Polynomial with parameter in coefficient

Questions where a constant appears as part of a coefficient expression (e.g., p+3 or p-15) requiring algebraic manipulation before solving simultaneous equations.

1 questions

Edexcel P2 2023 January Q5
5. $$f ( x ) = x ^ { 3 } + ( p + 3 ) x ^ { 2 } - x + q$$ where \(p\) and \(q\) are constants and \(p > 0\)
Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\)
  1. show that $$9 p + q = - 51$$ Given also that when \(\mathrm { f } ( x )\) is divided by ( \(x + p\) ) the remainder is 9
  2. show that $$3 p ^ { 2 } + p + q - 9 = 0$$
  3. Hence find the value of \(p\) and the value of \(q\).
  4. Hence find a quadratic expression \(\mathrm { g } ( x )\) such that $$f ( x ) = ( x - 3 ) g ( x )$$