Verify factor then sketch or analyse curve

Questions that verify a factor, factorise completely, then sketch the curve or find/analyse turning points and other curve properties.

2 questions

AQA C1 2006 January Q6
6 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 10 x + 8$$
    1. Using the factor theorem, show that \(x - 2\) is a factor of \(\mathrm { p } ( x )\).
    2. Hence express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Sketch the curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 10 x + 8\), showing the coordinates of the points where the curve cuts the axes.
    (You are not required to calculate the coordinates of the stationary points.)
Edexcel C2 Q9
9. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\).
  1. Show that \(( x - 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Using your answer to part (b), write down the coordinates of one of the turning points of the curve \(y = \mathrm { f } ( x )\) and give a reason for your answer.
  4. Using differentiation, find the \(x\)-coordinate of the other turning point of the curve \(y = \mathrm { f } ( x )\).