Algebraic Fractions (Edexcel IGCSE Maths): Revision notes

Section 1

How do you simplify an algebraic fraction?

An algebraic fraction is simplified by factorising the numerator and denominator fully, then cancelling any factors common to both.

Steps:

Factorise the top (numerator) completely.
Factorise the bottom (denominator) completely.
Cancel any factor that appears in both — never cancel individual terms, only whole factors.

Example: $\dfrac{x^2-9}{x^2+5x+6} = \dfrac{(x-3)(x+3)}{(x+2)(x+3)} = \dfrac{x-3}{x+2}$

Always state any values of $x$ that make the original denominator zero — these must be excluded (e.g. $x \neq -2, -3$ above).

Key termsalgebraic fractionfactorisecancel

Common mistake

Cancelling terms instead of factors is the most common error, e.g. wrongly cancelling the $x$ in $\dfrac{x+3}{x}$ to get $3$ . You can only cancel a factor that multiplies the WHOLE numerator and WHOLE denominator.

Exam tip

Always fully factorise first — look for a common factor, then difference of two squares, then quadratic factorising — before attempting to cancel anything.

Section 2

How do you add or subtract algebraic fractions?

Just like with numerical fractions, you need a common denominator before adding or subtracting.

Find the lowest common denominator (LCD) — usually the product of the two denominators, or their LCM if they share factors.
Write each fraction with the common denominator by multiplying top and bottom by the missing factor.
Combine the numerators over the single denominator, expanding brackets carefully.
Simplify the resulting numerator, then factorise and cancel if possible.

Example: $\dfrac{2}{x+1} + \dfrac{3}{x-2} = \dfrac{2(x-2) + 3(x+1)}{(x+1)(x-2)} = \dfrac{2x-4+3x+3}{(x+1)(x-2)} = \dfrac{5x-1}{(x+1)(x-2)}$

Key termscommon denominatorlowest common denominator (LCD)

Common mistake

Forgetting to multiply the WHOLE numerator by the missing factor — e.g. writing $\dfrac{2}{x+1} = \dfrac{2(x-2)}{(x+1)(x-2)}$ is correct, but a common slip is only multiplying part of the numerator.

Example

$\dfrac{1}{x} - \dfrac{1}{x+2} = \dfrac{(x+2)-x}{x(x+2)} = \dfrac{2}{x(x+2)}$

Section 3

How do you multiply or divide algebraic fractions?

Multiplication and division of algebraic fractions follow the same rules as numerical fractions, but factorising first makes cancelling much easier.

Multiplying: factorise every numerator and denominator, cancel common factors across the whole calculation, then multiply what remains: $\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{ac}{bd}$

Dividing: flip (take the reciprocal of) the second fraction and multiply: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}$

Example: $\dfrac{x^2-4}{x+3} \div \dfrac{x-2}{x^2-9} = \dfrac{(x-2)(x+2)}{x+3} \times \dfrac{(x-3)(x+3)}{x-2} = (x+2)(x-3)$

Key termsreciprocal

Think of it like this

Think of it like cancelling in a numerical fraction calculation, e.g. $\dfrac{4}{9} \times \dfrac{3}{8}$ — you cancel common factors across the whole multiplication before multiplying out, not just within one fraction.

Exam tip

Never cancel before factorising — an unfactorised expression like $x^2-4$ hides the factor $(x-2)$ that you need to cancel.

Section 4

How do you solve equations containing algebraic fractions?

To solve an equation with algebraic fractions, clear the denominators first, then solve the resulting equation.

Multiply every term on both sides by the LCD of all the fractions.
Cancel denominators — this leaves a normal linear or quadratic equation.
Expand any brackets and simplify.
Solve for $x$ (factorise if quadratic).
Check your answer(s) work in the original equation, and reject any value that makes a denominator zero.

Example: solve $\dfrac{3}{x} + \dfrac{2}{x-1} = 1$ .

Multiply by $x(x-1)$ : $3(x-1) + 2x = x(x-1)$ $3x-3+2x = x^2-x$ $x^2 -6x+3=0$ Solve the resulting quadratic using the quadratic formula, then check both solutions are valid (i.e. $x \neq 0$ and $x \neq 1$ ).

Key termsclearing the denominatorextraneous solution

Common mistake

Forgetting to multiply EVERY term (including any non-fraction term or the number on the other side) by the LCD — a very common lost mark in exams.

Exam tip

Fraction equations involving $x$ in the denominator often lead to a quadratic — expect to factorise or use the quadratic formula, and always check solutions are valid.

Must Know

Only cancel whole factors, never individual terms, and always factorise fully first.
Adding/subtracting fractions requires a common denominator; multiplying/dividing does not.
To divide by a fraction, multiply by its reciprocal (flip the second fraction).
To solve an equation with fractions, multiply every term by the LCD to clear denominators before solving.
State any restrictions (values that make a denominator zero) and check solutions are valid, rejecting extraneous ones.
Difference of two squares ( $a^2-b^2=(a-b)(a+b)$ ) is a very common factorising tool in this topic.

That's the notes covered.

Carry on to the next subtopic.