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WMA12/01 Oct 2021, Question 4: Cubic Roots after Division

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

The question, in short

For f(x) = (x² − 2)(2x − 3) − 21, find a remainder, verify a linear factor, then factorise and count real roots of f(x) = 0.

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

Part (a) · 1 mark

Give the remainder left when f(x) is divided by 2x − 3.

  1. 1f(3/2) = (9/4 − 2)(0) − 21
  2. 2Remainder = −21

Answer−21

Part (b) · 2 marks

Using the factor theorem, demonstrate that x − 3 divides f(x).

  1. 1f(3) = (3² − 2)(2×3 − 3) − 21
  2. 2= 7 × 3 − 21 = 0
  3. 3f(3) = 0, so (x − 3) is a factor of f(x)

AnswerShown: f(3) = (9 − 2)(3) − 21 = 0, so (x − 3) is a factor

Part (ci) · 3 marks

Write f(x) as a product of its factors.

  1. 1f(x) = 2x³ − 4x − 3x² + 6 − 21 = 2x³ − 3x² − 4x − 15
  2. 2Divide by (x − 3): 2x³ − 3x² − 4x − 15 = (x − 3)(2x² + 3x + 5)

Answerf(x) = (x − 3)(2x² + 3x + 5)

Part (cii) · 2 marks

Prove that f(x) = 0 possesses exactly one real solution.

  1. 1For 2x² + 3x + 5: b² − 4ac = 3² − 4(2)(5) = −31
  2. 2−31 < 0, so the quadratic has no real roots
  3. 3So f(x) = 0 has only one real root, x = 3

AnswerShown: discriminant of 2x² + 3x + 5 is 9 − 40 = −31 < 0, so x = 3 is the only real root

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Also searched as: WMA12 October 2021 · WMA12/01 · wma12 oct21 · IAS Maths Pure Mathematics 2 October 2021 · question 4.
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