Binomial expansion — A/L Combined Mathematics

What must I remember about Binomial expansion?

• Binomial Theorem for positive integer index n: (a + b)ⁿ = ∑_r=0ⁿ nCr * a^n-r * b^r = nC0 * aⁿ + nC1 * a^n-1b + ... + nCn * bⁿ.
• Total number of terms: The expansion of (a + b)ⁿ has exactly (n + 1) terms.
• General Term formula: The (r + 1)-th term is T_r+1 = nCr * a^n-r * b^r where r = 0, 1, 2, ..., n.
• Symmetry of binomial coefficients: nCr = nC(n-r) and nC0 = nCn = 1; Pascal's Triangle identity: (n-1)C(r-1) + (n-1)Cr = nCr.
• Sum of binomial coefficients: Setting a = 1, b = 1 gives nC0 + nC1 + nC2 + ... + nCn = 2ⁿ; alternating sum: nC0 - nC1 + nC2 - ... = 0, meaning sum of odd-position coefficients equals sum of even-position coefficients = 2^n-1.

Test yourself (1 minute)

1. Middle term determination

   If n is even, there is ONE middle term at T_{n/2 + 1}; if n is odd, there are TWO middle terms at T_(n+1)/2 and T_(n+3)/2.

2. Greatest coefficient vs Greatest term

   Greatest coefficient is always the middle coefficient nC_n/2 (for even n); Greatest numerical term depends on values of a and b, found by solving T_r+1 / T_r ≥ 1.

3. Independent of x (constant) term

   Set the overall power of x in T_r+1 to 0 and solve for integer r (0 ≤ r ≤ n).

Practise Binomial expansion questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 22

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Where does Binomial expansion fit in the syllabus?

Year 2 · Term 1