Algebraic Fractions

Algebraic fractions behave in the same way as numerical fractions. You need to simplify, add, subtract, multiply and divide them fluently at A-Level. The Edexcel 9MA0 specification requires confident manipulation of algebraic fractions, including improper fractions from polynomial division.

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Simplifying Algebraic Fractions

To simplify an algebraic fraction, factorise the numerator and denominator, then cancel common factors.

Example: Simplify (x² − 9)/(x² + 5x + 6).

• Numerator: x² − 9 = (x + 3)(x − 3)
• Denominator: x² + 5x + 6 = (x + 2)(x + 3)
• Cancel (x + 3): (x − 3)/(x + 2)

Example: Simplify (2x² + 7x + 3)/(x² + 3x).

• Numerator: 2x² + 7x + 3 = (2x + 1)(x + 3)
• Denominator: x² + 3x = x(x + 3)
• Cancel (x + 3): (2x + 1)/x

Exam Tip: Never cancel individual terms — only cancel common factors. For instance, in (x + 3)/(x + 5), you cannot cancel the x's.

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Multiplying Algebraic Fractions

Multiply numerators together and denominators together, cancelling common factors first.

Example: Simplify (x² − 4)/(x + 1) × (x + 1)/(x + 2).