FROM A GIVEN CUBIC EQUATION FRAME A CUBIC EQUATION HAVING GIVEN ROOTS

If α, β and γ are the roots of the cubic equation

x³ + 2x² + 3x + 4 = 0

form a cubic equation whose roots are 

Problem 1 :

 2α, 2β and 2γ

Solution :

Here α = 2α, β = 2β and γ = 2γ

Considering the given cubic equation as

ax³ + bx² + cx + d = 0

x³ + 2x² + 3x + 4 = 0

a = 1, b = 2, c = 3 and d = 4

Coefficient of x² = (α + β + γ) = -b/a ==> -2

Coefficient of x =  α β + βγ + γα = c/a ==> 3

Constant term = α β γ = -d/a ==> -4

Coefficient of x² of new equation :

α + β + γ = 2α + 2β + 2γ

= 2(α+β+γ)

= 2(-2)

= -4

Coefficient of x of new equation :

α β + βγ + γα = 2α (2β) + (2β)(2γ) + (2γ)(2α)

= 4(α β + βγ + γα)

= 4(3)

= 12

Constant of new equation :

α β γ = 2α (2β) (2γ)

= 8α β γ

= 8(-4)

= -32

Required equation,

x³ - 4x² + 12x - 32 = 0

Problem 2 :

1/α, 1/β and 1/γ

Solution :

The required equation :

Problem 3 :

-α, -β and -γ

Solution :

The required equation :