Question

if alpha beta gamma are the zeroes of the given cubic Polynomials , find the values of the expressions given in the table


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Answer 1

Answer:

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Polynomials

Introduction to Polynomials

If alpha, beta, gamma are t...

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Asked on December 20, 2019 by

Singhan Kallepalli

If α,β,γ are the zeroes of the cubic polynomial x

3

+4x+2, then find the value of:

α+β

1

+

β+γ

1

+

γ+α

1

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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.