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WMA12/01 Jan 2022, Question 5: The Factor and Remainder Theorems

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 8 marks

The question, in short

Cubic f(x) = 3x³ + Ax² + Bx − 10 with integer A, B; use two remainder conditions involving k and a factor to find A, B, then the quadratic factor.

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

Part (a) · 3 marks

Use the remainders when f(x) is divided by (x − 1) and (x + 1), with k, to show 11A + 9B = 83.

  1. 1f(1) = 3 + A + B − 10 = k, so k = A + B − 7
  2. 2f(−1) = −3 + A − B − 10 = −10k, so A − B − 13 = −10k
  3. 3A − B − 13 = −10(A + B − 7) = −10A − 10B + 70
  4. 411A + 9B = 83

AnswerShown: A + B − 7 = k and A − B − 13 = −10k give 11A + 9B = 83

Part (b) · 3 marks

Given (3x − 2) is a factor of f(x), find the integer values of A and B.

  1. 1f(2/3) = 0: 3(8/27) + A(4/9) + B(2/3) − 10 = 0
  2. 2Multiply by 9: 8 + 4A + 6B − 90 = 0, so 2A + 3B = 41 (or 12A + 18B = 246)
  3. 3Solve with 11A + 9B = 83: 6A + 9B = 123, so 5A = −40
  4. 4A = −8, B = 19

AnswerA = −8, B = 19

Part (c) · 2 marks

Using your A and B, find the quadratic g(x) with f(x) = (3x − 2)g(x).

  1. 1f(x) = 3x³ − 8x² + 19x − 10
  2. 2Divide by (3x − 2): f(x) = (3x − 2)(x² − 2x + 5)
  3. 3g(x) = x² − 2x + 5

Answerg(x) = x² − 2x + 5

Stuck on a step? Read the free lesson: The Factor and Remainder Theorems

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Also searched as: WMA12 January 2022 · WMA12/01 · wma12 jan22 · IAS Maths Pure Mathematics 2 January 2022 · question 5.
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