Expand and simplify \({\left( {x - \frac{2}{x}} \right)^4}\).

Hence determine the constant term in the expansion \((2{x^2} + 1){\left( {x - \frac{2}{x}} \right)^4}\).

\({\left( {x - \frac{2}{x}} \right)^4} = {x^4} + 4{x^3}\left( { - \frac{2}{x}} \right) + 6{x^2}{\left( { - \frac{2}{x}} \right)^2} + 4x{\left( { - \frac{2}{x}} \right)^3} + {\left( { - \frac{2}{x}} \right)^4}\)     (A2)

 Note: Award (A1) for 3 or 4 correct terms. 

Note: Accept combinatorial expressions, e.g. \(\left( {\begin{array}{*{20}{c}}
  4 \\
  2
\end{array}} \right)\) for 6.

\( = {x^4} - 8{x^2} + 24 - \frac{{32}}{{{x^2}}} + \frac{{16}}{{{x^4}}}\)     A1

[3 marks]

constant term from expansion of \((2{x^2} + 1){\left( {x - \frac{2}{x}} \right)^4} = -64 + 24 = -40\)     A2

Note: Award A1 for –64 or 24 seen.

[2 marks]

It was disappointing to see many candidates expanding \({\left( {x - \frac{2}{x}} \right)^4}\) by first expanding \({\left( {x - \frac{2}{x}} \right)^2}\) and then either squaring the result or multiplying twice by \(\left( {x - \frac{2}{x}} \right)\), processes which often resulted in arithmetic errors being made. Candidates at this level are expected to be sufficiently familiar with Pascal’s Triangle to use it in this kind of problem. In (b), some candidates appeared not to understand the phrase ‘constant term’.

It was disappointing to see many candidates expanding \({\left( {x - \frac{2}{x}} \right)^4}\) by first expanding \({\left( {x - \frac{2}{x}} \right)^2}\) and then either squaring the result or multiplying twice by \(\left( {x - \frac{2}{x}} \right)\), processes which often resulted in arithmetic errors being made. Candidates at this level are expected to be sufficiently familiar with Pascal's Triangle to use it in this kind of problem. In (b), some candidates appeared not to understand the phrase "constant term".