Year 10: Simplification of rational algebraic expressions

Key concepts

• Rational expression: An expression that can be written as a fraction where the numerator and denominator are polynomials.
• Simplifying: Reducing a rational expression to its simplest form, meaning the numerator and denominator have no common factors.

Steps for simplification

1. Factor the Numerator and Denominator: Factor each polynomial expression fully. This is crucial!
2. Identify Common Factors: Look for factors that appear in both the numerator and denominator.
3. Cancel Common Factors: Divide out any common factors. Remember, you're dividing by a factor, not plugging it in.
4. Check for Extraneous Solutions: After simplifying, make sure your solution makes the original denominator zero. If it does, the solution is extraneous and invalid.

Examples

Example 1:$(x\mathrm{²}-9)/(x+3)$

Factor: $\left(\right(x+3\left)\right(x-3\left)\right)/(x+3)$

Cancel: $x-3$

Example 2:$(x\mathrm{²}+5x+6)/(x+2)$

Factor: $(x+2)(x+3)/(x+2)$

Cancel: $x+3$