FRAMING A CUBIC EQUATION WITH GIVEN THE ROOTS

Framing a Cubic Equation with Given the Roots - Practice questions

Question 1 :

If α , β and γ are the roots of the cubic equation x³ + 2x² + 3x + 4 = 0 , form a cubic equation whose roots are

(i) 2α , 2β , 2γ

Solution :

x³ + 2x² + 3x + 4 = 0

By comparing the  given equation with the general form of cubic equation, we get

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

Coefficient of x²  =  α + β + γ   =  2

Coefficient of x  =   α β + β γ + γα   =  3

Constant term  =  α β γ   =  4

Here α = 2α, β = 2β, γ = 2γ

Coefficient of x²  =  2α + 2β + 2γ  

2(α + β + γ)  =  2(2)  =  4

Coefficient of x  =  2α (2β) + 2β(2γ) + 2γ(2α)  

=  4 (αβ) + 4(βγ) + 4(γα)

  =  4 (αβ + βγ + γα)

=  4(3)

=  12

Constant term  =  2α (2β) (2γ)

=  8 α β γ

=  8(4)

=  32

Hence the required equation is 

x³ + 4x² + 12x + 32 = 0

(ii)  1/α, 1/β, 1/γ

Coefficient of x²  =  (1/α) + (1/β) + (1/γ)

(β γ + α γ + α β)/α β γ  =  3/4

Coefficient of x  =  (1/α) (1/β) + (1/β) (1/γ) + (1/γ)(1/α)

  =  (1/α β) + (1/βγ) + (1/γα)

  =  (γ + α + β)/α β γ

  =  2/4  =  1/2

Constant term  =   (1/α) (1/β) (1/γ)

=  1/αβγ

=  1/4 

By applying the above values in the general form of the cubic equation, we get

x³ + (3/4)x² + (1/2)x + (1/4) = 0

4x³ + 3x² + 2x + 1 = 0

Hence the required cubic equation is 4x³ + 3x² + 2x + 1 = 0.

(iii) −α, -β, -γ

Coefficient of x²  =  −α + (-β) + (-γ)

  =  - (α + β + γ)

  =  -2

Coefficient of x  =  −α (-β) + (-β)(-γ) + (-γ) (−α)

  =  α β + β γ + α γ

  =  3

Constant term  =  −α (-β) (-γ)

  =  - αβγ

=  -4

By applying the above values in the general form of the cubic equation, we get

x³ + (-2)x² + 3x + (-4) = 0

x³ - 2x² + 3x - 4 = 0

Hence the required cubic equation is x³ - 2x² + 3x - 4 = 0.

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