Problem 89 - Algebra and Trigonometry

Question

Explain how to simplify a rational expression.

Step-by-Step Solution

Answer

To simplify a rational expression, factor the numerator and the denominator, cancel out common factors, and write down the simplified result. The result is '1' if all factors were removed, the remaining factors in the numerator if only the numerator has remaining factors, '1' divided by the remaining factors in the denominator if only the denominator has remaining factors, and the remaining factors in the numerator divided by the remaining factors in the denominator if both the numerator and the denominator have remaining factors.

Step 1: Factoring the Numerator and the Denominator

Begin by factoring the numerator and the denominator separately. This breaks down the expression into its simplest components.

Step 2: Identify Common Factors

After factoring, both the numerator and the denominator should be composed of a number of simpler factors. Check if there are any factors that appear both in the numerator and denominator. These are the common factors.

Step 3: Cancel Out Common Factors

Common factors that appear in both the numerator and the denominator can be cancelled. This means if the same factor appears in both the numerator and the denominator, it can be removed.

Step 4: Write the Simplified Expression

After removing all the common factors, write down the simplified expression. If every factor was removed, the result is simply '1'. If there are remaining factors in the numerator only, the result is just those factors. If there are remaining factors in the denominator only, the result is '1' divided by those factors. If there are remaining factors in both the numerator and the denominator, the result is those factors in the numerator divided by those factors in the denominator.