Factor condition (zero remainder)

Questions where a linear expression is stated to be a factor of the polynomial, requiring the polynomial to equal zero at a specific value.

2 questions

OCR MEI Paper 1 2018 June Q1
1 Show that ( \(x - 2\) ) is a factor of \(3 x ^ { 3 } - 8 x ^ { 2 } + 3 x + 2\).
AQA Paper 1 2022 June Q11
11 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + ( b + 2 ) x ^ { 2 } + 2 ( b + 2 ) x + 8$$ where \(b\) is a constant.
11
  1. Use the factor theorem to prove that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) for all values of \(b\).
    11
  2. The graph of \(y = \mathrm { p } ( x )\) meets the \(x\)-axis at exactly two points.
    11
    1. Sketch a possible graph of \(y = \mathrm { p } ( x )\)
      \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-20_1084_965_1619_532} 11
  4. (ii) Given \(\mathrm { p } ( x )\) can be written as $$\mathrm { p } ( x ) = ( x + 2 ) \left( x ^ { 2 } + b x + 4 \right)$$ find the value of \(b\). Fully justify your answer.