p\right)\left(3p^2-3q\right)-{\left(-p\right)}^3$

$r=3pq-2p^3$