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0606/12 Feb/Mar 2026, Question 1: Solving Cubic Equations

Cambridge IGCSE Additional Mathematics 0606 · Paper 12 (Non-calculator) · Non-calculator · 4 marks

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The question, in short

For the cubic p(x) = 4x³ + 8x² − x − 2, demonstrate that x + 2 divides it exactly, then determine every root of p(x) = 0.

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Part (a) · 1 mark

Verify that substituting x = −2 into 4x³ + 8x² − x − 2 gives zero, confirming x + 2 is a factor.

  1. 1p(−2) = 4(−2)³ + 8(−2)² − (−2) − 2
  2. 2= −32 + 32 + 2 − 2 = 0
  3. 3Remainder is 0, so x + 2 is a factor

Answerp(−2) = 0, so x + 2 is a factor (shown)

Part (b) · 3 marks

Find all values of x satisfying 4x³ + 8x² − x − 2 = 0, using the known factor to obtain the remaining roots.

  1. 1p(x) = (x + 2)(4x² − 1)
  2. 24x² − 1 = (2x + 1)(2x − 1)
  3. 3x = −2, x = 1/2, x = −1/2

Answerx = −2, x = 1/2, x = −1/2

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