Factor and rescale

Use a standard expansion like (1-y)^n and factor constants from the denominator or argument, then rescale to find the expansion of expressions like 1/(3-2x) from 1/(1-x).

3 questions · Standard +0.6

1.02y Partial fractions: decompose rational functions 1.04c Extend binomial expansion: rational n, |x|<1 1.04d Binomial expansion validity: convergence conditions

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Edexcel AEA 2017 Specimen Q1

8 marks Challenging +1.2

1.(a)For $| y | < 1$ ,write down the binomial series expansion of $( 1 - y ) ^ { - 2 }$ in ascending powers of $y$ up to and including the term in $y ^ { 3 }$ (b)Show that when it is convergent,the series $$1 + \frac { 2 x } { x + 3 } + \frac { 3 x ^ { 2 } } { ( x + 3 ) ^ { 2 } } + \ldots + \frac { r x ^ { r - 1 } } { ( x + 3 ) ^ { r - 1 } } + \ldots$$ can be written in the form $( 1 + a x ) ^ { n }$ ,where $a$ and $n$ are constants to be found.
(c)Find the set of values of $x$ for which the series in part(b)is convergent.

AQA C4 Q5

10 marks Standard +0.3

1. Obtain the binomial expansion of $( 1 - x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of $x$.
Obtain the binomial expansion of $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
Given that $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ can be written in the form $\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$, find the values of $A , B$ and $C$.
Hence find the binomial expansion of $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.

AQA C4 2006 January Q5

15 marks Standard +0.3

1. Obtain the binomial expansion of $( 1 - x ) ^ { - 1 }$ up to and including the term in $x ^ { 2 }$.
  (2 marks)
2. Hence, or otherwise, show that $$\frac { 1 } { 3 - 2 x } \approx \frac { 1 } { 3 } + \frac { 2 } { 9 } x + \frac { 4 } { 27 } x ^ { 2 }$$ for small values of $x$.
Obtain the binomial expansion of $\frac { 1 } { ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.
Given that $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ can be written in the form $\frac { A } { ( 3 - 2 x ) } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 1 - x ) ^ { 2 } }$, find the values of $A , B$ and $C$.
Hence find the binomial expansion of $\frac { 2 x ^ { 2 } - 3 } { ( 3 - 2 x ) ( 1 - x ) ^ { 2 } }$ up to and including the term in $x ^ { 2 }$.