#### Question

Construct a new equation of which the roots are twice of those of the equation x^3-px^2+qx-r=0

Question

Construct a new equation of which the roots are twice of those of the equation x^3-px^2+qx-r=0

Let x_1,x_2 and x_3 are three roots of the cubic equation x^3-px^2+qx-r=0

Using the Vieta's formula, the following equations are established.

x_1+x_2+x_3=p

x_1x_2+x_1x_3+x_2x_3=q

x_1x_2x_3=r

Since the roots of the new equation are twice of x_1,x_2 and x_3, we can obtain the similar equations by applying the Vieta's formula.

2x_1+2x_2+2x_3=2p

2x_12x_2+2x_12x_3+2x_22x_3=4q

2x_12x_22x_3=8r

The values on the right hand side are just coefficients of the new equation.

Then, we get the new equation as follow.

x^3-2px^2+4qx-8r=0