Moderate -0.5 This is a straightforward application of the binomial expansion for negative/fractional powers. Part (a) requires routine substitution into the formula with basic algebraic manipulation to simplify coefficients. Part (b) tests standard knowledge that the expansion is valid when |2x/3| < 1. While it requires careful arithmetic, it involves no problem-solving or novel insight—slightly easier than average due to its mechanical nature.
6.
$$f ( x ) = ( 3 - 2 x ) ^ { - 4 }$$
a)
Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each coefficient as a simplified fraction.
(3)
b) For what values of \(x\) is the expansion valid?
6.
$$f ( x ) = ( 3 - 2 x ) ^ { - 4 }$$
a)
Find the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, giving each coefficient as a simplified fraction.\\
(3)\\
b) For what values of $x$ is the expansion valid?\\
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q6 [4]}}