Partial fractions showing coefficient is zero

A question is this type if and only if it asks to find partial fraction constants and specifically prove/show that one of the constants (typically A or B) equals zero.

5 questions · Standard +0.3

1.02y Partial fractions: decompose rational functions 1.04c Extend binomial expansion: rational n, |x|<1

Sort by: Default | Easiest first | Hardest first

Edexcel C34 2015 June Q2

9 marks Standard +0.3

2. Given that $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { ( 1 - 2 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } }$$

find the values of the constants $A$ and $C$ and show that $B = 0$ (4)
Hence, or otherwise, find the series expansion of $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 2 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 2 }$, simplifying each term.
(5)

Edexcel C4 2006 January Q5

11 marks Standard +0.3

5. $$f ( x ) = \frac { 3 x ^ { 2 } + 16 } { ( 1 - 3 x ) ( 2 + x ) ^ { 2 } } = \frac { A } { ( 1 - 3 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 3 } .$$

Find the values of $A$ and $C$ and show that $B = 0$.
Hence, or otherwise, find the series expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Simplify each term.

OCR MEI C4 Q6

8 marks Standard +0.3

Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x }$$ where $A , B$ and $C$ are constants, find $B$ and $C$, and show that $A = 0$.
Given that $x$ is sufficiently small, find the first three terms of the binomial expansions of $( 1 + x ) ^ { - 2 }$ and $( 1 - 4 x ) ^ { - 1 }$. Hence find the first three terms of the expansion of $\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }$.

OCR MEI C4 2006 June Q2

11 marks Standard +0.3

Given that $$\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) } = \frac { A } { 1 + x } + \frac { B } { ( 1 + x ) ^ { 2 } } + \frac { C } { 1 - 4 x } ,$$ where $A , B$ and $C$ are constants, find $B$ and $C$, and show that $A = 0$.
Given that $x$ is sufficiently small, find the first three terms of the binomial expansions of $( 1 + x ) ^ { - 2 }$ and $( 1 - 4 x ) ^ { - 1 }$. Hence find the first three terms of the expansion of $\frac { 3 + 2 x ^ { 2 } } { ( 1 + x ) ^ { 2 } ( 1 - 4 x ) }$.

SPS SPS FM Pure 2025 January Q2

8 marks Standard +0.3

Given that $$\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)} = \frac{A}{1 + x} + \frac{B}{(1 + x)^2} + \frac{C}{1 - 4x},$$ where $A$, $B$ and $C$ are constants, find $B$ and $C$, and show that $A = 0$. [4]
Given that $x$ is sufficiently small, find the first three terms of the binomial expansions of $(1 + x)^{-2}$ and $(1 - 4x)^{-1}$. Hence find the first three terms of the expansion of $\frac{3 + 2x^2}{(1 + x)^2(1 - 4x)}$. [4]