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WMA12/01 Jan 2020, Question 3: Dividing Polynomials

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

The question, in short

For the cubic f(x) = 6x³ + 17x² + 4x − 12, verify a given linear factor, factorise fully, then solve a related equation in tan θ for π/2 < θ < π to 3 significant figures.

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

Part (a) · 2 marks

Show, using the factor theorem, that 2x + 3 divides f(x) exactly.

  1. 1f(−3/2) = 6(−27/8) + 17(9/4) + 4(−3/2) − 12
  2. 2= −20.25 + 38.25 − 6 − 12 = 0
  3. 3Since f(−3/2) = 0, (2x + 3) is a factor of f(x)

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

Part (b) · 4 marks

Use algebra to express f(x) as three linear factors multiplied together.

  1. 1Divide f(x) by (2x + 3): quotient 3x² + 4x − 4
  2. 2f(x) = (2x + 3)(3x² + 4x − 4)
  3. 33x² + 4x − 4 = (3x − 2)(x + 2)
  4. 4f(x) = (2x + 3)(3x − 2)(x + 2)

Answerf(x) = (2x + 3)(3x − 2)(x + 2)

Part (c) · 2 marks

Find θ between π/2 and π satisfying 6 tan³θ + 17 tan²θ + 4 tan θ − 12 = 0, to 3 significant figures.

  1. 1With x = tan θ: (2 tan θ + 3)(3 tan θ − 2)(tan θ + 2) = 0
  2. 2tan θ = −3/2, 2/3 or −2
  3. 3In π/2 < θ < π, tan θ is negative, so use tan θ = −3/2 and tan θ = −2
  4. 4θ = π − arctan(3/2) = 2.16
  5. 5θ = π − arctan 2 = 2.03

Answerθ = 2.03, 2.16

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Also searched as: WMA12 January 2020 · WMA12/01 · wma12 jan20 · IAS Maths Pure Mathematics 2 January 2020 · question 3.
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