If zeros of the polynomial f (x) = x3 – 3px2 + qx – r are in A.P., then
Given;
f(x) = x3 – 3px2 + qx – r
Let a, b and c be the zeroes of the polynomial x3 – 3px2 + qx – r
a + b + c = 3 p
ab + bc + ac = q
abc = r
a + b = 2b ⇒ b = p ⇒ a + c = 2p
b(a + c) + ac = q
⇒ 2p2 + ac = q
⇒ ac = r/p
∴ 2p2 + r/p = q
2p3 = pq – r