Question

If a, b. g are the zeros of the polynomial x + 4x + 1 then (a + b)^(-1)+ (b+g)^(-1) +(g+a)^(-1)=​

Answer 1

Answer:

We have,

P(x)=x

3

+4x+2

Since, α,β,γ are the zeroes of this polynomial.

Now,

α+β+γ=−

1

0

=0

αβ+βγ+γα=

1

4

=4

αβγ=−

1

2

=−2

Since,

α+β

1

+

β+γ

1

+

γ+α

1

⇒

(α+β)(β+γ)(γ+α)

(β+γ)(γ+α)+(α+β)(γ+α)+(α+β)(β+γ)

⇒

(α+β)(β+γ)(γ+α)

βγ+αβ+γ

2

+αγ+αγ+α

2

+βγ+αβ+αβ+αγ+β

2

+βγ

⇒

(α+β)(β+γ)(γ+α)

α

2

+β

2

+γ

2

+3(αβ+βγ+γα)

⇒

(α+β)(β+γ)(γ+α)

α

2

+β

2

+γ

2

+2(αβ+βγ+γα)+(αβ+βγ+γα)

⇒

(α+β)(β+γ)(γ+α)

(α+β+γ)

2

+(αβ+βγ+γα)

⇒

(α+β)(β+γ)(γ+α)

0+0

⇒0

Hence, this is the answer.