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4PM1/02 Jun 2021, Question 6: Simultaneous Equations with a Curve

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 13 marks

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The question, in short

Cubic f(x) = x³ + (p + 1)x² − 10x + q with integer p, q; use two given factors to find p and q, then factorise f(x) fully.

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Part (a) · 3 marks

Given (x − 3) is a factor, show that 9p + q + 6 = 0.

  1. 1f(3) = 0
  2. 227 + 9(p + 1) − 30 + q = 0
  3. 327 + 9p + 9 − 30 + q = 0 ⇒ 9p + q + 6 = 0

Answer9p + q + 6 = 0 shown

Part (b) · 3 marks

Given (x + p), p > 0, is also a factor, show that p² + 10p + q = 0.

  1. 1f(−p) = 0
  2. 2(−p)³ + (p + 1)p² + 10p + q = 0
  3. 3−p³ + p³ + p² + 10p + q = 0 ⇒ p² + 10p + q = 0

Answerp² + 10p + q = 0 shown

Part (c) · 5 marks

Hence find the values of p and q.

  1. 1q = −9p − 6
  2. 2p² + 10p − 9p − 6 = 0 ⇒ p² + p − 6 = 0
  3. 3(p + 3)(p − 2) = 0
  4. 4p > 0 so p = 2
  5. 5q = −18 − 6 = −24

Answerp = 2, q = −24

Part (d) · 2 marks

Using your p and q, factorise f(x) completely.

  1. 1f(x) = x³ + 3x² − 10x − 24
  2. 2(x − 3) and (x + 2) are factors, so f(x) = (x + a)(x − 3)(x + 2)
  3. 3Constant term: −6a = −24 ⇒ a = 4
  4. 4f(x) = (x + 4)(x − 3)(x + 2)

Answerf(x) = (x + 4)(x − 3)(x + 2)

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