1.2PureCore

Algebraic Fractions

Year 2 pushes algebraic fractions further, with more involved factorising and polynomial numerators. The four operations work exactly as with number fractions — the priority is always to factorise so you can cancel.

25 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel A level Maths: 1.2 Algebraic FractionsWatch the full walkthrough before the notes below.

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What you'll be able to do

Simplify algebraic fractions by factorising and cancelling
Multiply and divide algebraic fractions
Add and subtract with a common denominator
Handle polynomial numerators confidently

Simplifying

Factorise the numerator and denominator fully, then cancel common $factors$ (never individual terms). With quadratics this often means factorising both top and bottom first.

1Factorise:

\frac{( x - 3 ) ( x + 3 )}{( x + 3 ) ( x + 4 )}

2Cancel the common

(x + 3)

Answer

\frac{x - 3}{x + 4}

Tip — Cancelling is only valid on factors. (x² − 9)/(x + 3) works because x² − 9 factorises.

Multiplying and dividing

Factorise everything, cancel across the fractions, then multiply tops and bottoms. To divide, multiply by the reciprocal (keep–change–flip).

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

Flip the divisor and multiply.

Adding and subtracting

Find a common denominator (usually the product of the denominators, or their LCM), rewrite each fraction, then combine the numerators carefully — watch signs when subtracting.

1Common denominator

(x - 1) (x + 1)

2Numerator:

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

Answer

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

Formula recap

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

Cancel common factors.

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

Divide = flip and multiply.

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

Common denominator to add/subtract.

Common mistakes to avoid

Cancelling terms instead of factors.

Factorise fully first; only cancel a factor common to top and bottom.

Sign slips when subtracting fractions.

Bracket the numerator being subtracted: −2(x − 1) = −2x + 2.

Key takeaways

Factorise, then cancel common factors.
Multiply: cancel and multiply across; divide: flip and multiply.
Add/subtract: common denominator, then combine numerators (mind the signs).

Test yourself

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