Simplify (x² − 16)/(x − 4), including restrictions.
Incorrect answer: x + 4, all real x
Cancellation works on factors, not terms joined by addition. A canceled denominator factor still excludes its zeros from the original expression’s domain.
Losing restrictions when canceling factors
Cancel common factors, but keep exclusions from the original denominator. Simplification does not fill a hole in the domain.
The correction
x + 4, with x ≠ 4
Factor the numerator as (x − 4)(x + 4).
- Set the original denominator unequal to zero and record excluded values.
- Factor both polynomials completely.
- Cancel common factors, not individual terms, while keeping the original restrictions.
Try the reasoning in another example
Simplify (6x²)/(3x).
- The original denominator excludes x = 0.
- Divide coefficients and cancel one factor of x.
Answer: 2x, with x ≠ 0
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