# Algebraic Fractions

> Mathematics · CIE IGCSE 0580 (2025-2027)
> Source: https://www.owlsprep.com/study/cie-0580-u2-algebraic-fractions/

This guide covers all Extended tier algebraic fraction skills required for CIE IGCSE Maths 0580 (2025-2027), including simplification, multiplication, division, addition and subtraction of rational expressions.

**Prerequisites:** [Factorising linear, quadratic and difference of two square expressions](https://www.owlsprep.com/study/cie-0580-u2-factorisation/); [Operations with numerical fractions](https://www.owlsprep.com/study/cie-0580-u1-fraction-operations/)

## Learning objectives

- Simplify algebraic fractions by factorising numerators and denominators and cancelling common factors
- Multiply and divide algebraic fractions correctly
- Add and subtract algebraic fractions by calculating the lowest common denominator
- Combine all operations to simplify complex rational expressions for Papers 2 and 4

## Simplifying Algebraic Fractions

**Simplifying Algebraic Fractions** — The process of reducing an algebraic fraction to its lowest form by cancelling common factors between the fully factorised numerator and denominator.

*Example:* \(\frac{2x + 4}{x^2 - 4} = \frac{2(x+2)}{(x+2)(x-2)} = \frac{2}{x-2}\) for \(x \neq -2, 2\)

To simplify any algebraic fraction, follow these two mandatory steps: first, fully factorise both the numerator and denominator, checking for common linear factors, differences of two squares, and quadratic factors. Second, cancel any identical factors that appear in both the numerator and denominator. You may only cancel factors, not standalone terms.

**Worked example:** Simplify \(\frac{x^2 - 9}{x^2 + 5x + 6}\)

1. Step 1: Factorise the numerator, which is a difference of two squares:
2. $$x^2 - 9 = (x + 3)(x - 3)$$
3. Step 2: Factorise the quadratic denominator:
4. $$x^2 + 5x + 6 = (x + 3)(x + 2)$$
5. Step 3: Rewrite the fraction with factorised forms, then cancel the common \((x+3)\) factor:
6. $$\frac{(x+3)(x-3)}{(x+3)(x+2)} = \frac{x-3}{x+2}$$

> **Exam tip:** You do not need to state excluded values (values of x that make the original denominator zero) unless explicitly asked, as marks are only awarded for the simplified final fraction in most 0580 questions.

*Calculator:* forbidden

## Multiplying & Dividing Algebraic Fractions

Multiplying algebraic fractions follows the same rule as multiplying numerical fractions: multiply all numerators together, multiply all denominators together, then simplify the result. For division, flip the second fraction (take its reciprocal) and change the operation to multiplication, then proceed as for multiplication. You can simplify before multiplying to reduce the size of polynomials you work with.

**Worked example:** Calculate \(\frac{3x^2 - 12x}{x^2 - 4x + 4} \div \frac{6x}{x^2 - 2x}\)

1. Step 1: Flip the second fraction and change the operation to multiplication:
2. $$\frac{3x^2 - 12x}{x^2 - 4x + 4} \times \frac{x^2 - 2x}{6x}$$
3. Step 2: Factorise all expressions fully:
4. $$3x^2 -12x = 3x(x - 4) \newline x^2 -4x +4 = (x-2)^2 \newline x^2 - 2x = x(x - 2)$$
5. Step 3: Rewrite the fraction with factorised terms, then cancel common factors:
6. $$\frac{3x(x-4)}{(x-2)^2} \times \frac{x(x-2)}{6x} = \frac{x(x-4)}{2(x-2)}$$

> **tip**
>
> Cancel factors before multiplying to avoid working with large polynomials later, this saves valuable time in non-calculator Paper 2 exams.

*Calculator:* forbidden

## Adding & Subtracting Algebraic Fractions

To add or subtract algebraic fractions, you first need to find the lowest common denominator (LCD) of all fractions, just like with numerical fractions. The LCD is the lowest common multiple of all the denominators. Once you rewrite each fraction with the LCD as its denominator, you can add or subtract the numerators directly, then simplify the result.

**Worked example:** Simplify \(\frac{2}{x + 1} + \frac{3}{x - 2}\)

1. Step 1: Find the LCD of the two denominators. Since \((x+1)\) and \((x-2)\) have no common factors, the LCD is \((x+1)(x-2)\)
2. Step 2: Rewrite each fraction with the LCD:
3. $$\frac{2(x-2)}{(x+1)(x-2)} + \frac{3(x+1)}{(x+1)(x-2)}$$
4. Step 3: Combine the numerators over the common denominator:
5. $$\frac{2(x-2) + 3(x+1)}{(x+1)(x-2)}$$
6. Step 4: Expand and simplify the numerator:
7. $$2x - 4 + 3x + 3 = 5x -1$$
8. Final simplified form:
9. $$\frac{5x - 1}{(x+1)(x-2)}$$

> **Exam tip:** Always put brackets around the numerator of the second fraction when subtracting, to avoid sign errors when distributing the negative sign across all terms.

*Calculator:* forbidden

## Combined Operations with Algebraic Fractions

Most exam questions for this topic ask you to combine multiple operations (add, subtract, multiply, divide) into one simplification task. Always follow the order of operations: brackets first, then multiplication/division, then addition/subtraction. Factorise at every step to catch common factors early and avoid unnecessary expansion work.

**Worked example:** Simplify \(\left( \frac{1}{x} - \frac{1}{y} \right) \times \frac{x^2 y}{y^2 - x^2}\)

1. Step 1: Simplify the expression inside the brackets first by subtracting the two fractions:
2. $$\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}$$
3. Step 2: Factorise the denominator of the second fraction, which is a difference of two squares:
4. $$y^2 - x^2 = (y - x)(y + x)$$
5. Step 3: Rewrite the full expression with simplified terms:
6. $$\frac{y - x}{xy} \times \frac{x^2 y}{(y - x)(y + x)}$$
7. Step 4: Cancel common factors across numerators and denominators:
8. $$\frac{(y-x) \cdot x^2 y}{xy \cdot (y-x)(y+x)} = \frac{x}{y + x}$$

**Summary**

- Always factorise all expressions first before cancelling any terms
- For division: flip the second fraction then multiply before simplifying
- For add/subtract: find the LCD first then combine numerators
- Follow order of operations (brackets first) for combined tasks

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Cancelling individual terms instead of factors, e.g. \(\frac{x + 2}{x + 4} \rightarrow \frac{2}{4}\)
  - Why it fails: You can only cancel identical factors that apply to the entire numerator or denominator, not standalone terms
  - Correct: Factorise the numerator and denominator fully first, only cancel shared factors
- **Wrong:** Forgetting to flip the second fraction when dividing algebraic fractions
  - Why it fails: Division of fractions requires multiplying by the reciprocal of the divisor
  - Correct: Rewrite division as multiplication by the inverted second fraction before any other steps
- **Wrong:** Distributing the negative sign incorrectly when subtracting fractions, e.g. \(\frac{3}{x} - \frac{x + 1}{2} \rightarrow \frac{6 - x + 1}{2x}\)
  - Why it fails: The negative sign applies to every term in the numerator of the second fraction
  - Correct: Put brackets around the second numerator when subtracting, then distribute the negative sign fully
- **Wrong:** Using the product of denominators instead of the LCD for addition/subtraction, leading to unnecessary complex expansion
  - Why it fails: Using the lowest common denominator reduces the number of terms you need to expand and simplify later
  - Correct: Identify the LCD by taking the highest power of each unique factor across all denominators
- **Wrong:** Forgetting to factorise quadratic or difference of two square expressions before simplifying
  - Why it fails: Common factors are often hidden in unfactorised polynomials, leading to an unsimplified final answer
  - Correct: Factorise every numerator and denominator fully at the start of any algebraic fraction problem

## Cheatsheet

| Operation | Rule | Example |
| --- | --- | --- |
| Simplify | Factorise numerator & denominator, cancel common factors | \(\frac{x^2 - 1}{x^2 + 3x + 2} = \frac{x - 1}{x + 2}\) |
| Multiply | Multiply numerators, multiply denominators, simplify | \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\) |
| Divide | Flip second fraction, multiply, simplify | \(\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}\) |
| Add/Subtract | Find LCD, rewrite fractions, combine numerators, simplify | \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\) |

## What's next

Mastering algebraic fractions is a critical foundation for more advanced Extended tier algebra topics in CIE IGCSE Maths 0580, including solving algebraic fraction equations and working with quadratic graphs. The skills you have learned here are frequently tested in both Paper 2 (non-calculator) and Paper 4 (calculator) exams, often as part of multi-step questions worth 3-5 marks. Next, you should apply these simplification skills to solving equations that contain algebraic fractions, and practice combining these operations with rearranging formulae to ensure you can answer even the most complex exam questions for this topic. Be sure to attempt past paper questions for this topic to get familiar with the common phrasing and trap questions used by examiners.

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