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Algebraic Fractions

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Cambridge IGCSE · International Maths Extended

Algebraic Fractions

The big idea: Algebraic fractions follow exactly the same rules as ordinary number fractions — the only new skill is that you have to factorise first before you can simplify, combine, multiply, or cancel anything. If you can do fractions with numbers, and you can factorise, you can do this whole chapter.

Summary — What This Chapter Covers

Simplifying: factorise the top and bottom fully, then cancel any common factor (a single term or a whole bracket).
Adding & subtracting: find the lowest common denominator (LCD), rewrite each fraction over it, combine into one fraction, then check if it simplifies.
Multiplying: factorise everything first, cancel common factors across the two fractions, then multiply tops together and bottoms together.
Dividing: flip (reciprocate) the second fraction and turn ÷ into ×, then multiply as above.
Solving equations: either combine fractions into one and cross-multiply, or clear every denominator by multiplying both sides by each one in turn — then solve the resulting equation (often a quadratic or cubic).

1. Simplifying Algebraic Fractions

An algebraic fraction is just a fraction where the top (numerator) and/or bottom (denominator) contains letters, e.g. 2x / (x²+3x). To simplify one, you're doing the exact same thing you'd do with a numeric fraction like 15/20 = 3/4 — you find what's common to top and bottom, and cancel it. The only extra step with algebra is that the "common factor" is often hidden inside an expression, so your first job is always to factorise fully.

The Method

Step 1: Factorise the numerator fully.
Step 2: Factorise the denominator fully.
Step 3: Cancel any factor common to both.

Think of it like reducing 15/20: you spot that both are multiples of 5, and divide both by 5. Here, instead of spotting a common number, you spot a common bracket or term after factorising.

Cancelling a single term

Worked Example

Simplify x(5x − 1) / 4x

The x on top and the x on bottom are common factors — cancel them.

Result: (5x − 1) / 4

Cancelling a whole bracket

Worked Example

Simplify x(x + 2) / [(x + 2)(x − 1)]

The bracket (x + 2) appears on both top and bottom — cancel the whole bracket, not just part of it.

Result: x / (x − 1)

A trickier one — factorise before you can even see the common factor

Worked Example

Simplify (4x + 6) / (2x² − 7x − 15)

Factorise the top using 2 as a common factor: 2(2x + 3)

Factorise the bottom (a quadratic). Knowing the top gave (2x+3) is a big hint that this bracket appears again: (2x + 3)(x − 5)

Now the fraction is 2(2x + 3) / [(2x + 3)(x − 5)] — cancel (2x + 3)

Result: 2 / (x − 5)

Examiner Tip

When asked to simplify, always factorise top and bottom first. It's very likely one factor will match on both sides — if you're stuck factorising a tricky quadratic, look at what the other side factorised to. It often gives away the answer!

Common Mistake

You can only cancel a factor that divides every single term on both top and bottom. The fraction 6x / (x + 1) cannot be simplified — x is not a factor of the whole denominator (x + 1 doesn't factorise), so there's nothing to cancel. Don't be tempted to cross out the x's just because they "look the same"!

Practice Question 1

Simplify: (3x + 12) / (x² + 7x + 12)

Practice Question 2

Simplify: (x² − 9) / (2x + 6)

2. Adding & Subtracting Algebraic Fractions

This works exactly like adding numeric fractions, e.g. 1/2 + 1/5. You can't just add tops and bottoms — you need a common denominator first. With algebra, the denominator is often already factorised (or needs to be), which actually makes finding the LCD easier, because you can literally see what's "missing" from each one.

The 4-Step Method

1. Find the Lowest Common Denominator (LCD)
2. Rewrite each fraction over the LCD
3. Combine into a single fraction — add or subtract the numerators
4. Simplify: check if the top factorises and cancels with the bottom

Finding the LCD — don't just multiply blindly

A common trap is to always multiply the two denominators together. That always gives you a common denominator, but not always the lowest one. Compare these:

Compare These LCDs

• LCD of 1/(x+2) and 1/(x+5) is (x+2)(x+5) — nothing in common, so multiply them.

• LCD of 1/x and 1/2x is just 2x (NOT 2x²) — because 2x already contains an x. This mirrors how the LCD of 1/2 and 1/4 is just 4, not 8.

• LCD of 1/(x+2) and 1/[(x+2)(x−1)] is (x+2)(x−1) — the second denominator already contains the first.

Worked Example — Subtracting

Express x/(x+4) − 3/(x−1) as a single fraction.

LCD = (x+4)(x−1)

Rewrite: x(x−1) / [(x+4)(x−1)] − 3(x+4) / [(x−1)(x+4)]

Combine: [x(x−1) − 3(x+4)] / [(x+4)(x−1)]

Expand carefully — watch the negative sign hitting both terms in the second bracket: (x² − x) − (3x + 12) = x² − 4x − 12

Factorise the top: (x + 2)(x − 6)

Nothing cancels with (x+4)(x−1), so final answer: (x+2)(x−6) / [(x+4)(x−1)]

The #1 Mistake in This Topic

When subtracting, the minus sign applies to every term in the second numerator, not just the first one. x(x−1) − 3(x+4) expands to x² − x − 3x − 12 (both terms in the second bracket flip sign) — not x² − x − 3x + 12. This single sign error is where most marks are lost in this topic.

Examiner Tip

Keep the top and bottom of your final answer in factorised form rather than expanding it all out. This makes it much easier to spot last-minute cancelling — and if nothing cancels, you know instantly you're done.

Practice Question 1

Express as a single fraction: 2/(x+1) + 3/(x−2)

Practice Question 2

Express as a single fraction: (x−4)/[2(x−3)] − (x−1)/(2x)

3. Multiplying & Dividing Algebraic Fractions

Good news — multiplying is actually the easiest of the four operations, because you don't need a common denominator at all. You just factorise everything, cancel anything common (even across the two different fractions), then multiply straight across.

Multiplying — 4 Steps

1. Factorise every numerator and denominator fully
2. Cancel common factors — top-and-bottom of either fraction
3. Multiply tops together, multiply bottoms together
4. Check for any final factorising/cancelling

Dividing — the "Flip and Multiply" Rule

a/b ÷ c/d = a/b × d/c

Flip (find the reciprocal of) the second fraction only, change ÷ to ×, then multiply as normal.

The Sneaky Trick: Negative Factoring

Sometimes two brackets look almost identical but with opposite signs, e.g. (2x − 3) and (3 − 2x). These are negatives of each other! You can rewrite 2x − 3 as −(3 − 2x) to reveal a hidden common factor and cancel it — just remember to carry that minus sign into your final answer.

Worked Example — Multiplying

Simplify: (2x²−11x+12)/(x²+3x−28) × (x²−5x−14)/(9−4x²)

Factorise all four quadratics:

2x²−11x+12 = (2x−3)(x−4)

x²+3x−28 = (x+7)(x−4)

x²−5x−14 = (x+2)(x−7)

9−4x² = (3−2x)(3+2x) (difference of two squares)

Rewrite 2x−3 as −(3−2x) so it can cancel with (3−2x) on the bottom.

Cancel (x−4) and (3−2x)-type brackets.

Final answer: −(x+2)(x−7) / [(x+7)(3+2x)]

Worked Example — Dividing

Divide (x+3)/(x−4) by (2x+6)/(x²−16)

Flip and multiply: (x+3)/(x−4) × (x²−16)/(2x+6)

Factorise: x²−16 = (x−4)(x+4) and 2x+6 = 2(x+3)

= (x+3)/(x−4) × [(x−4)(x+4)]/[2(x+3)]

Cancel (x+3) and (x−4)

Final answer: (x+4)/2

Practice Question 1

Simplify: x/(3x+6) × (2x+4)/(x+7)

Practice Question 2

Divide: (x²−25)/(x+2) ÷ (x−5)/(3x+6)

4. Solving Equations with Algebraic Fractions

This is where everything comes together — and it's usually where the harder exam marks live. There are two valid routes to the same answer. Pick whichever feels more natural, but be consistent so you don't mix up steps halfway through.

Method A — Combine First

1. Add/subtract the fractions into one single fraction (as in Topic 2)
2. Cross-multiply to clear the remaining denominator
3. Expand, rearrange to = 0, and solve

Method B — Clear Denominators First

Multiply every term on both sides of the equation by each denominator, one at a time, cancelling as you go

This avoids fraction-adding altogether — you're removing the fractions from the start.

Worked Example — Method A

Solve: 8/(x+1) − 5/(x+2) = 1

Combine the left side: (3x+11)/[(x+1)(x+2)] = 1

Cross-multiply: 3x+11 = (x+1)(x+2)

Expand: 3x+11 = x²+3x+2

Rearrange: 0 = x² − 9

Factorise: 0 = (x−3)(x+3)

Solutions: x = 3 or x = −3

Worked Example — Method B

Solve: 4/(x−3) + 5/(x+1) = 5

Multiply every term by (x−3): 4 + 5(x−3)/(x+1) = 5(x−3)

Multiply every term by (x+1): 4(x+1) + 5(x−3) = 5(x−3)(x+1)

Expand and simplify: 9x − 11 = 5x² − 10x − 15

Rearrange: 0 = 5x² − 19x − 4

Factorise: 0 = (5x+1)(x−4)

Solutions: x = −1/5 or x = 4

Why This Often Ends Up a Quadratic (or Cubic!)

Once you clear all the denominators, you're left with an ordinary polynomial equation. Because you multiplied out brackets like (x−3)(x+1), you'll almost always land on a quadratic — and occasionally, if there are three separate denominators, a cubic. Don't panic when you see x³ appear — just rearrange to = 0 and factorise as usual (often by grouping).

Examiner Tip

When multiplying by an algebraic expression, always put brackets around it, e.g. 5(x + 4), not 5x + 4. This is the single biggest source of expansion errors. Multiplying by both denominators in one combined step can save time, but only do this once you're confident — it's easy to forget a term.

Practice Question 1

Solve: 3/(x+2) + 4/(x−1) = 2

Practice Question 2

Show that 3/(p+2) − 4/p = 2p can be rearranged to 2p³ + 4p² + 4p + 12 = 0

What to Memorise

Simplifying rule

Factorise top AND bottom fully first. Only cancel factors common to every term — never cancel a "matching letter" that isn't a genuine common factor.

LCD is not always "multiply the denominators"

Check if one denominator is a multiple of the other, or already contains the other, before multiplying blindly — this avoids extra simplifying work later.

Dividing = flip and multiply

a/b ÷ c/d = a/b × d/c — only flip the second fraction, never the first.

Difference of two squares

a² − b² = (a − b)(a + b). This shows up constantly disguised as things like 9 − 4x² = (3−2x)(3+2x).

Negative brackets are secretly the same factor

(2x − 3) = −(3 − 2x). Spotting this reveals hidden cancellations in multiply/divide questions.

Solving fractional equations

Either combine fractions then cross-multiply, OR clear each denominator by multiplying both sides by it — always end by rearranging to = 0 and factorising.

Concepts Checklist

I can factorise the top and bottom of a fraction before simplifying

I can cancel both single-term and whole-bracket common factors

I know a term can only be cancelled if it's a factor of the WHOLE top or bottom

I can find the lowest common denominator, not just multiply denominators together

I can add and subtract algebraic fractions, being careful with negative signs

I can multiply algebraic fractions by factorising and cancelling first

I can divide algebraic fractions using "flip and multiply"

I can spot when two brackets are negatives of each other (e.g. 2x−3 and 3−2x)

I can solve equations with fractions by combining then cross-multiplying

I can solve equations with fractions by clearing denominators first

I can rearrange a fractional equation into a quadratic or cubic and solve it

I always use brackets when multiplying by an algebraic expression

Exam Tips & Common Traps

Trap 1: Fake cancelling

Students often cancel an x that appears in one term of the denominator but not all of them, e.g. wrongly "simplifying" 6x/(x+1). Always ask: is this factor present in every term on top, and every term on bottom? If not — it cannot be cancelled.

Trap 2: Sign errors when subtracting

When you subtract two numerators, the minus sign distributes across every term in the second bracket. x(x−1) − 3(x+4) becomes x² − x − 3x − 12, not x² − x − 3x + 12. Write out the expansion in full rather than doing it in your head.

Trap 3: Wrong LCD

Multiplying the two denominators together always works, but if one denominator already contains a factor of the other (like x and 2x, or (x+2) and (x+2)(x−1)), you'll create unnecessary extra factors to cancel later. Spend a moment checking for shared factors before you commit to an LCD.

What Examiners Actually Look For

Method marks are awarded for: correctly identifying the LCD, correctly cross-multiplying or clearing denominators, and correctly expanding brackets. Even if your final factorisation is wrong, you can pick up most of the marks by showing clean, correct algebraic steps up to that point. Show every line of working.

Recognising Question Patterns

"Simplify" → factorise and cancel. "Express as a single fraction" → find LCD, combine (do NOT solve — there's no equals sign to solve). "Solve" → there IS an equals sign; combine or clear fractions, then solve the resulting polynomial equation. "Show that..." → you're given the answer, so work forwards methodically and make every algebraic step explicit so the marker can follow your logic to that exact result.

Final Checklist Before You Submit an Answer

1) Did I factorise fully everywhere I could? 2) Did I cancel only genuine common factors? 3) If subtracting, did I distribute the negative sign correctly? 4) Is my final fraction in its simplest possible form? 5) For equations, did I check both/all solutions make sense (i.e. they don't make an original denominator zero)?

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2. Adding & Subtracting Algebraic Fractions
3. Multiplying & Dividing Algebraic Fractions
Exam Tips & Common Traps

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Algebraic Fractions

Algebraic Fractions

Summary — What This Chapter Covers

1. Simplifying Algebraic Fractions

Cancelling a single term

Cancelling a whole bracket

A trickier one — factorise before you can even see the common factor

2. Adding & Subtracting Algebraic Fractions

Finding the LCD — don't just multiply blindly

3. Multiplying & Dividing Algebraic Fractions

4. Solving Equations with Algebraic Fractions

What to Memorise

Concepts Checklist

Exam Tips & Common Traps

More International Mathematics topics