Simplifying to Lowest Terms (Without Losing the Holes)

Cancel factors, never terms

Reducing a numeric fraction means dividing top and bottom by a common factor: 6/8 = 3/4. With rational expressions it is identical, except the common factor is a whole polynomial factor. So the first move is always factoring the numerator and denominator completely, then cancelling any factor they share.

A worked reduction

Simplify   (x^2 - 9) / (x^2 + 7x + 12)

1. State excluded values from the ORIGINAL denominator:
     x^2 + 7x + 12 = (x + 3)(x + 4) = 0  ->  x ≠ -3, x ≠ -4

2. Factor top and bottom:
     numerator   x^2 - 9      = (x - 3)(x + 3)   [difference of two squares]
     denominator x^2 + 7x + 12 = (x + 3)(x + 4)

3. Cancel the common factor (x + 3):
     (x - 3)(x + 3) / [(x + 3)(x + 4)] = (x - 3) / (x + 4)

Result:  (x - 3)/(x + 4),  with  x ≠ -3  and  x ≠ -4

The factor (x−3) on top is a difference of two squares, a special product worth recognizing on sight. After cancelling, the simplified form (x−3)/(x+4) is an equivalent expression — equal to the original at every input where both are defined.

Why the hole must be carried along

The original expression had two excluded values, −3 and −4. The simplified (x−3)/(x+4) looks like it only forbids −4. But the two forms are only truly equal where the original lived, so x = −3 stays excluded — it is a removable gap, a hole. Writing the restriction next to the answer keeps the lowest-terms form honest.

1. Record excluded values from the original denominator first.
2. Factor numerator and denominator completely.
3. Cancel only shared whole factors.
4. Report the reduced expression together with all original restrictions.