Binomial Expansion: Concepts & Theory

If n is a positive integer, (a + x)n can be expanded as
  • (a + x)n = an + nC1 an-1 x + nC2 an-2x2 + nC3 an-3x3 + ... + xn.
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    Applying binomial on (a + b)2
    Using general expansion equation:
    a2-0 + 2c1(a)2-1(b)1 + b2-0 = a2+2ab+ b2
    Applying Binomial on (a + b)3
    a3-0 + 3c1a3-1b1 + 3c2 a3-2b2 + b3-0 = a3 + 3a2b + 3ab2 + b3
    And if in this case if particular term say ‘r’ is asked, then instead of applying the whole expansion, the following direct formula can be applied to find the ‘r’th term
  • ‘r’th term = xn–r+1ar – 1[{n(n – 1)(n – 2) ... (n – r + 2)} ÷ (r – 1)!]. Similarly (x + a)n can be expanded as
  • (x + a)n = xn + nC1 xn–1a + nC1 xn–2 a2 + nC1 xn–3 a3 + ... + an In case of a binomial expansion, if any particular term say ‘r’ is asked, then instead of applying the whole expansion, the following direct formula can be applied to find the ‘r’th term.
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  • ‘r’th term = an–r+1 xr–1 [{n(n–1) (n – 2) ... (n – r + 2)} ÷ (r – 1)!] So order of the term to be expanded is very important i.e., whether it is in the form (a + x)n or (x + a)n
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