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WMA12/01 Jun 2025, Question 4: The Factor and Remainder Theorems

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

The question, in short

A cubic f(x) = ax³ + bx² + 5x − 3 has (x + 3) as a factor and leaves remainder 7/4 on division by (2x − 1); use these to relate a and b, then divide by (x − 2).

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 linear relation connecting the constants a and b.

  1. 1(x + 3) is a factor, so f(−3) = 0
  2. 2a(−27) + b(9) + 5(−3) − 3 = 0
  3. 3−27a + 9b − 18 = 0, so −27a + 9b = 18
  4. 4Divide by 9: −3a + b = 2, so b = 3a + 2

AnswerShown: f(−3) = 0 gives −27a + 9b = 18, so b = 3a + 2

Part (b) · 4 marks

Apply the remainder condition for division by (2x − 1) together with part (a) to determine a and b.

  1. 1Remainder theorem: f(½) = 7/4
  2. 2a/8 + b/4 + 5/2 − 3 = 7/4
  3. 3Multiply by 8: a + 2b + 20 − 24 = 14, so a + 2b = 18
  4. 4Substitute b = 3a + 2: a + 6a + 4 = 18, so a = 2
  5. 5b = 3(2) + 2 = 8

Answera = 2, b = 8

Part (c) · 3 marks

By algebraic division of the cubic by (x − 2), state the quotient and the remainder.

  1. 1f(x) = 2x³ + 8x² + 5x − 3
  2. 22x³ ÷ x = 2x²; subtract 2x²(x − 2) = 2x³ − 4x² to leave 12x² + 5x − 3
  3. 312x² ÷ x = 12x; subtract 12x(x − 2) = 12x² − 24x to leave 29x − 3
  4. 429x ÷ x = 29; subtract 29(x − 2) = 29x − 58 to leave 55
  5. 5Quotient 2x² + 12x + 29, remainder 55

AnswerQuotient 2x² + 12x + 29, remainder 55

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Also searched as: WMA12 June 2025 · WMA12/01 · wma12 jun25 · IAS Maths Pure Mathematics 2 June 2025 · question 4.
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