If the zeroes of the polynomial f(x) = x3 – 12x2 + 39x + k are in A.P. Find the value of k.

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

(A – D) + A + (A + D) = –12

⇒ 3A = –12

⇒ A = –4 ……[4]

From [2], we have

((A – D))A((A + D)) = –k

⇒ A(A² – D²) = –k

⇒ 4(A² – D²) = k

From [3], we have

⇒ 2A² + A² – D² = 39

From [4] and [5]

⇒ k = 28

If the zeroes of the polynomial f(x) = x³ – 12x² + 39x + k are in A.P. Find the value of k.