Find the term in ${x^4}$ in the expansion of ${\left( {3{x^2} - \frac{2}{x}} \right)^5}$ .

evidence of substituting into binomial expansion     (M1)

e.g. ${a^5} + \left( {\begin{array}{*{20}{c}} 5\\ 1 \end{array}} \right){a^4}b + \left( {\begin{array}{*{20}{c}} 5\\ 2 \end{array}} \right){a^3}{b^2} +  \ldots$

identifying correct term for ${x^4}$    (M1)

evidence of calculating the factors, in any order     A1A1A1

e.g. $\left( {\begin{array}{*{20}{c}} 5\\ 2 \end{array}} \right),27{x^6},\frac{4}{{{x^2}}}$ ; $10{(3{x^2})^3}{\left( {\frac{{ - 2}}{x}} \right)^2}$

Note: Award A1 for each correct factor.

${\rm{term}} = 1080{x^4}$     A1     N2

Note: Award M1M1A1A1A1A0 for 1080 with working shown.

[6 marks]

Although a great number of students recognized they could use the binomial theorem, fewer were successful in finding the term in ${x^4}$ .

Candidates showed various difficulties when trying to solve this problem:

• choosing the incorrect term
• attempting to expand ${\left( {3{x^2} - \frac{2}{x}} \right)^5}$ by hand
• finding only the coefficient of the term
• not being able to determine which term would yield an ${x^4}$
• errors in the calculations of the coefficient.