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O-LevelAdditional MathematicsFactors of polynomials4037/21 · May/June 2013 · Q129 Marks

The function f(x) = x³ + x² + ax + b is divisible by x - 3 and leaves a remainder of 20 when divided by x + 1.

  1. (i)Show that b = 6 and find the value of a.[4]
  2. (ii)Using your value of a and taking b as 6, find the non-integer roots of the equation f(x) = 0 in the form p ± √q, where p and q are integers.[5]

Mark scheme

Marking points from the official mark scheme (May/June 2013). The mark or mark type is shown in brackets.

  1. [1] (i) f(3) = (27 + 9 + 3a + b) = 0 or 3a + b = -36
  2. [1] (i) f(-1) = (-1 + 1 – a + b) = 20 or -a + b = 20
  3. [1] (i) Solve equations
  4. [1] (i) a = -14, b = 6
  5. [1] (ii) Find quadratic factor
  6. [1] (ii) x² - 4x - 2
  7. [1] (ii) Use quadratic formula or completing square on relevant 3 term quadratic
  8. [1] (ii) (-4 ± √16+8) / 2 or better
  9. [1] (ii) 2 ± √6 isw

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About this question

Topic
Factors of polynomials, O-Level Additional Mathematics (syllabus 4037). Structured question, 9 marks.
Source
O-Level Additional Mathematics, May/June 2013 Paper 2 Variant 1, question 12.

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