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WMA12/01 Jan 2023, Question 5: Cubic Roots after Division

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 9 marks

The question, in short

A cubic f(x) = x³ + (p + 3)x² − x + q has constants p > 0 and q, with (x − 3) a factor. Use this and a remainder condition to find p, q, and a quadratic factor.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 2 marks

Using the factor (x − 3), derive the equation 9p + q = −51.

  1. 1Factor theorem: f(3) = 0
  2. 227 + 9(p + 3) − 3 + q = 0
  3. 327 + 9p + 27 − 3 + q = 0 ⇒ 9p + 51 + q = 0
  4. 49p + q = −51

AnswerShown: f(3) = 27 + 9(p + 3) − 3 + q = 0 ⇒ 9p + q = −51

Part (b) · 2 marks

Given the remainder 9 when dividing by (x + p), derive 3p² + p + q − 9 = 0.

  1. 1Remainder theorem: f(−p) = 9
  2. 2(−p)³ + (p + 3)(−p)² − (−p) + q = 9
  3. 3−p³ + p³ + 3p² + p + q = 9
  4. 43p² + p + q − 9 = 0

AnswerShown: f(−p) = −p³ + (p + 3)p² + p + q = 9 ⇒ 3p² + p + q − 9 = 0

Part (c) · 3 marks

Solve the two equations together to determine the values of p and q.

  1. 1q = −51 − 9p
  2. 23p² + p − 51 − 9p − 9 = 0 ⇒ 3p² − 8p − 60 = 0
  3. 3(3p + 10)(p − 6) = 0, and p > 0 so p = 6
  4. 4q = −51 − 54 = −105

Answerp = 6, q = −105

Part (d) · 2 marks

Using the found values, write a quadratic g(x) with f(x) = (x − 3)g(x).

  1. 1f(x) = x³ + 9x² − x − 105
  2. 2f(x) = (x − 3)(x² + bx + c): constant −3c = −105 ⇒ c = 35
  3. 3x² terms: b − 3 = 9 ⇒ b = 12
  4. 4g(x) = x² + 12x + 35

Answerg(x) = x² + 12x + 35

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