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Algebraic Fractions

Algebraic fractions follow exactly the same rules as ordinary number fractions — you just work with expressions instead of digits. The single most important habit: factorise first, because cancelling is only ever allowed on common factors.

30 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel AS Level Maths: 7.1 Algebraic FractionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Simplify an algebraic fraction by cancelling common factors
Multiply and divide algebraic fractions
Add and subtract fractions with a common denominator
Know when (and when not) cancelling is allowed

Simplifying — factorise then cancel

To simplify an algebraic fraction, factorise the numerator and denominator fully, then cancel any factor that appears in both. You can only cancel $factors$ , never individual terms.

\frac{x ^{2} - 9}{x + 3} = \frac{( x + 3 ) ( x - 3 )}{x + 3} = x - 3

Factorise the difference of two squares, then cancel the common

(x + 3)

1Factorise the top:

(x + 2) (x + 3)

2Cancel the common

(x + 2)

Answer

x + 3

Tip — Cancelling is for factors only. In (x + 3)/(x + 2) nothing cancels — the x’s are terms, not factors.

Multiplying and dividing

To multiply, factorise everything, cancel across the fractions, then multiply the tops and the bottoms. To divide, $flip$ the second fraction and multiply (keep–change–flip).

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Dividing by a fraction is multiplying by its reciprocal.

1Flip and multiply:

\frac{x}{x + 1} \times \frac{x + 1}{x ^{2}}

2Cancel

(x + 1)

and one

x

\frac{1}{x}

Answer

\frac{1}{x}

Adding and subtracting

As with numbers, you need a $common denominator$ before adding or subtracting. Multiply each fraction so the denominators match, then combine the numerators.

1Common denominator

x (x + 1)

2Rewrite:

\frac{2 ( x + 1 )}{x ( x + 1 )} + \frac{3 x}{x ( x + 1 )}

3Combine the numerators:

2 (x + 1) + 3 x = 5 x + 2

Answer

\frac{5 x + 2}{x ( x + 1 )}

Tip — Only the numerators add — keep the common denominator the same underneath.

Formula recap

\frac{a c}{b c} = \frac{a}{b}

Cancel common factors only.

\frac{a}{b} \div \frac{c}{d} = \frac{a d}{b c}

Divide = flip and multiply.

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

Common denominator to add.

Common mistakes to avoid

Cancelling terms: (x + 3)/3 = x + 1.

You can only cancel factors, not terms — (x + 3)/3 does not simplify.

Adding fractions by adding tops and bottoms separately.

Find a common denominator first; only the numerators combine.

Key takeaways

Factorise fully, then cancel common factors (never terms).
Multiply: cancel and multiply across; divide: flip and multiply.
Add/subtract: get a common denominator, then combine numerators.

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