Question

Find the zeroes of the polynomial f(x) = x3 – px2 + qx – r, if it is given that the sum of two zeroes is zero.

Answer 1

Given : f(x) = x³ – px² + qx – r,   sum of two zeroes is zero.

To find : zeroes of the polynomial  

Solution:

Let say three zeroes are

α ,  β , γ

and α +   β = 0

f(x) = x³ – px² + qx – r,

Sum of zeroes = α +  β +  γ   = -(-p)/1  = p

=> 0 +  γ   = p

Hence one zero is p

αβ +  βγ  +  αγ  = q  

=> αβ + γ (β + α)  = q

=> αβ  + 0 = q

=>   αβ  = q

αβγ   =  -(-r/1) = r  

αβγ= r

=> pq   = r

α +   β = 0     &  αβ  = q

=> x²  - 0*x + q  = 0

=> x² = - q

=> x =  ±√-q

Zeroes  are   p  ,  ±√-q  

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Answer 2

let's take alpha and beta zeroes of polynomial

1. alpha beta and gamma