Non-zero remainder condition

Questions where division by a linear expression gives a specified non-zero remainder, requiring the polynomial to equal that remainder value at a specific point.

3 questions · Moderate -0.2

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CAIE P2 2023 November Q1
2 marks Easy -1.2
1 When the polynomial $$a x ^ { 3 } + 4 a x ^ { 2 } - 7 x - 5$$ is divided by \(( x + 2 )\), the remainder is 33 .
Find the value of the constant \(a\).
Edexcel C2 Q2
7 marks Standard +0.3
\(f(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.
  1. Given that f(n) has a remainder of 3 when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
  2. Show that f(n) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
  3. Hence show that f(n) is divisible by 3 for all positive integer values of \(n\). [2]
Edexcel C2 Q3
7 marks Standard +0.3
\(\text{f}(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.
  1. Given that f\((n)\) has a remainder of \(3\) when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
  2. Show that f\((n)\) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
  3. Hence show that f\((n)\) is divisible by \(3\) for all positive integer values of \(n\). [2]