9.
$$f ( x ) = \frac { 50 x ^ { 2 } + 38 x + 9 } { ( 5 x + 2 ) ^ { 2 } ( 1 - 2 x ) } \quad x \neq - \frac { 2 } { 5 } \quad x \neq \frac { 1 } { 2 }$$
Given that \(\mathrm { f } ( x )\) can be expressed in the form
$$\frac { A } { 5 x + 2 } + \frac { B } { ( 5 x + 2 ) ^ { 2 } } + \frac { C } { 1 - 2 x }$$
where \(A\), \(B\) and \(C\) are constants
- find the value of \(B\) and the value of \(C\)
- show that \(A = 0\)
- Use binomial expansions to show that, in ascending powers of \(x\)
$$f ( x ) = p + q x + r x ^ { 2 } + \ldots$$
where \(p , q\) and \(r\) are simplified fractions to be found.
- Find the range of values of \(x\) for which this expansion is valid.