Algebraic Fractions

Q: Can I cancel individual terms in an algebraic fraction?

No. You can only cancel common factors that apply to the entire numerator or denominator. Always factorise both fully before cancelling any terms.

Mathematics· E2.3· 15 min read

1. Simplifying Algebraic Fractions★★☆☆☆⏱ 3 min

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📘 Definition

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.

2. Multiplying & Dividing Algebraic Fractions★★★☆☆⏱ 4 min

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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})

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Step 1: Flip the second fraction and change the operation to multiplication:
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$\frac{3x^2 - 12x}{x^2 - 4x + 4} \times \frac{x^2 - 2x}{6x}$
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
$\frac{3x(x-4)}{(x-2)^2} \times \frac{x(x-2)}{6x} = \frac{x(x-4)}{2(x-2)}$

3. Adding & Subtracting Algebraic Fractions★★★★☆⏱ 4 min

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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
$\frac{2(x-2)}{(x+1)(x-2)} + \frac{3(x+1)}{(x+1)(x-2)}$
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Step 3: Combine the numerators over the common denominator:
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$\frac{2(x-2) + 3(x+1)}{(x+1)(x-2)}$
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Step 4: Expand and simplify the numerator:
7
$2x - 4 + 3x + 3 = 5x -1$
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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.

4. Combined Operations with Algebraic Fractions★★★★★⏱ 4 min

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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
$\frac{y - x}{xy} \times \frac{x^2 y}{(y - x)(y + x)}$
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Step 4: Cancel common factors across numerators and denominators:
8
$\frac{(y-x) \cdot x^2 y}{xy \cdot (y-x)(y+x)} = \frac{x}{y + x}$

5. Common Pitfalls

Wrong move:

Cancelling individual terms instead of factors, e.g. (\frac{x + 2}{x + 4} \rightarrow \frac{2}{4})

Why:

You can only cancel identical factors that apply to the entire numerator or denominator, not standalone terms

Correct move:

Factorise the numerator and denominator fully first, only cancel shared factors

Wrong move:

Forgetting to flip the second fraction when dividing algebraic fractions

Why:

Division of fractions requires multiplying by the reciprocal of the divisor

Correct move:

Rewrite division as multiplication by the inverted second fraction before any other steps

Wrong move:

Distributing the negative sign incorrectly when subtracting fractions, e.g. (\frac{3}{x} - \frac{x + 1}{2} \rightarrow \frac{6 - x + 1}{2x})

Why:

The negative sign applies to every term in the numerator of the second fraction

Correct move:

Put brackets around the second numerator when subtracting, then distribute the negative sign fully

Wrong move:

Using the product of denominators instead of the LCD for addition/subtraction, leading to unnecessary complex expansion

Why:

Using the lowest common denominator reduces the number of terms you need to expand and simplify later

Correct move:

Identify the LCD by taking the highest power of each unique factor across all denominators

Wrong move:

Forgetting to factorise quadratic or difference of two square expressions before simplifying

Why:

Common factors are often hidden in unfactorised polynomials, leading to an unsimplified final answer

Correct move:

Factorise every numerator and denominator fully at the start of any algebraic fraction problem

6. Quick Reference 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})

7. Frequently Asked

Can I cancel individual terms in an algebraic fraction?

No. You can only cancel common factors that apply to the entire numerator or denominator. Always factorise both fully before cancelling any terms.

Is algebraic fractions tested in Core tier papers?

No, this topic is exclusively tested in Extended tier Papers 2 and 4 for CIE IGCSE Maths 0580.

Going deeper

syllabus_documentCIE IGCSE 0580 2025-2027 SyllabusSee section E2.3 for algebraic fractions specification

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.