Binomial theorem — A/L Combined Mathematics

How do I practise Binomial theorem?

2 questions with worked solutions

Check before you answer: Power Extraction Equating: Collecting all powers of x into a single exponent expression and setting it to 0 for independent terms or k for specified powers.

1. What is the coefficient of x³ in the binomial expansion of (1 + 2x)⁵?
   1. A) 160
   2. B) 80
   3. C) 10
   4. D) 40
   Check my answer

   Correct answer: B) 80

   The general term is T_r+1 = ⁵C_r (1)^5-r (2x)^r = ⁵C_r · 2^r · x^r. For the term in x³, set r = 3: T₄ = ⁵C₃ · 2³ · x³ = [5! / (3! 2!)] · 8 · x³ = 10 · 8 · x³ = 80 x³. Thus, the coefficient is 80.

   Show worked solution

   Step 1: General term: T_r+1 = ⁵C_r · 2^r · x^r. Step 2: For x³, r = 3 => Coefficient = ⁵C₃ × 2³ = 10 × 8 = 80. Answer: 80.
   Modelled on: the A/L Combined Mathematics examination pattern
2. Find the term independent of x in the binomial expansion of (x + 1/x)⁶.
   1. A) 15
   2. B) 1
   3. C) 20
   4. D) 6
   Check my answer

   Correct answer: C) 20

   The general term is T_r+1 = ⁶C_r · x^6-r · (1/x)^r = ⁶C_r · x^6-2r. The power of x is zero when 6 - 2r = 0, that is r = 3. So the term independent of x is T₄ = ⁶C₃ = 20.

   Show worked solution

   Step 1: General term: T_r+1 = ⁶C_r · x^6-r · (1/x)^r = ⁶C_r · x^6-2r. Step 2: The term is independent of x when 6 - 2r = 0, so r = 3. Step 3: T₄ = ⁶C₃ = 6! / (3! · 3!) = 20. Answer: 20.
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does Binomial theorem fit in the syllabus?

Year 2 · Term 1