Sheet № 22 · Higher only · AQA · Edexcel · OCR
Algebraic Fractions –
Algebraic fractions work exactly like numerical fractions — the only difference is that the numerators and denominators contain algebraic expressions instead of plain numbers. This topic sits on the Higher tier and draws together skills from factorising, expanding, and solving equations. AQA, Edexcel, and OCR all test algebraic fractions,
§Key definitions
Question:
Simplify (x² − 4)/(x² + 5x + 6).
Answer:
(x − 2)/(x + 3)
Q1:
Simplify (6x²)/(9x³).
Q2:
Simplify (x² − x − 6)/(x² − 9).
Q3:
Solve 5/(x − 2) − 3/(x + 1) = 1.
§Formulas to memorise
To simplify: factorise numerator and denominator, then cancel common factors.
To add or subtract: find a common denominator, rewrite each fraction, then combine numerators.
To multiply: multiply numerators together and denominators together, simplifying where possible.
To divide: flip the second fraction and multiply.
Factorise the numerator — fully.
Factorise the denominator — fully.
Cancel any factors — that appear in both.
Find the lowest common denominator (LCD). — If the denominators are (x + 1) and (x − 2), the LCD is (x + 1)(x − 2).
Multiply each fraction — so that both have the LCD.
Expand the numerators — if needed.
Worked example
Simplify (x² − 4)/(x² + 5x + 6).
Working:
⚠ Common mistakes
- ✗Cancelling terms instead of factors. In (x + 5)/x, you cannot cancel the x. You can only cancel when the numerator and denominator are fully factorised and share a common factor.
- ✗Forgetting to multiply every term by the LCD. When clearing fractions in an equation, the right-hand side (even if it is just a number) must also be multiplied.
- ✗Sign errors when expanding. In 2/(x − 3) becoming 2(x + 1)/[(x + 1)(x − 3)], make sure you expand 2(x + 1) correctly as 2x + 2, not 2x + 1.
- ✗Not checking for excluded values. If your solution makes a denominator zero, it must be rejected. State this explicitly in your answer.
- ✗Leaving the answer un-simplified. After adding fractions, check whether the resulting numerator and denominator share a common factor.
✦ Exam tips
- →Factorise before you cancel. Never try to cancel terms by crossing out parts of expressions. Always factorise fully first.
- →For simplification questions, show each factorisation step. Examiners want to see that you have factorised, not just written the final answer.
- →When solving equations with algebraic fractions, the resulting equation is often quadratic. Be ready to factorise or use the quadratic formula.
- →These questions often carry 4-5 marks on Higher papers. The marks are distributed across identifying the LCD, forming the equation, and solving correctly. Do not skip steps.