Question

if sum of two zeros of x³-mx²+nx-p then show that mn=p
​

Answer 1

Given---> x³ - mx² + nx - p is a polynomial and sum of two zeroes is zero.

To prove----> mn = p

Proof----> x³ - mx² + nx - p

Let zeroes of given polynomial is α , β , γ

ATQ, sum of two zeroes = 0

=> α + β = 0

=> β = - α

Now,

α + β + γ = - Coefficient of x²/Coefficient of x³

=> α + β + γ = - ( - m ) / 1

=> α + β + γ = m

Putting β = - α , in it we get,

=> α - α + γ = m

α and ( - α ) cancel out each other , we get,

=> γ = m

Now,

αβ + βγ + γα = coefficient of x / coefficient of x³

=>αβ + βγ + γ ( - β ) = n / 1

=> αβ + βγ - βγ = n

- βγ and βγ cancel out each other and , we get ,

=> αβ = n

Now,

α β γ = - Costant term / Coefficient of x³

=> α β γ = - ( - p ) / 1

=> α β γ = p

=> ( α β ) ( γ ) = p

Putting αβ = n and γ = p , we get,

=> ( n ) ( m ) = p

=> mn = p

Hence proved

#Answerwithquality

#BAL