Binomial Theorem

Expand binomials with positive integral indices using the Binomial Theorem

  • Theory
  • Examples
    It is given that \({(1 + 4x)^n} - {(1 - 3x)^n} = 7nx + 196{x^2} + \) terms involving higher powers of x.
    (a) If n is a positive integer, find the value of n.
    (b) Hence find the coefficient of \({x^3}\).
  • Solutions
    (a) \({(1 + 4x)^n} = 1 + C_1^n(4x) + C_2^n{(4x)^2} + \) terms involving higher powers of x
                            \( = 1 + 4nx + 8n(n - 1){x^2} + \) terms involving higher powers of x
         \({(1 - 3x)^n} = 1 + C_1^n( - 3x) + C_2^n{( - 3x)^2} + \) terms involving higher powers of x
                            \( = 1 - 3nx + \frac{{9n(n - 1)}}{2}{x^2} + \) terms involving higher powers of x
    ∴ \({(1 + 4x)^n} - {(1 - 3x)^n} = [4n - ( - 3n)]x + [8n(n - 1) - \frac{{9n(n - 1)}}{2}]{x^2}\) + terms involving higher powers of x                                              \( = 7nx + \frac{{7n(n - 1)}}{2}{x^2} + \) terms involving higher powers of x
    By comparing the coefficient of \({x^2}\), we have
    \(\begin{array}{1}\frac{{7n(n - 1)}}{2} = 196\\n(n - 1) = 56\\{n^2} - n - 56 = 0\\(n - 8)(n + 7) = 0\end{array}\)
    \(n = \underline{\underline 8} \) or \(n = - 7\) (rejected)