Question

if zeros of the polynomial FX = x cube minus 3px square + qx minus r, are in A.P. then​

Answer 1

Step-by-step explanation:

Denoting the three zeros by α,β,γ, we have:

f(x)=x3−3px2+qx−r

f(x)=(x−α)(x−β)(x−γ)

f(x)=x3−(α+β+γ)x2+(αβ+βγ+γα)x−αβγ

In particular, equating the coefficient of x2 we have:

α+β+γ=3p

If α,β and γ are in arithmetic progression with common difference δ, then:

α=β−δ

γ=β+δ

So:

3β=(β−δ)+β+(β+δ)=α+β+γ=3p

So:

β=p

That is: p is one of the zeros of f(x)

So:

0=f(p)=p3−3pp