AQA A-Level Mathematics: Binomial Expansion and Approximation — mark scheme explained

The question

Expand (3 - 2x) 4 using the binomial theorem. [Paraphrased for study — not reproduced from any exam paper.]

Mark scheme, decoded

What each mark is really for — in plain English — and the wording trap that loses it.

• S1

  (3 - 2x) 4 = Σ r=0 4 (4Cr) 3 4-r (-2x) r

• S2

  = (4C0) 3 4 (-2x) 0 + (4C1) 3 3 (-2x) 1 + (4C2) 3 2 (-2x) 2 + (4C3) 3 1 (-2x) 3 + (4C4) 3 0 (-2x) 4

• S3

  = 1 × 81 × 1 + 4 × 27 × -2x + 6 × 9 × 4x 2 + 4 × 3 × -8x 3 + 1 × 1 × 16x 4

• S4

  = 81 - 216x + 216x 2 - 96x 3 + 16x 4

Model answer

Worked through, with each step tagged to the mark it earns.

1. S1

   (3 - 2x) 4 = Σ r=0 4 (4Cr) 3 4-r (-2x) r

2. S2

   = (4C0) 3 4 (-2x) 0 + (4C1) 3 3 (-2x) 1 + (4C2) 3 2 (-2x) 2 + (4C3) 3 1 (-2x) 3 + (4C4) 3 0 (-2x) 4

3. S3

   = 1 × 81 × 1 + 4 × 27 × -2x + 6 × 9 × 4x 2 + 4 × 3 × -8x 3 + 1 × 1 × 16x 4

4. S4

   = 81 - 216x + 216x 2 - 96x 3 + 16x 4

5. Final answer: 81 - 216x + 216x 2 - 96x 3 + 16x 4

Common mistakes

• Forgetting to include the factorial in the binomial coefficient formula. — Always write out the full formula: nCr = n! / [r!(n-r)!].
• Using the wrong sign when expanding (a - bx) n . — Write out the first few terms to see the pattern: a n , -n × a n-1 × bx, + (n(n-1)/2!) × a n-2 × (bx) 2 , etc.
• Not checking the validity condition for rational n expansions. — Always verify the validity condition before using the binomial expansion for rational n: for (1 + x) n it is |x| n it is |bx/a| < 1.
• Using the wrong value for 0! in calculations. — Remember that 0! = 1 and use this in your calculations.
• Forgetting to include the constant term when expanding (a + bx) n . — Always start with the constant term a n and then proceed to the other terms.
• Not simplifying fractions in the binomial coefficients. — Simplify fractions in the binomial coefficients before substituting them into the expansion.
• Using the wrong number of terms for approximation. — Use enough terms to achieve the desired level of accuracy, but not so many that it becomes overly complex.
• Forgetting to apply the binomial theorem for probability calculations. — Always use the formula P(X = r) = nCr × p r × (1-p) n-r for binomial probability calculations.

Where the marks go

• Full worked solution (all marking points)5 marks

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