Algebraic fractions
Objective. Simplify and combine algebraic fractions with stated restrictions.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
An algebraic fraction has variables in the numerator, denominator, or both, and this topic covers simplifying, adding and multiplying them like numerical fractions. To simplify, factorise the numerator and denominator fully and cancel any common factors, remembering that cancelling is only valid for common multiplying factors, not terms being added. To add fractions with different denominators, find a common denominator just as with numbers, then combine numerators. State any values of x that would make a denominator zero as restrictions.
Worked example
Simplify the algebraic fraction (x² − 9)/(x² + x − 12) fully, stating any restriction on x.
- 1Factorise the numerator: x² − 9 = (x − 3)(x + 3).
- 2Factorise the denominator: x² + x − 12 = (x + 4)(x − 3).
- 3Cancel the common factor (x − 3) from top and bottom.
- 4Result: (x + 3)/(x + 4), with x ≠ 3 and x ≠ −4.
Answer: (x + 3)/(x + 4), x ≠ 3, x ≠ −4
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils cancel terms that are added rather than multiplied, such as cancelling the x in (x + 3)/x to leave just 3. Only common factors that multiply the whole numerator and denominator can be cancelled.
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