Question

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

Answer 1

condition is 2p³ - 3pq + r = 0

let zeroes of given cubic polynomial are ; (a - d), a , (a + d).

sum of zeroes = - coefficient of x²/coefficient of x³

⇒(a - d) + a + (a + d) = -3p/1

⇒3a = -3p

⇒a = -p ..........(1)

products of zeroes = - constant/coefficient of x³

⇒(a - d)a(a + d) = -r/1

⇒a³ - ad² = -r

⇒(-p)³ - (-p)d² = -r

⇒-p³ + pd² = -r

⇒(p³ - r)/p = d² ............(2)

sum of products of two consecutive zeroes = coefficient of x/coefficient of x³

⇒(a - d)a + a(a + d) + (a - d)(a + d) = 3q

⇒a² - ad + a² + ad + a² - d² = 3q

⇒3a² - d² = 3q

from equations (1) and (2),

⇒3(-p)² - (p³ - r)/p = 3q

⇒3p³ - p³ + r = 3pq

⇒2p³ + r - 3pq = 0 [this is required condition ]