Question

if alpha beta and gamma are the three zeros of the polynomial of p of x = 2 x cube minus 3 X square + 2 x minus 8 than one by Alpha beta + 1 by beta gamma + 1 by Kama beta equals to​

Answer 1

Step-by-step explanation:

Given that α,β,γ are the roots of x

3

+x

2

+2x+3=0

so we get α+β+γ=−1,αβ+βγ+γα=2,αβγ=−3

Given that α+β,β+γ,γ+α are the roots of f(x)=0

let f(x)=x

3

+bx

2

+cx+d=0

−b=2(α+β+γ)=−2

⇒b=2

⇒c=(α+β)(β+γ)+(α+γ)(β+γ)+(α+β)(α+γ)=α

2

+β

2

+γ

2

+3(αβ+βγ+αγ)

⇒c=(α+β+γ)

2

+αβ+βγ+αγ=3

⇒−d=(α+β)(β+γ)(γ+α)=(α+β+γ)(αβ+βγ+γα)−αβγ=(−1)(2)−(−3)=1

⇒d=−1

Therefore f(x)=x

3

+2x

2

+3x−1