Standard +0.3 This is a straightforward application of the binomial expansion for negative integer powers followed by algebraic manipulation. Part (a) is routine recall of the formula with simple arithmetic for coefficients. Part (b) requires recognizing that the expression can be rewritten as (2-x)²(1-3x)^(-2) and multiplying two polynomial expansions, which is mechanical but requires care with arithmetic. The 'show that' format makes it easier as students know the target answer.
2. (a) Expand \(( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(b) Hence, or otherwise, show that for small \(x\),
$$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$
2. (a) Expand $( 1 - 3 x ) ^ { - 2 } , | x | < \frac { 1 } { 3 }$, in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, simplifying each coefficient.\\
(b) Hence, or otherwise, show that for small $x$,
$$\left( \frac { 2 - x } { 1 - 3 x } \right) ^ { 2 } \approx 4 + 20 x + 85 x ^ { 2 } + 330 x ^ { 3 }$$
\hfill \mbox{\textit{Edexcel C4 Q2 [7]}}