Product expressions with roots

A question is this type if and only if it asks to evaluate products like (α+k)(β+k)(γ+k) or similar expressions involving all roots without solving the equation.

2 questions · Standard +0.8

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OCR FP1 2013 June Q8
6 marks Standard +0.8
8 The cubic equation \(k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0\), where \(k\) is a non-zero constant, has roots \(\alpha , \beta\) and \(\gamma\).
Find the value of \(( \alpha + 1 ) ( \beta + 1 ) + ( \beta + 1 ) ( \gamma + 1 ) + ( \gamma + 1 ) ( \alpha + 1 )\) in terms of \(k\).
CAIE FP1 2017 November Q4
8 marks Standard +0.8
4 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 10 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
  2. Find the value of \(( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )\).