Verified answer

EXPLANATION.

α, β, γ are the roots of the equation.

⇒ x³ - px² + qx + r = 0.

As we know that,

Sum of the zeroes of the cubic polynomial.

⇒ α + β + γ = - b/a.

⇒ α + β + γ = -(-p)/1 = p.

Sum of the zeroes of the cubic polynomial two at a time.

⇒ αβ + βγ + γα = c/a.

⇒ αβ + βγ + γα = q/1 = q.

Products of the zeroes of the cubic polynomial.

⇒ αβγ = - d/a.

⇒ αβγ = -(r/1) = - r.

To find value of :

⇒ (α + β)(β + γ)(γ + α).

As we know that,

We can write equation as,

⇒ [αβ + αγ + β² + βγ] x (γ + α).

⇒ γ(αβ + αγ + β² + βγ) + α(αβ + αγ + β² + βγ).

⇒ (αβγ + αγ² + β²γ + βγ²) + (α²β + α²γ + αβ² + αβγ).

⇒ αγ² + β²γ + βγ² + α²β + α²γ + αβ² + αβγ + αβγ.

⇒ (α²β + α²γ + αβγ) + (β²γ + β²α + αβγ) + (γ²α + γ²β + αβγ) - αβγ.

⇒ α(αβ + αγ + βγ) + β(βγ + βα + αγ) + γ(γα + γβ + αβ) - αβγ.

⇒ (α + β + γ)(αβ + βγ + γα) - αβγ.

Put the values in the equation, we get.

⇒ (p)(q) - (-r).

⇒ pq + r.

Values of : (α + β)(β + γ)(γ + α) = pq + r.

Answer:

pq + r

Step-by-step explanation:

Given

α, β, γ are the roots of the equation x³ - px² + qx + r = 0

Comparing the given equation with ax³ + bx² + cx + d = 0

• a = 1

• b = - p

• c = q

• d = r

Sum of roots = - b/a

⇒ α + β + γ = - ( - p) /1

⇒α + β + γ = p

To solve the question we will write the above equation as

• α + β = p - γ

• β + γ = p - α

• γ + α = p - β

Sum of product of roots taken two at a time = c/a

⇒ αβ + βγ + γα = q/1 = q

Product of roots = - d/a

⇒ αβγ = - r/1

⇒ αβγ = - r

Now, coming to the question

( α + β )( β + γ )( γ + α )

= ( p - γ )( p - α)( p - β )

= p³ - ( α + β + γ )p² + ( αβ + βγ + γα )p - αβγ

= p³ - ( p)p² + ( q)p - ( - r)

= p³ - p³ + pq + r

= pq + r

Therefore the value of the given expression is pq + r.