What if the roots of this equation are in geometric progression?

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1. Show that, if the roots of the equation \[x^3-5x^2+qx-8=0\] are in geometric progression, then \(q=10\).

If numbers are in geometric progression, how can we express them algebraically?

If we know the roots, can we write down the equation?

2. If \(\alpha\), \(\beta\), \(\gamma\) are the roots of the equation \[x^3-x^2+4x+7=0,\] find the equation whose roots are \(\beta+\gamma\), \(\gamma+\alpha\), \(\alpha+\beta\).

How are these roots connected to the coefficients of the polynomial?

Can we work with these connections to build the required equation without working out what the roots actually are?