6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\).
- Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
- Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
- Express \(\mathrm { p } ( x )\) as the product of linear factors.
- The curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) passes through the point \(( 0 , k )\). State the value of \(k\).
- Sketch the graph of \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\), indicating the values of \(x\) where the curve touches or crosses the \(x\)-axis.