Find the condition that the zeros of the polynomial f(x) = x³ + 3px² + 3qx + r may be in A.P.

Given: f(x) = x³ + 3px² + 3qx + r

Concept Used:

For a cubic polynomial: ax³ + bx² + cx + d = 0

Sum of roots

The product of roots taken two at a time

The product of roots

Explanation:

Let the roots be,

α = a – d

β = a

γ = a + d

Sum of roots = (a – d) + a + (a + d) = 3a

Also,

Sum of roots

Therefore,

3a = –3p

a = –p

Since a is the zero of the polynomial, therefore f(a) = 0

⇒ f(a) = a³ + 3pa² + 3qa + r = 0

⇒ a³ + 3pa² + 3qa + r = 0

Substitute a = –p

⇒ –p³ + 3p³ – 3pq + r = 0

⇒ 2p³ – 3pq + r = 0