Watch out
Cancelling terms across addition or subtraction.
Scottish National 5 Mathematics
National 5 Maths Algebraic fractions
Simplify and use all four operations with algebraic fractions while respecting excluded values.
Formula reference
Check the National 5 rules and formulae linked to this topic.
Which method?
Match exam clues to a suitable method.
Before you start
- Factorise numerators and denominators where possible.
- State excluded values when a denominator could be zero.
- Use common denominators for adding and subtracting.
Explanation
Algebraic fractions behave like number fractions, but the top and bottom contain algebra.
Simplifying usually starts by factorising, then cancelling common factors. You may cancel factors, not individual terms.
When adding or subtracting, use a common denominator before combining the numerator.
Visual support
Cancel factors, not terms
= (x − 3)
= x + 3
x ≠ 3
Key formulae and rules
- + = (ad + bc) / bd
- = x + 3, x ≠ 3
Check
Multiply the simplified expression back by the cancelled factors and keep every excluded denominator value.
Exam tip
Only cancel common factors. If there is a plus or minus sign, factorise before cancelling.
Worked examples
Worked example 1
Simplify by factorising
Simplify
- Factorise the numerator: x² − 9 = (x − 3)(x + 3)
- Write the fraction as (x − 3)
- Cancel the common factor x − 3.
Answer: x + 3, where x ≠ 3
Worked example 2
Multiply algebraic fractions
Simplify ×
- Multiply numerators and denominators.
- Numerator: 3x × 8 = 24x
- Denominator: 4 × 6x = 24x
- = 1, assuming x ≠ 0
So: 1.
Worked example 3
Add with a common denominator
Simplify +
- Common denominator is x(x + 1).
- = 2(x + 1)
- = (x + 1)
- Add numerators: 2x + 2 + 3x.
Answer: (x + 1)
Watch out
- Cancelling terms across addition or subtraction.
- Forgetting to factorise first.
- Not using brackets around a whole numerator.