Simplifying a Rational Expression

Simplifying a rational expression means factoring the numerator and the denominator completely, then canceling the factors they share.

Simplify (x² − 9)/(x² + 5x + 6). Factor both floors: the numerator is a difference of squares, (x − 3)(x + 3); the denominator's pair hunt gives (x + 2)(x + 3). The shared factor (x + 3) cancels: (x − 3)/(x + 2). Spot-check with a number: at x = 1, the original is −8/12 = −2/3 and the simplified form is −2/3 — the same value in a smaller dress. Factoring first is not optional: cancellation reads factors only, and unfactored floors show none. A different number works as well as x = 1 for the spot-check; any value that keeps both denominators alive settles agreement.