Find constant then solve inequality or further work

Given one factor, find the single unknown constant, then solve an inequality, find a remainder, or perform other further work beyond simple factorisation.

4 questions · Moderate -0.6

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CAIE P2 2023 June Q4
8 marks Moderate -0.3
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + k x - 30$$ where \(k\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(k\).
  2. Hence find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) and factorise \(\mathrm { p } ( x )\) completely.
  3. It is given that \(a\) is one of the roots of the equation \(\mathrm { p } ( x ) = 0\). Given also that the equation \(| 4 y - 5 | = a\) is satisfied by two real values of \(y\), find these two values of \(y\).
CAIE P3 2016 March Q4
7 marks Moderate -0.5
4 The polynomial \(4 x ^ { 3 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\).
  2. When \(a\) has this value,
    1. factorise \(\mathrm { p } ( x )\),
    2. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.
CAIE P3 2004 November Q3
6 marks Moderate -0.8
3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 4\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).
  1. Find the value of \(a\). When \(a\) has this value,
  2. factorise \(\mathrm { p } ( x )\),
  3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.
Edexcel C2 Q1
4 marks Moderate -0.8
\(f(x) = 2x^3 - x^2 + px + 6\), where \(p\) is a constant. Given that \((x - 1)\) is a factor of \(f(x)\), find
  1. the value of \(p\), [2]
  2. the remainder when \(f(x)\) is divided by \((2x + 1)\). [2]