1 The polynomial \(\mathrm { p } ( x )\) is given by
$$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$
where \(k\) is a constant.
- Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
- Express \(\mathrm { p } ( x )\) as the product of three linear factors.
- Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
- Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)