Question

find the condition that the zeroes of the polynomial x cube +3px square+ 3qx +rmay be in ap

Answer 1

Answer :

Given :

x3 + 3px2 + 3qx + r = 0

Let α , β and γ are three roots for the equation then :
By relationship between zeros and coefficient , we get

α + β + γ = - 3p ---------1

αβ + β γ + γ α = 3q ----------2

α β γ = - r ----------3

if roots α , β and γ are in AP then:
we know difference is same As :
β - α = γ - β

⇒2β = α + γ

From equation 1 , we get

⇒2β + β = - 3p

⇒β = - p --------------4

Put the value of β in equation 3 , and get

⇒α γ ( - p ) ​= - r

⇒α γ = rp -------------- 5

From equation 2 , we get

⇒β ( α + γ ) + rp ​= 3q

⇒-p (-2 p)+rp ​= 3q

⇒2p2 + rp ​= 3q
⇒2p3 + r = 3pq (answer)
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