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WMA12/01 Jan 2025, Question 5: 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) = 3x³ + ax² − 10x + b has constants a and b, with (3x − 4) as a factor; prove a relation between a and b, then use a remainder condition to find a and b and factorise fully.

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

Part (a) · 2 marks

Using the factor (3x − 4), derive the equation 16a + 9b = 56.

  1. 1(3x − 4) is a factor, so f(4/3) = 0
  2. 23(4/3)³ + a(4/3)² − 10(4/3) + b = 0
  3. 364/9 + 16a/9 − 40/3 + b = 0
  4. 4Multiply by 9: 64 + 16a − 120 + 9b = 0, so 16a + 9b = 56

AnswerShown: f(4/3) = 0 gives 64 + 16a − 120 + 9b = 0, so 16a + 9b = 56

Part (b) · 4 marks

Given the remainder b when f(x) is divided by (x − 2), determine the values of a and b.

  1. 1f(2) = b: 24 + 4a − 20 + b = b
  2. 24a + 4 = 0, so a = −1
  3. 3Substitute into 16a + 9b = 56: −16 + 9b = 56
  4. 49b = 72, so b = 8

Answera = −1, b = 8

Part (c) · 3 marks

With these values, factorise f(x) completely using algebra.

  1. 1f(x) = 3x³ − x² − 10x + 8
  2. 2Divide by (3x − 4): f(x) = (3x − 4)(x² + x − 2)
  3. 3Factorise the quadratic: x² + x − 2 = (x + 2)(x − 1)
  4. 4f(x) = (3x − 4)(x + 2)(x − 1)

Answerf(x) = (3x − 4)(x + 2)(x − 1)

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