Find quotient and remainder by division

A question is this type if and only if you must explicitly find both the quotient and remainder when dividing a polynomial by a linear or quadratic expression.

2 questions

OCR C2 Q8
8. \(\mathrm { p } ( x ) = x ^ { 4 } - ( x - 2 ) ^ { 4 }\).
  1. Show that ( \(x - 1\) ) is a factor of \(\mathrm { p } ( x )\).
  2. Show that $$\mathrm { p } ( x ) = 8 x ^ { 3 } - 24 x ^ { 2 } + 32 x - 16$$
  3. Find the quotient and remainder when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ).
OCR C2 2016 June Q7
7 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - x + 3\).
  1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\).
  2. Hence find the three roots of the equation \(\mathrm { f } ( x ) = 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{555f7205-5e2a-4471-901d-d8abc9dd4b4a-3_540_718_1466_660} The diagram shows the curve \(C\) with equation \(y = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + 12 x + 9\).
  3. Show that the \(x\)-coordinates of the stationary points on \(C\) are given by \(x ^ { 3 } - 3 x ^ { 2 } - x + 3 = 0\).
  4. Use integration to find the exact area of the region enclosed by \(C\) and the \(x\)-axis.