# Algebra and Functions: Partial Fractions (P4)

> Edexcel International A-Level Mathematics · IAL Maths P4
> Source: https://www.owlsprep.com/study/edexcel-ial-math-p4-algebra-and-functions/

This guide teaches partial fraction decomposition for Edexcel IAL P4, including handling improper fractions via polynomial division, distinct linear factors, and repeated linear factors, plus common exam applications.

**Prerequisites:** [Polynomial long division (Edexcel IAL P3)](https://www.owlsprep.com/study/edexcel-ial-math-p3-polynomial-division/); [Simplifying rational expressions (Edexcel IAL P2)](https://www.owlsprep.com/study/edexcel-ial-math-p2-algebraic-fractions/)

## Learning objectives

- Perform polynomial division to rewrite improper rational functions as a polynomial plus a proper rational function
- Decompose proper rational functions with distinct linear denominators into partial fractions
- Decompose proper rational functions with repeated linear denominators into partial fractions
- Apply partial fractions to support integration, differentiation and binomial expansion tasks

## Step 1: Identify and Rewrite Improper Rational Functions

**Improper Rational Function** — A rational function where \(\deg(\text{numerator}) \geq \deg(\text{denominator})\). Must be rewritten via polynomial division as a polynomial plus a proper rational function before partial fraction decomposition.

*Example:* \(\frac{2x^3 + 5x^2 - x + 1}{x^2 + 2x - 3}\) is improper as \(\deg(\text{numerator})=3 > \deg(\text{denominator})=2\)

**Worked example:** Rewrite \(\frac{2x^3 + 5x^2 - x + 1}{x^2 + 2x - 3}\) as a polynomial plus a proper rational function.

1. Confirm the function is improper: \(\deg(\text{numerator})=3\), \(\deg(\text{denominator})=2\), so polynomial division is required.
2. $$2x^3 + 5x^2 - x + 1 = (2x + 1)(x^2 + 2x - 3) + (3x + 4)$$
3. The rewritten form is \(2x + 1 + \frac{3x + 4}{x^2 + 2x - 3}\), where \(\frac{3x + 4}{x^2 + 2x - 3}\) is a proper rational function.

> **Exam tip:** Always check the degree of numerator and denominator first: method marks are deducted for skipping polynomial division for improper fractions.

*Calculator:* allowed

## Partial Fractions: Distinct Linear Factors Denominator

**Partial Fractions (Distinct Linear Factors)** — Decomposition of a proper rational function into sum of fractions with constant numerators and unique linear denominators. Constants \(A, B\) are solved via substitution or equating coefficients.

*Notation:* \frac{px + q}{(ax + b)(cx + d)} = \frac{A}{ax + b} + \frac{B}{cx + d}

**Worked example:** Decompose \(\frac{3x + 4}{(x - 1)(x + 3)}\) into partial fractions.

1. Set up the decomposition form for distinct linear factors: \(\frac{3x +4}{(x-1)(x+3)} = \frac{A}{x-1} + \frac{B}{x+3}\)
2. Multiply both sides by the denominator to eliminate fractions: \(3x + 4 = A(x + 3) + B(x - 1)\)
3. Substitute \(x=1\) (root of first factor): \(3(1) + 4 = A(4) \implies 7 = 4A \implies A = \frac{7}{4}\)
4. Substitute \(x=-3\) (root of second factor): \(3(-3) + 4 = B(-4) \implies -5 = -4B \implies B = \frac{5}{4}\)
5. Final decomposition: \(\frac{7}{4(x-1)} + \frac{5}{4(x+3)}\)

> **tip**
>
> Substitution of roots of the denominator is the fastest method to find constants for linear factors, and is preferred over equating coefficients in exams to save time.

*Calculator:* allowed

## Partial Fractions: Repeated Linear Factors Denominator

**Partial Fractions (Repeated Linear Factor)** — Decomposition for denominators including a squared linear term: you need one fraction for each power of the repeated factor up to its exponent, not just the highest power.

*Notation:* \frac{px + q}{(ax + b)(cx + d)^2} = \frac{A}{ax + b} + \frac{B}{cx + d} + \frac{C}{(cx + d)^2}

**Worked example:** Decompose \(\frac{7x^2 + 19x + 16}{(x +1)(x + 2)^2}\) into partial fractions.

1. Set up the decomposition form correctly, including terms for both the first and second power of the repeated factor: \(\frac{7x^2 +19x +16}{(x+1)(x+2)^2} = \frac{A}{x+1} + \frac{B}{x+2} + \frac{C}{(x+2)^2}\)
2. Multiply through by the denominator: \(7x^2 + 19x + 16 = A(x+2)^2 + B(x+1)(x+2) + C(x+1)\)
3. Substitute \(x=-1\) (root of distinct factor): \(7(1) + 19(-1) + 16 = A(1) \implies 4 = A\)
4. Substitute \(x=-2\) (root of repeated factor): \(7(4) + 19(-2) +16 = C(-1) \implies 6 = -C \implies C = -6\)
5. Equate coefficients of \(x^2\) to find \(B\): \(7 = A + B \implies 7 = 4 + B \implies B = 3\)
6. Final decomposition: \(\frac{4}{x+1} + \frac{3}{x+2} - \frac{6}{(x+2)^2}\)

> **Exam tip:** A very common mistake is omitting the linear power term for the repeated factor: always remember you need \(n\) terms for a factor raised to the \(n\)th power.

*Calculator:* allowed

## Exam Applications of Partial Fractions

Partial fractions are not tested in isolation in P4: they are a supporting technique for three core topics: binomial expansion of rational functions, integration of rational functions, and simplifying differentiation of complex rational functions to avoid repeated quotient rule use.

**Worked example:** Show that \(\int \frac{3x +4}{(x-1)(x+3)} dx = \frac{7}{4}\ln|x-1| + \frac{5}{4}\ln|x+3| + c\)

1. First decompose the integrand into partial fractions (from earlier example): \(\frac{3x+4}{(x-1)(x+3)} = \frac{7}{4(x-1)} + \frac{5}{4(x+3)}\)
2. Integrate term by term using the standard rule \(\int \frac{k}{ax + b} dx = \frac{k}{a}\ln|ax + b| + c\)
3. $$\int \left( \frac{7}{4(x-1)} + \frac{5}{4(x+3)} \right) dx = \frac{7}{4}\ln|x-1| + \frac{5}{4}\ln|x+3| + c$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Skipping polynomial division for improper rational functions before starting partial fraction decomposition.
  - Why it fails: Partial fraction decomposition only works for proper rational functions; you will get an incorrect result and lose method marks if you skip this step.
  - Correct: Always compare the degree of numerator and denominator first; if \(\deg(\text{numerator}) \geq \deg(\text{denominator})\), perform polynomial long division first to get a polynomial plus a proper rational function.
- **Wrong:** Omitting the linear power term for a repeated linear factor, e.g. writing \(\frac{A}{x+1} + \frac{B}{(x+2)^2}\) for a denominator \((x+1)(x+2)^2\).
  - Why it fails: The decomposition is incomplete, so the constants you calculate will not satisfy the original equation, leading to full loss of marks for the question.
  - Correct: Include one term for each power of the repeated factor up to its exponent: for \((cx + d)^2\), include both \(\frac{B}{cx +d}\) and \(\frac{C}{(cx +d)^2}\).
- **Wrong:** Forgetting to include the polynomial term from polynomial division in your final partial fraction decomposition.
  - Why it fails: Your final result will only represent the proper fraction part, not the original improper function, leading to incorrect answers in follow-up applications (integration, binomial expansion).
  - Correct: Add the polynomial obtained from division to your partial fraction decomposition of the proper remainder fraction to get the full decomposition of the original function.
- **Wrong:** Using the same constant letter for multiple terms, or mixing up constants when substituting values.
  - Why it fails: This leads to arithmetic errors when solving for constants, and examiners will penalize incorrect constant values even if your set-up is correct.
  - Correct: Use distinct letters for each unknown constant, and double-check your substitution calculations before proceeding.
- **Wrong:** Attempting to decompose fractions with irreducible quadratic denominators (e.g. \(x^2 + 2\)) for P4 questions.
  - Why it fails: Irreducible quadratic denominators are explicitly out of scope for Edexcel IAL P4; any such question would be for Further Maths, and attempting to decompose them will waste time and lead to errors.
  - Correct: Only process denominators that are products of linear factors, including at most one repeated linear factor, as per P4 specification.

## Cheatsheet

| Denominator Type | Decomposition Form | First Step Required? |
| --- | --- | --- |
| Distinct linear factors: \((ax + b)(cx + d)(ex + f)\) | \(\frac{A}{ax+b} + \frac{B}{cx+d} + \frac{C}{ex+f}\) | No, if function is proper |
| Repeated linear factor: \((ax + b)(cx + d)^2\) | \(\frac{A}{ax+b} + \frac{B}{cx+d} + \frac{C}{(cx+d)^2}\) | No, if function is proper |
| Improper rational function (\(\deg(N) \geq \deg(D)\)) | Polynomial \(Q(x)\) + partial fractions of remainder fraction | Yes: perform polynomial division first to get \(Q(x) + \frac{R(x)}{D(x)}\) where \(\deg(R) < \deg(D)\) |

## What's next

Now that you have mastered partial fraction decomposition for Edexcel IAL P4, you can apply this technique to the three core topics it supports in the unit. Partial fractions are a prerequisite for binomial expansion of rational functions, where they allow you to break complex fractions into simpler forms that fit the standard binomial expansion formula for negative and fractional exponents. They are also essential for integrating rational functions, where each partial fraction term can be integrated directly using the natural logarithm rule, and can simplify differentiation tasks by avoiding the need for repeated quotient rule applications. Make sure you practice combining partial fractions with these follow-up topics to score full marks on P4 exam questions.

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