1.5PureStretch

Algebraic Division

Partial fractions only work on PROPER fractions. When the numerator’s degree is as big as the denominator’s, you must divide first to split off a polynomial part, leaving a proper fraction to decompose.

25 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel A level Maths: 1.5 Algebraic Division (Partial Fractions)Watch the full walkthrough before the notes below.

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What you'll be able to do

Recognise an improper algebraic fraction
Use polynomial division to make it proper
Write an improper fraction as polynomial + proper fraction
Combine division with partial fractions

Proper vs improper

A fraction is $improper$ if the degree of the numerator is greater than or equal to the degree of the denominator. Partial fractions need a $proper$ fraction, so an improper one must be divided out first.

de g (numerator) \geq de g (denominator) \Rightarrow improper

Divide before decomposing.

Dividing out

Use polynomial long division (or compare coefficients) to write the improper fraction as a $polynomial part plus a proper remainder fraction$ .

\frac{F ( x )}{D ( x )} = Q (x) + \frac{R ( x )}{D ( x )}

Quotient

Q

plus a proper fraction with remainder

R

1Divide:

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

2So

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

Answer

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

Tip — Quick degree check first: if the top is the same degree or higher than the bottom, divide before anything else.

Then partial fractions

Once the leftover fraction is proper, split it with partial fractions as usual. The full answer is the polynomial part plus the partial fractions — exactly the form needed for integration in Year 2.

Formula recap

\frac{F ( x )}{D ( x )} = Q (x) + \frac{R ( x )}{D ( x )}

Improper ⟶ polynomial + proper.

de g (num) \geq de g (den) \Rightarrow divide first

When to divide.

Common mistakes to avoid

Applying partial fractions straight to an improper fraction.

Divide out the polynomial part first to leave a proper fraction.

Forgetting the polynomial part in the final answer.

The answer is quotient + partial fractions, not just the fractions.

Key takeaways

Improper: numerator degree ≥ denominator degree.
Divide to write it as polynomial + proper fraction.
Then decompose the proper fraction with partial fractions.

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