Algebraic fractions
Cancel factors, and retain the values excluded by the original denominator.
A factor multiplies the whole numerator or denominator. A term belongs to a sum. You can cancel the common factor x − 1 in a product, but you cannot cancel individual x terms across addition.
For addition, use a common denominator and expand the complete new numerator. For division, multiply by the reciprocal, checking that the divisor is not zero. Simplification can conceal a hole in the original function, so state exclusions before cancellation.
Worked example
Simplify (x² − 1)/(x² + x).
- Factor the numerator as (x − 1)(x + 1).
- Factor the denominator as x(x + 1); exclude x = 0 and x = −1.
- Cancel x + 1 where it is nonzero.
Answer: (x − 1)/x, with x ≠ 0, −1
Revise first: Expanding and factorising.
Practise algebraic fractions
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 2.9: Algebraic fractions
- 7M20 · core · A2.1: Algebraic fractions
- 6993 · core · AL2: Algebraic fractions
- 8365 · core · 2.9: Add rational expressions with original exclusions
- 7M20 · core · A2.1: Add rational expressions with original exclusions
- 6993 · core · AL2: Add rational expressions with original exclusions
- 8365 · core · 2.9: Subtract rational expressions with original exclusions
- 7M20 · core · A2.1: Subtract rational expressions with original exclusions
- 6993 · core · AL2: Subtract rational expressions with original exclusions
- 8365 · core · 2.9: Multiply rational expressions with original exclusions
- 7M20 · core · A2.1: Multiply rational expressions with original exclusions
- 6993 · core · AL2: Multiply rational expressions with original exclusions
- 8365 · core · 2.9: Divide rational expressions and exclude a zero divisor
- 7M20 · core · A2.1: Divide rational expressions and exclude a zero divisor
- 6993 · core · AL2: Divide rational expressions and exclude a zero divisor
Next practice: Domain and range.