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 Algebraic Methods

Edexcel A level Maths: 1.2 Algebraic FractionsCurated by Kepler, not produced by us. Watch it 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 $\textbf{factors}$ (never individual terms). With quadratics this often means factorising both top and bottom first.

Example 1Simplify

\dfrac{x^2 - 9}{x^2 + 7x + 12}

1Factorise:

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

2Cancel the common

(x+3)

Answer

\dfrac{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).

\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{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.

Example 1Write

\dfrac{3}{x - 1} - \dfrac{2}{x + 1}

as a single fraction.

1Common denominator

(x-1)(x+1)

2Numerator:

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

Answer

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

Formula recap

\dfrac{ac}{bc} = \dfrac{a}{b}

Cancel common factors.

\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{ad}{bc}

Divide = flip and multiply.

\dfrac{a}{b} \pm \dfrac{c}{d} = \dfrac{ad \pm bc}{bd}

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

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Simplify $\dfrac{x^{2} - 9}{x + 3}$ .
- $x - 3$
- $x + 3$
- $x - 9$
- $x^{2}$
Show answer and explanation
Answer: $x - 3$
(x−3)(x+3)/(x+3).
Simplify $\dfrac{x^{2} - 9}{x^{2} + 7x + 12}$ .
- $\dfrac{x - 3}{x + 4}$
- $\dfrac{x + 3}{x + 4}$
- $x - 3$
- $\dfrac{x-3}{x+3}$
Show answer and explanation
Answer: $\dfrac{x - 3}{x + 4}$
Cancel (x + 3).
$\dfrac{a}{b} \div \dfrac{c}{d}$ equals:
- $\dfrac{ad}{bc}$
- $\dfrac{ac}{bd}$
- $\dfrac{bc}{ad}$
- $\dfrac{a+d}{b+c}$
Show answer and explanation
Answer: $\dfrac{ad}{bc}$
Flip and multiply.
A common denominator for $\dfrac{3}{x-1} - \dfrac{2}{x+1}$ is:
- $(x-1)(x+1)$
- $x - 1$
- $x + 1$
- $x^{2}$
Show answer and explanation
Answer: $(x-1)(x+1)$
Product of the denominators.
$\dfrac{3}{x-1} - \dfrac{2}{x+1}$ has numerator:
- $x + 5$
- $x + 1$
- $5x$
- $x - 5$
Show answer and explanation
Answer: $x + 5$
3(x+1) − 2(x−1) = x + 5.
You may cancel in a fraction only when you have a common:
- Factor
- Term
- Power
- Constant
Show answer and explanation
Answer: Factor
Factors, not terms.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

All 5 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.