Use the Binomial Theorem to expand \(( x+2) ^{5}\).

Solution We use the Binomial Theorem with \(a=2\) and \(n=5\). Then \begin{eqnarray*} ( x+2) ^{5} &=& {{5}\choose{0}} x^{5}+ {{5}\choose{1}} 2x^{4}+ {{5}\choose{2}} 2^{2}x^{3}+ {{5}\choose{3}} 2^{3}x^{2}+ \displaystyle{{5}\choose{4}} 2^{4}x+ {{5}\choose{5}} 2^{5} \nonumber \\[4pt] &=&1\cdot x^{5}+5\cdot 2 x^{4}+10\cdot 4 x^{3}+10\cdot 8 x^{2}+5\cdot 16 x+1\cdot 32 \nonumber \\[4pt] &=&x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x+32 \end{eqnarray*}

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