4.3PureStretch

Using Partial Fractions

A rational function with a factorised denominator can be split into partial fractions, each of which is easy to expand binomially. Combining the expansions gives the series for the whole expression.

30 min Video by Zeeshan Zamurred Binomial Expansion

Edexcel A level Maths: (Part 1) 4.3 Using Partial Fractions in Binomial ExpansionWatch the full walkthrough before the notes below.

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What you'll be able to do

Split a rational expression into partial fractions
Rewrite each fraction with a negative power
Expand each part binomially
Combine and state the overall range of validity

The strategy

First decompose into partial fractions. Then write each denominator as a bracket raised to a negative power, e.g. $\frac{A}{1 - x} = A (1 - x)^{- 1}$ , and expand each binomially. Add the series term by term.

\frac{f ( x )}{( 1 + a x ) ( 1 + b x )} = \frac{A}{1 + a x} + \frac{B}{1 + b x}

Decompose, then expand each part.

Overall validity

Each expansion has its own range of validity. The combined series is valid only where $all$ of them hold — take the most restrictive (smallest) range.

Tip — Overall validity = the strictest (smallest) of the individual ranges.

Worked example

Expand $\frac{3 x}{( 1 + x ) ( 1 - 2 x )}$ .

1Partial fractions:

\frac{- 1}{1 + x} + \frac{1}{1 - 2 x}

- (1 + x)^{- 1} = - (1 - x + \dots)

;

(1 - 2 x)^{- 1} = 1 + 2 x + \dots

3Sum:

(- 1 + 1) + (1 + 2) x + \dots = 3 x + \dots

Answer

3 x + \dots

, valid for

∣ x ∣ < \frac{1}{2}

Formula recap

\frac{A}{1 + a x} = A (1 + a x)^{- 1}

Negative-power form for expansion.

validity = min of each part

Strictest range wins.

Common mistakes to avoid

Taking the widest range of validity.

The combined series needs every part to converge — use the smallest range.

Expanding before decomposing.

Split into partial fractions first; each part is then a simple binomial.

Key takeaways

Decompose into partial fractions first.
Write each as a negative power and expand binomially.
Combine series; validity is the strictest individual range.

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