FInd the condition that the zeroes of the polynomial `f(x) =x^3+3px^2+3qx+r` may be in A.P

Explanation

Let `alpha-d,alpha and alpha+d` are the zeroes of the given polynomial.

Then, sum of zeroes ` = alpha-d+alpha+alpha+d = 3alpha`

`=> 3alpha = -3p => alpha = -p->(1)`

Product of zeroes taking two at a time ` = alpha(alpha-d)+ alpha(alpha+d)+(alpha-d)(alpha+d)`

`=alpha^2-alphad+alpha^2+alphad+alpha^2-d^2`

`=3alpha^2-d^2`

`=> 3alpha^2-d^2 = 3q->(2)`

Now, product of zeroes ` = alpha(alpha^2-d^2)`

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