Question

If zeros of the polynomial f(x) = x³ − 3px² + qx − r are in A.P., then

पर्याय

• 2p³ = pq − r

• 2p³ = pq + r

•  p³ = pq − r

• None of these

Answer 1

Let `a-d,a,a+d` be the zeros of the polynomial  f(x) = x³ − 3px² + qx − r  then

`\text{sum of zero }= - (text{coefficient of x})/(text{coefficient of } x^2)`

`(a - d) + a(a +b)= -(-3p)/1`

`a - cancel(d)+a+a+cancel(d)= 3p`

`3a = 3p`

`a = 3/3p`

`a = p`

Since a is a zero of the polynomial `f(x)`

Therefore,

`f(a)= 0`

`a ^3 - 3pa^2 + qa - r=0`

Substituting  `a=p`.we get

`p^3 - 3p(p)^2 + q xxp -r =0`

`p^3 - 3p^3 + qp-r =0`

`-2p^3 + qp - r =0`

` qp-r = 2p^3`

Hence, the correct choice is (a).