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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 Algebraic Methods
Edexcel A level Maths: 1.5 Algebraic Division (Partial Fractions)Curated by Kepler, not produced by us. Watch it 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
1

Proper vs improper

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

deg⁡(numerator)≥deg⁡(denominator)⇒improper\deg(\text{numerator}) \ge \deg(\text{denominator}) \Rightarrow \text{improper}
Divide before decomposing.
2

Dividing out

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

F(x)D(x)=Q(x)+R(x)D(x)\dfrac{F(x)}{D(x)} = Q(x) + \dfrac{R(x)}{D(x)}
Quotient QQ plus a proper fraction with remainder RR.
Example 1Write x2+1x−1\dfrac{x^2 + 1}{x - 1} as polynomial + proper fraction.
1Divide: x2+1=(x−1)(x+1)+2x^2 + 1 = (x - 1)(x + 1) + 2.
2So x2+1x−1=x+1+2x−1\dfrac{x^2 + 1}{x - 1} = x + 1 + \dfrac{2}{x - 1}.
Answer
x+1+2x−1x + 1 + \dfrac{2}{x - 1}

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

3

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

F(x)D(x)=Q(x)+R(x)D(x)\dfrac{F(x)}{D(x)} = Q(x) + \dfrac{R(x)}{D(x)}
Improper ⟶ polynomial + proper.
deg⁡(num)≥deg⁡(den)⇒divide first\deg(\text{num}) \ge \deg(\text{den}) \Rightarrow \text{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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. A fraction is improper when the numerator degree is:

    • ≥ the denominator degree
    • < the denominator degree
    • Always 1
    • Zero
    Show answer and explanation

    Answer: ≥ the denominator degree

    Then you must divide first.

  2. Before partial fractions, an improper fraction must be:

    • Divided out
    • Squared
    • Differentiated
    • Cancelled
    Show answer and explanation

    Answer: Divided out

    To make it proper.

  3. x2+1x−1\dfrac{x^{2} + 1}{x - 1} equals:

    • x+1+2x−1x + 1 + \dfrac{2}{x-1}
    • x−1+2x−1x - 1 + \dfrac{2}{x-1}
    • x+1x + 1
    • x+1x−1x + \dfrac{1}{x-1}
    Show answer and explanation

    Answer: x+1+2x−1x + 1 + \dfrac{2}{x-1}

    x² + 1 = (x−1)(x+1) + 2.

  4. An improper fraction can be written as:

    • Polynomial + proper fraction
    • Two proper fractions
    • A single linear factor
    • A product
    Show answer and explanation

    Answer: Polynomial + proper fraction

    Quotient plus remainder/divisor.

  5. Is x2x+3\dfrac{x^{2}}{x + 3} proper or improper?

    • Improper
    • Proper
    • Neither
    • Both
    Show answer and explanation

    Answer: Improper

    Degree 2 over degree 1.

  6. The final answer for an improper fraction includes:

    • The polynomial part too
    • Only the fractions
    • Only the quotient
    • No constants
    Show answer and explanation

    Answer: The polynomial part too

    Quotient + partial fractions.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.