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MathidiaMathematics Academy
Advanced algebra and functions
Lesson 3 of 75
Standard practice

Algebraic fractions

Objective. Simplify and solve rational expressions with exclusions.

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Learn: the key idea

Algebraic fractions are ratios of polynomials that are simplified by factorising numerator and denominator then cancelling common factors, while excluding values that make any denominator zero. Example: (x²−9)/(x²+x−6) factorises to (x−3)(x+3)/((x+3)(x−2)), cancelling (x+3) to give (x−3)/(x−2), valid for x≠−3,2. When solving equations with fractions, multiply through by the lowest common denominator first, then solve the resulting polynomial, always rejecting any solution that was excluded.

Worked example

Simplify the algebraic fraction (x²−4)/(x²+5x+6) and state excluded values.

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Illustrated card showing 4 … 5
  1. 1Factorise numerator: x²−4=(x−2)(x+2)
  2. 2Factorise denominator: x²+5x+6=(x+2)(x+3)
  3. 3Cancel the common factor (x+2) from numerator and denominator
  4. 4State exclusions where original denominator is zero: x≠−2,−3

Answer: (x−2)/(x+3), x≠−2,−3

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Common mistake. Students cancel terms that are added rather than factors, e.g. wrongly simplifying (x+3)/x to 1+3 instead of leaving it unsimplified, since only multiplied factors can cancel.

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