Using Partial Fractions

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1. Before binomially expanding a rational function, you should:

   • split into partial fractions
   • differentiate it
   • square it
   • integrate it
   Show answer and explanation

   Answer: split into partial fractions

   Each part is then a simple binomial.

2. $\tfrac{A}{1-x}$ is rewritten for expansion as:

   • $A(1-x)^{-1}$
   • $A(1-x)$
   • $A(1+x)^{-1}$
   • $A(1-x)^{1}$
   Show answer and explanation

   Answer: $A(1-x)^{-1}$

   Negative power.

3. If one part is valid for $|x|<1$ and another for $|x|<\tfrac12$, the combined series is valid for:

   • $|x| < \tfrac12$
   • $|x| < 1$
   • $|x| < \tfrac32$
   • all $x$
   Show answer and explanation

   Answer: $|x| < \tfrac12$

   Strictest range.

4. The expansion of $(1-2x)^{-1}$ begins:

   • $1 + 2x + \cdots$
   • $1 - 2x + \cdots$
   • $1 + x + \cdots$
   • $1 - x + \cdots$
   Show answer and explanation

   Answer: $1 + 2x + \cdots$

   nx with n=−1, X=−2x gives +2x.

5. Partial fractions are useful here because each piece:

   • is a simple binomial to expand
   • cancels to zero
   • has no x
   • is already a series
   Show answer and explanation

   Answer: is a simple binomial to expand

   Single bracket to a negative power.