- 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?

- 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?