Find the coefficient of \({x^8}\) in the expansion of \({\left( {{x^2} - \frac{2}{x}} \right)^7}\).

each term is of the form \(\left( {\begin{array}{*{20}{c}}  7 \\   r \end{array}} \right){({x^2})^{7 - r}}{\left( {\frac{{ - 2}}{x}} \right)^\prime }\)     (M1)

\( = \left( {\begin{array}{*{20}{c}}  7 \\   r \end{array}} \right){x^{14 - 2r}}{( - 2)^r}{x^{ - r}}\)

so \(14 - 3r = 8\)     (A1)

\(r = 2\)

so require \(\left( {\begin{array}{*{20}{c}}  7 \\   2 \end{array}} \right){({x^2})^{}}^5{\left( {\frac{{ - 2}}{x}} \right)^2}\) (or simply \(\left( {\begin{array}{*{20}{c}}  7 \\   2 \end{array}} \right){( - 2)^2}\))     A1

\( = 21 \times 4\)

\( = 84\)     A1

Note:     Candidates who attempt a full expansion, including the correct term, may only be awarded M1A0A0A0.

[4 marks]