IB Mathematics HL May 2015 P1 TZ2 Q12

The cubic equation \(x^3+px^2+qx+c=0\), has roots \(\alpha,\beta,\gamma\). By expanding \((x-\alpha)(x-\beta)(x-\gamma)\) show that

1. 1. \(p=-(\alpha+\beta+\gamma)\);
   2. \(q=\alpha\beta+\beta\gamma+\gamma\alpha\);
   3. \(c=-\alpha\beta\gamma\).[3]

It is now given that \(p=-6\) and \(q=18\) for parts (b) and (c) below.

2. 1. In the case that the three roots \(\alpha,\beta,\gamma\) form an arithmetic sequence, show that one of the roots is \(2\).
   2. Hence determine the value of \(c\).[5]
3. In another case the three roots \(\alpha,\beta,\gamma\) form a geometric sequence. Determine the value of \(c\).[6]