8. A curve \(C\) has equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 5 x + 2\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2 . The \(x\)-coordinate of \(P\) is 3 .
- Find the \(x\)-coordinate of \(Q\).
- Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
This tangent intersects the coordinate axes at the points \(R\) and \(S\).
- Find the length of \(R S\), giving your answer as a surd.