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Answer 1

Answer:

q²x³ - 2pqx² + p²x + q

Step-by-step explanation:

α , β & γ are roots of  x³ + px + q = 0

For a³ + bx² + cx + d  here a = 1  , b = 1 , c = p  d = q

α + β + γ = -b/a  = 0

α β γ = -d/a  = q

α β + β γ + α γ  = c/a  =  p

(1/α + 1/β) , ((1/β + 1/γ) , (1/γ + 1/α)

Sum of roots = 2 (1/α + 1/β + 1/γ)  = 2 (β γ + α γ + α β)/α β γ = 2p/q   = -b/a

=> a = q  , b = -2p

Products of roots = (1/α + 1/β)* ((1/β + 1/γ) *(1/γ + 1/α)

= (α + β)(β + γ )(α +  γ)/(α β γ)²

= (-γ)(-α)(-β)/(α β γ)²

= -1/α β γ

= -1/q  = -d/a   => d = 1

(1/α + 1/β) *(1/γ + 1/α) + (1/α + 1/β)* (1/β + 1/γ) + (1/β + 1/γ) *(1/γ + 1/α)

=   (α + β)(α +  γ)/α β αγ  + (α + β)(β +  γ)/α β βγ  + (β +  γ)(α +  γ)/α β γγ

= (-γ)(-β)/α β αγ + (-γ)(-α)/α β βγ + (-α)(-β)/α β γγ

= 1/α² + 1/β² + 1/γ²

= ( (αβ)² + (αγ)² + (βγ)²) / (αβγ)²

= ( (αβ + αγ + βγ)² - 2(αβαγ +  αββγ + αγβγ) ) /(αβγ)²

=  ( p² - 2αβγ(α + β + γ) )/ q²

=   p²/q²  

=   (p²/q)/q   = c/a   => c = p²/q

qx³ -2px² + (p²/q)x + 1

= q²x³ - 2pqx² + p²x + q