15.2 Simplifying algebraic fractions — Learn

Prerequisites: factorise quadratics and cancel common numerical factors.

Objectives: factorise a numerator or denominator, cancel common factors, and state excluded values.

An algebraic fraction has an algebraic expression in its numerator, denominator, or both. Factorising writes an expression as a product. You can cancel common factors, but you cannot cancel separate terms joined by addition or subtraction.

Example 1. Simplify $\frac{x^2-9}{x^2+6x+9}$.

Factorise the numerator: $x^2-9=(x-3)(x+3)$. Factorise the denominator: $x^2+6x+9=(x+3)(x+3)$. Cancel the common factor $x+3$: $\frac{x-3}{x+3}$. The original denominator is zero at $x=-3$, so $x\neq-3$.

Example 2. Simplify $\frac{6x^2}{9x}$.

Write numerator and denominator as products: $\frac{2\cdot3\cdot x\cdot x}{3\cdot3\cdot x}$. Cancel common factors: $\frac{2x}{3}$. The original denominator requires $x\neq0$.

Watch out. You cannot cancel the $x$ terms in $\frac{x+4}{x+2}$. They are parts of sums, not common factors. At $x=1$, the fraction is $5/3$, not $1$. Cancel whole common factors only.

Progress check

Explain why the simplified expression can look defined at an excluded input. The restriction comes from the original denominator.

You should now be able to

• factorise the numerator and denominator;
• cancel whole common factors;
• state values excluded by the original denominator.