SPEC A4 • ALGEBRA — ALGEBRAIC FRACTIONS [HIGHER]
Algebraic Fractions
Four operations on fractions where the numerator and denominator contain letters. Simplify, add, subtract, multiply and divide — all built on one principle: factorise first, then cancel (or combine).
What You'll Learn HIGHER
Algebraic fractions behave like numerical fractions — but with one extra skill. Whenever you want to cancel or combine, you must factorise first.
Four operations
Simplify — factorise top and bottom, cancel common factors
Add & subtract — rewrite over a common denominator, then combine numerators
Multiply — factorise, cross-cancel, then multiply straight across
Divide — "keep, change, flip", then multiply
Authentic AQA evidence
Algebraic fractions appeared on Jun 2024 Paper 2 Q19(a) and Nov 2022 Paper 3 Q22 (both Higher). The AQA examiner reports flag these as "challenging for many" — largely due to sign errors when combining and students cancelling terms rather than factors.
Factors vs Terms — The Cancellation Rule
This is the single most important idea in the whole lesson. Get it wrong once and the entire question collapses.
The Rule
You can only cancel factors — things that multiply the whole numerator or denominator. You can never cancel terms — parts of a sum.
CANCEL ✓ (factors)(x+3)(x−3)(x+3)(x+2) = x−3x+2
(x + 3) is a factor (multiplies the whole top and whole bottom) — you can cancel it.
DO NOT CANCEL ✗ (terms)x² + 4x² + 2 ≠ 42 = 2
x² is a term (added to something) — you cannot cancel it. Test: at x = 1, the LHS = 5/3, not 2.
Quick sanity check
Before you cancel anything, ask: "Is this thing multiplying the entire numerator?" If you can see a plus or minus sign next to it, it's a term — leave it alone.
Part 1 Simplifying
The Method
Factorise the numerator (common factor, DOTS, quadratic — any technique from 2.03 and 2.04).
Factorise the denominator.
Cancel any factor appearing in both.
Tools you'll need — from previous lessons
Common factor (Lesson 2.03): e.g. 2x² + 6x = 2x(x + 3)
DOTS (Lesson 2.03): e.g. x² − 9 = (x + 3)(x − 3)
Quadratic factorisation (Lesson 2.04): e.g. x² + 5x + 6 = (x + 2)(x + 3)
Combined skills — very common AQA pattern
Numerator with a common factor + denominator that's a quadratic:
2x + 4x² + 5x + 6 = 2(x + 2)(x + 2)(x + 3) = 2x + 3
Pull out the 2 on top; factorise the quadratic on the bottom; cancel (x + 2). The leftover 2 stays in the numerator.
01Worked Example — Simplify
Simplify:x² − 9x² + 5x + 6[3 marks]
Step 1 — Factorise numerator
x² − 9 = x² − 3² = (x + 3)(x − 3) (DOTS)
Step 2 — Factorise denominator
x² + 5x + 6 → product 6, sum 5 → 2 and 3 → (x + 2)(x + 3)
Step 3 — Cancel the common factor (x + 3)
(x + 3)(x − 3)(x + 2)(x + 3) = x − 3x + 2
Mark scheme split
M1 for factorising numerator OR denominator correctly. M1 for factorising both. A1 for the fully simplified final form.
MCE Worked Example — Simplify
Simplify:
x² − 4x² − x − 6
[3 marks]
Method
Factorise the numerator (DOTS) and the denominator (monic quadratic). Cancel the shared factor.
Calculate
Top: x² − 4 = (x + 2)(x − 2).
Bottom: x² − x − 6 → product −6, sum −1 → −3 and +2 → (x + 2)(x − 3).
Evaluate
= (x + 2)(x − 2)(x + 2)(x − 3) = x − 2x − 3
Note on domain
The cancellation is valid only when x + 2 ≠ 0, i.e. x ≠ −2. AQA doesn't usually require this to be stated, but it's worth knowing why the original fraction is undefined at x = −2.
Part 2 Adding & Subtracting
Just like numerical fractions: you can only combine numerators when the denominators match.
The Method
Factorise every denominator (so the true common denominator is visible).
Find the common denominator — the product of all distinct factors.
Rewrite each fraction over the common denominator by multiplying top and bottom by whatever's missing.
Combine numerators over the common denominator — then simplify if possible.
Analogy — numerical fractions
13 + 14 = 412 + 312 = 712
Same moves with algebra: find a common denominator (here 12 = 3 × 4, the product of denominators), rewrite numerators, add.
02Worked Example — Add Two Fractions
Express as a single fraction in its simplest form:3x + 1 + 2x − 1[3 marks]JUN 2024 • P2 • Q19 PATTERN
Step 1 — Identify the common denominator
The two denominators (x + 1) and (x − 1) share no common factor, so the common denominator is their product: (x + 1)(x − 1).
Step 2 — Rewrite both fractions with this denominator
3(x − 1)(x + 1)(x − 1) + 2(x + 1)(x + 1)(x − 1)
Step 3 — Combine numerators; expand
3(x − 1) + 2(x + 1)(x + 1)(x − 1) = 3x − 3 + 2x + 2(x + 1)(x − 1)
Step 4 — Collect like terms in the numerator
= 5x − 1 (x + 1)(x − 1)
Sign-tracking warning
The Jun 2024 P2 Q19(a) examiner report flagged sign errors when distributing across brackets as the dominant reason students lost marks. Use brackets when rewriting numerators — write 3(x − 1), not 3x − 1, then expand carefully.
03Worked Example — Subtract Two Fractions
Express as a single fraction:3x + 2 − 2x + 1[3 marks]
Step 1 — Common denominator = (x + 2)(x + 1)
3(x + 1)(x + 2)(x + 1) − 2(x + 2)(x + 2)(x + 1)
Step 2 — Combine numerators (mind the minus!)
3(x + 1) − 2(x + 2)(x + 2)(x + 1)
Step 3 — Expand carefully (the −2 multiplies BOTH terms)
3x + 3 − 2x − 4 = x − 1= x − 1 (x + 2)(x + 1)
The subtraction trap
When subtracting, the minus sign belongs to every term in the second numerator. Write − 2(x + 2) with the bracket, expand as −2x − 4. A common slip is writing −2x + 4 (forgetting to distribute the minus).
Part 3 Multiplying
The Method
Factorise every numerator and denominator.
Cross-cancel — any factor in a numerator can cancel with the same factor in any denominator.
Multiply straight across (numerators together, denominators together).
Example: x² − 4 x + 1 × x + 1 x − 2
Factorise: x² − 4 = (x + 2)(x − 2). Rewrite:
(x + 2)(x − 2)x + 1 × x + 1x − 2
Cross-cancel: (x + 1) appears in a numerator and a denominator. So does (x − 2). Cancel both pairs:= x + 2
Why cancel before multiplying?
If you multiply out first, you get (x² − 4)(x + 1) / [(x + 1)(x − 2)] and then still have to factorise and cancel. Cross-cancelling early keeps numbers small and mistakes few.
Part 4 Dividing
"Keep, Change, Flip"
Division is multiplication by the reciprocal — identical to numerical fractions.
Keep the first fraction.
Change ÷ to ×.
Flip the second fraction (swap numerator and denominator).
Then factorise, cross-cancel, and multiply straight across as usual.
Example: x² − 9 x² − 4 ÷ x + 3 x − 2
Keep, change, flip:
x² − 9x² − 4 × x − 2x + 3
Factorise x² − 9 and x² − 4 (both DOTS):
(x + 3)(x − 3)(x + 2)(x − 2) × x − 2x + 3
Cross-cancel (x + 3) and (x − 2):= x − 3x + 2
MCE Worked Example — Divide
Simplify:
x² + 5x + 6x² − 4 ÷ x + 3x − 2
[3 marks]
Method
Keep-change-flip to turn division into multiplication. Then factorise every numerator and denominator and cross-cancel.
Calculate
KCF: x²+5x+6x²−4 × x−2x+3
Factorise:
x²+5x+6 = (x+2)(x+3)
x²−4 = (x+2)(x−2)
(x+2)(x+3)(x+2)(x−2) × x−2x+3
Evaluate
Cross-cancel (x+2), (x+3), (x−2).
= 1
All factors cancelled?
Yes — every factor on the top had a match on the bottom, leaving just 1. This can happen with cleverly-designed questions and is a legitimate final answer. The answer is a number, not an expression.
Examiner Alerts
Alert 1 — Cancelling terms, not factors
Students write:x² + 4x² + 2 = 42 = 2 — cancelled x² across sums.
Write instead: this expression does not simplify. x² + 4 and x² + 2 share no common factor. Only cancel when you can see (factor)(factor) top and bottom.
Alert 2 — Denominator not factorised before adding
Students write:3x + 1 + 2x² − 1 using (x + 1)(x² − 1) as the common denominator.
Write instead: first factorise x² − 1 = (x + 1)(x − 1). The true common denominator is (x + 1)(x − 1) — much smaller. Always factorise every denominator first.
Alert 3 — Losing the minus sign on subtraction
Students write: 3(x + 1) − 2(x + 2) = 3x + 3 − 2x + 4 — forgot to distribute the minus.
Write instead: −2(x + 2) = −2x − 4. The minus belongs to both terms. Sign errors when expanding brackets are consistently flagged in AQA examiner reports on algebraic fractions questions — including Jun 2024 P2 Q19(a).
Common Errors
Students often writeCorrect versionWhy it matters
x² + 4 x² + 2 = 2Does not simplifyx² is a term, not a factor — can never be cancelled across a sum
3 x+1 + 2 x−1 = 5 (x+1)+(x−1)= 5x − 1 (x+1)(x−1)Can't "add denominators" — must use a common one and rewrite numerators
3(x + 1) − 2(x + 2) = 3x + 3 − 2x + 4= 3x + 3 − 2x − 4Minus belongs to every term of the second bracket
For 3 x+1 + 2 x²−1 , used (x+1)(x²−1) as the common denominatorFactorise x²−1 = (x+1)(x−1) first; common denom = (x+1)(x−1)Missing the factorisation step makes the denominator bigger than it needs to be
Flipped the wrong fraction on divisionKeep the first, flip the secondA ÷ B = A × (1/B), not (1/A) × B
Practice Question 1
Simplify:
x² + 7x + 12x² + 5x + 4
[3 marks]
M
Method. Factorise both quadratics. Cancel the common factor.
C
Calculate.
Top: x² + 7x + 12 → product 12, sum 7 → 3 and 4 → (x + 3)(x + 4).
Bottom: x² + 5x + 4 → product 4, sum 5 → 1 and 4 → (x + 1)(x + 4).
Rewrite: (x + 3)(x + 4) / [(x + 1)(x + 4)].
Cancel (x + 4).
E
Evaluate. Final answer: (x + 3)/(x + 1).
Practice Question 2
Express as a single fraction in its simplest form:
2x + 3 + 5x + 1
[3 marks]
M
Method. Common denominator = (x + 3)(x + 1). Rewrite each fraction. Combine numerators.
C
Calculate.
= 2(x + 1)/[(x + 3)(x + 1)] + 5(x + 3)/[(x + 3)(x + 1)]
= [2(x + 1) + 5(x + 3)] / [(x + 3)(x + 1)]
= [2x + 2 + 5x + 15] / [(x + 3)(x + 1)]
= (7x + 17) / [(x + 3)(x + 1)]
E
Evaluate. Final answer: (7x + 17) / [(x + 3)(x + 1)].
Practice Question 3
Simplify:
x² − 9x + 2 × x + 2x − 3
[3 marks]
M
Method. Factorise everything. Cross-cancel. Multiply what remains.
C
Calculate.
x² − 9 = (x + 3)(x − 3) (DOTS).
Rewrite: [(x + 3)(x − 3)]/(x + 2) × (x + 2)/(x − 3).
Cross-cancel (x + 2) and (x − 3).
E
Evaluate. Final answer: x + 3.
Lesson Summary HIGHER
The golden rule
Factorise before cancelling. Only cancel factors (things multiplying the whole numerator/denominator), neverterms (parts of a sum).
The four operations
OperationMethod
SimplifyFactorise top + bottom, cancel common factors
Add / subtractFactorise every denominator, find common denominator, rewrite, combine numerators
MultiplyFactorise all, cross-cancel, multiply straight across
DivideKeep, change, flip — then multiply as above
Top 3 traps to avoid
Cancelling terms, not factors — (x² + 4)/(x² + 2) does NOT equal 2
Losing the minus sign on subtraction — −2(x + 2) = −2x − 4, not −2x + 4
Denominator not factorised first — missing a smaller common denominator makes the whole thing harder
Next lesson
Lesson 2.06 — Substitution & Using Formulae: back to F+H territory. We leave algebraic manipulation behind and move on to putting numerical values into algebraic expressions and formulae.
Next: What You'll Learn HIGHER