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4MA1/1HR Jan 2019, Question 15: Algebraic Fractions

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 1HR (Higher) · Calculator · 8 marks

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The question, in short

Simplify, expand, factorise and reduce four separate algebraic expressions involving powers, brackets, a difference of squares and a rational expression.

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

Part (a) · 2 marks

Simplify the expression (3x²y⁵)⁴ by raising each factor to the power 4.

  1. 1Raise each part to the power 4: 3⁴ = 81
  2. 2(x²)⁴ = x⁸ and (y⁵)⁴ = y²⁰
  3. 3So the answer is 81x⁸y²⁰

Answer81x⁸y²⁰

Part (b) · 2 marks

Expand and simplify 4n(n − 3)(n + 5) into a single polynomial.

  1. 1Expand the two brackets first: (n − 3)(n + 5) = n² + 5n − 3n − 15 = n² + 2n − 15
  2. 2Multiply by 4n: 4n(n² + 2n − 15)
  3. 34n × n² = 4n³, 4n × 2n = 8n², 4n × (−15) = −60n
  4. 4So the answer is 4n³ + 8n² − 60n

Answer4n³ + 8n² − 60n

Part (c) · 1 mark

Factorise the difference of two squares 4c² − 9d².

  1. 14c² = (2c)² and 9d² = (3d)², so this is a difference of two squares
  2. 2So 4c² − 9d² = (2c − 3d)(2c + 3d)

Answer(2c − 3d)(2c + 3d)

Part (d) · 3 marks

Simplify fully the fraction (x² − 7x + 12)/(4x − x²) by factorising and cancelling.

  1. 1Factorise the top: x² − 7x + 12 = (x − 3)(x − 4)
  2. 2Factorise the bottom: 4x − x² = x(4 − x)
  3. 3x − 4 = −(4 − x), so the fraction is −(x − 3)(4 − x) / (x(4 − x))
  4. 4Cancel (4 − x): −(x − 3)/x = (3 − x)/x

Answer(3 − x)/x

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