Question

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

Answer 1

Let α = a - d, β = a and y = a + d are the zeros of the given polynomial.

Sum of the zeros

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

⇒ f(a) = a³ + 3pa² + 3qa + r = 0

⇒ a³ + 3pa² + 3qa + r = 0

On substituting a = - p, we get 

⇒ -p³ + 3p³ - 3pq + r = 0

⇒ 2p³ - 3pq + r = 0