Question

If zeros of the polynomial f (x) = x³ – 3px² + qx – r are in A.P., then

A. 2p³ = pq – r

B. 2p³ = pq + r

C. p³ = pq – r

D. None of these

Answer 1

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 = 3p

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