Partial fractions with validity range

10 Let \(\mathrm { f } ( x ) = \frac { 24 x + 13 } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
3. State the set of values of \(x\) for which the expansion in (b) is valid.

9 Let \(\mathrm { f } ( x ) = \frac { 17 x ^ { 2 } - 7 x + 16 } { \left( 2 + 3 x ^ { 2 } \right) ( 2 - x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
3. State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.

8 Let \(\mathrm { f } ( x ) = \frac { 7 a ^ { 2 } } { ( a - 2 x ) ( 3 a + x ) }\), where \(a\) is a positive constant.

1. Express \(\mathrm { f } ( x )\) in partial fractions. \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-12_2718_40_107_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-13_2726_33_97_22}
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). [4]
3. State the set of values of \(x\) for which the expansion in part (b) is valid.

1. $$f ( x ) = \frac { 5 x + 10 } { ( 1 - x ) ( 2 + 3 x ) }$$

1. Write \(\mathrm { f } ( x )\) in partial fraction form.
2. 1. Hence find, in ascending powers of \(x\) up to and including the terms in \(x ^ { 2 }\), the binomial series expansion of \(\mathrm { f } ( x )\). Give each coefficient as a simplified fraction.
   2. Find the range of values of \(x\) for which this expansion is valid.

3

1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
   constants to be determined. constants to be determined. State the set of values of \(x\) for which the expansion is valid.

6

1. Express \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) }\) in partial fractions.
2. Hence use binomial expansions to show that \(\frac { x } { ( 1 + x ) ( 1 - 2 x ) } = a x + b x ^ { 2 } + \ldots\), where \(a\) and \(b\) are
   constants to be determined. State the set of values of \(x\) for which the expansion is valid.

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

1. 1. find the value of \(B\) and the value of \(C\)
   2. show that \(A = 0\)
2. 1. 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.
   2. Find the range of values of \(x\) for which this expansion is valid.

3

1. Express \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) in the form \(\frac { A } { 1 + x } + \frac { B } { 3 + 5 x }\), where \(A\) and \(B\) are integers.
2. Hence, or otherwise, find the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 + 9 x } { ( 1 + x ) ( 3 + 5 x ) }\) is valid.
   (2 marks)

2

1. 1. Find the binomial expansion of \(( 1 + x ) ^ { - 1 }\) up to the term in \(x ^ { 3 }\).
   2. Hence, or otherwise, obtain the binomial expansion of \(\frac { 1 } { 1 + 3 x }\) up to the term in \(x ^ { 3 }\).
2. Express \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) in partial fractions.
3. 1. Find the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) up to the term in \(x ^ { 3 }\).
   2. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 1 + 4 x } { ( 1 + x ) ( 1 + 3 x ) }\) is valid.

3

1. Find the binomial expansion of \(( 1 - x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
2. 1. Express \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) in the form \(\frac { A } { 1 - x } + \frac { B } { 2 - 3 x }\), where \(A\) and \(B\) are integers.
   2. Find the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) up to and including the term in \(x ^ { 2 }\).
3. Find the range of values of \(x\) for which the binomial expansion of \(\frac { 3 x - 1 } { ( 1 - x ) ( 2 - 3 x ) }\) is valid.

$$f(x) = \frac{5 - 8x}{(1 + 2x)(1 - x)^2}.$$

1. Express \(f(x)\) in partial fractions. [5]
2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying each coefficient. [6]
3. State the set of values of \(x\) for which your expansion is valid. [1]