Year 10: Simplification of rational algebraic expressions

What is a rational algebraic expression?

A rational algebraic expression is an expression that can be written as a fraction where the numerator and denominator are polynomials. It's essentially a fraction involving variables and exponents. Example: $(x^2+2x)/(x+1)$

Steps to simplify

1. Factor the numerator and denominator separately: This is the MOST important step. Find the greatest common factor (GCF) of all terms.
2. Cancel Common Factors: Once factored, identify and cancel out any factors that appear in both the numerator and denominator.
3. Check for Extraneous Solutions (If Applicable): This is crucial when dealing with rational equations. Ensure you're not cancelling factors that would make the denominator zero, leading to undefined expressions.

Examples

Example 1: Simplify $(x^2-9)/(x+3)$

1. Factor: $(x+3)(x-3)$

2. Cancel: $(x-3)$

Result: $x-3$

Example 2: Simplify $(2x^2+6x)/(x^2-1)$

1. Factor: $2(x+3)(x-1)$

2. Cancel: $\left(2\right(x-1\left)\right)$

Result: $2(x-1)$

Important notes

• Always simplify before any calculations.
• Be careful when factoring. Mistakes in factoring lead to incorrect simplification.
• Remember that you can only cancel factors that are in both the numerator and denominator.