5 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\).
- Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
- Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
- Express \(\mathrm { p } ( x ) = x ^ { 3 } - 2 x ^ { 2 } + 3\) in the form \(( x + 1 ) \left( x ^ { 2 } + b x + c \right)\), where \(b\) and \(c\) are integers.
- Hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.