Tag: Binomial theorem

  • Master Binomial Theorem: Essential Strategies to Excel in A Math

    Master Binomial Theorem: Essential Strategies to Excel in A Math

    Question

    In the binomial expansion of \(\left(x+\frac{k}{x}\right)^7\), where \(k\) is a positive constant, the coefficients of \( x^3 \) and \(x\)are the same.

    (i) Find the value of \(k\).

    (ii) Using the value of \(k\) found in part (i), find the coefficient of \(x^7\) in the expansion of \(\left(1-5 x^2\right)\left(x+\frac{k}{x}\right)^7\).

    Recognise that \(x\) is present in both terms within the brackets, this is a clue to use the general term $${T}_{r+1}={n \choose r} a^{n-r} b^r$$

    $$\begin{aligned} & T_{r+1} \text { in }\left(x+\frac{k}{x}\right)^7 \\ & =\binom{7}{r}(x)^{7-r}\left(\frac{k}{x}\right)^r \\ & =\binom{7}{r}(x)^{7-r}(k)^r(x)^{-r} \\ & =\binom{7}{r} k^r(x)^{7-2 r}\end{aligned}$$

    To be able to simplify the \({T}_{r+1}\) term correctly requires good grasp of indices

    For term in \(x^3, 7-2 r=3\)
    \(
    r=2
    \)

    Term in \(x^3=\binom{7}{2} k^2(x)^3\)
    \(
    =21 k^2 x^3
    \)

    For term in \(x, 7-2 r=1\)
    \(
    r=3
    \)

    Term in \(x=\binom{7}{3} k^3(x)^1\) \(=35 k^3 x\)

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