If the zeroes of a polynomial f(x) = ax3 +3bx2 + 3cx+d are in A.P. prove that 2b3 – 3abc + a2d = 0

Question

Philoid · Accepted Answer

We know, if p, q and r are the roots of a cubic equation then

For the given equation,

……[1]

……[2]

……[3]

As, roots are in AP we can let roots as (A – D), A, (A + D) [Where D is common difference]

From [1], we have

……[4]

From [2], we have

……[5]

From [3], we have

From [4] and [5]

⇒ 2b³ – 3abc + a²d = 0

Hence Proved!

If the zeroes of a polynomial f(x) = ax³ +3bx² + 3cx+d are in A.P. prove that 2b³ – 3abc + a²d = 0