Answer the question
Answers
EXPLANATION.
α, β, γ are the roots of the cubic equations.
⇒ 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.
Products of the zeroes of the cubic polynomial two at a time.
⇒ αβ + βγ + γα = c/a.
⇒ αβ + βγ + γα = q.
Product of the zeroes of the cubic polynomial.
⇒ αβγ = - d/a.
⇒ αβγ = r.
(a) If roots are : (βγ + 1/α), (γα + 1/β), (αβ + 1/γ).
As we know that,
Sum of the zeroes of the cubic polynomial.
⇒ βγ + 1/α + γα + 1/β + αβ + 1/γ.
⇒ 1/α + 1/β + 1/γ + αβ + βγ + γα.
⇒ [(αβ + βγ + γα)/αβγ] + αβ + βγ + γα.
Put the values in the equation, we get.
⇒ q/r + q = (q + qr)/r.
Products of the zeroes of the cubic polynomial two at a time.
⇒ (βγ + 1/α)(γα + 1/β) + (γα + 1/β)(αβ + 1/γ) + (αβ + 1/γ)(βγ + 1/α).
⇒ [αβγ² + γ + γ + 1/αβ] + [α²βγ + α + α + 1/βγ] + [αβ²γ + β + β + 1/γα].
⇒ α²βγ + αβ²γ + αβγ² + 2α + 2β + 2γ + 1/αβ + 1/βγ + 1/γα.
⇒ αβγ(α + β + γ) + 2(α + β + γ) + [(α + β + γ)/αβγ].
Put the values in the equation, we get.
⇒ r(p) + 2(p) + (p/r).
⇒ [(r²p + 2pr + p)/r].
Products of the zeroes of the cubic polynomial.
⇒ (βγ + 1/α) x (γα + 1/β) x (αβ + 1/γ).
⇒ [αβγ² + γ + γ + 1/αβ] x (αβ + 1/γ).
⇒ [αβγ² + 2γ + 1/αβ] x (αβ + 1/γ).
⇒ α²β²γ² + αβγ + 2αβγ + 2 + 1 + 1/αβγ.
Put the values in the equation, we get.
⇒ (αβγ)² + αβγ + 2(αβγ) + 3 + 1/αβγ.
⇒ (αβγ)² + 3(αβγ) + 1/αβγ + 3.
⇒ (r)² + 3(r) + 1/r + 3.
⇒ (r³ + 3r² + 3r + 1)/r = (r + 1)³/r.
As we know that,
Formula of cubic polynomial.
⇒ x³ - (α + β + γ)x² + (αβ + βγ + γα)x - αβγ.
Put the values in the equation, we get.
⇒ x³ - [(q + qr)/r]x² + [(r²p + 2pr + p)/r]x - [(r + 1)³/r] = 0.
⇒ rx³ - q(r + 1)x² + p(r + 1)² - (r + 1)³ = 0.
Theory of Equations
Answer:
a) rx³ - q( r + 1 )x² + p( r + 1 )²x - ( r + 1 )³
b) x³ - px² + ( 4q - p² )x + ( 8r - 4pq + p³ ) = 0
Step-by-step explanation:
Refer to attachment
