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WMA12/01 Oct 2024, Question 3: The Factor and Remainder Theorems

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 2 (WMA12) · Calculator · 7 marks

The question, in short

Use a cubic with unknown integer coefficients, a remainder condition and a linear factor to find the unknowns, then divide.

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

Part (a) · 2 marks

Apply the remainder theorem with divisor x + 3 and remainder 55 to derive the stated equation linking A and B.

  1. 1f(−3) = 2(−3)³ − (−3)² + A(−3) + B = 55
  2. 2−54 − 9 − 3A + B = 55
  3. 3−3A + B = 118, so 3A − B = −118

AnswerShown: f(−3) = −54 − 9 − 3A + B = 55, so 3A − B = −118

Part (b) · 3 marks

Use the factor theorem with 2x − 5 as a factor, solve simultaneously with the earlier equation for A and B.

  1. 1f(5/2) = 0: 2(5/2)³ − (5/2)² + (5/2)A + B = 0
  2. 2125/4 − 25/4 + 5A/2 + B = 0, so 5A + 2B = −50
  3. 3Solve with 3A − B = −118: 2 × (3A − B) gives 6A − 2B = −236
  4. 4Adding: 11A = −286, A = −26; then B = 3(−26) + 118 = 40

AnswerA = −26, B = 40

Part (c) · 2 marks

Divide the resulting cubic by x − 7 and state the quotient.

  1. 1f(x) = 2x³ − x² − 26x + 40
  2. 2Divide by (x − 7): 2x² first, 2x³ − 14x² leaves 13x² − 26x
  3. 313x next, 13x² − 91x leaves 65x + 40
  4. 465 next, 65x − 455 leaves remainder 495
  5. 5Quotient = 2x² + 13x + 65

Answer2x² + 13x + 65

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

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Also searched as: WMA12 October 2024 · WMA12/01 · wma12 oct24 · IAS Maths Pure Mathematics 2 October 2024 · question 3.
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