Cancel Factors while Keeping the Hole

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Simplify (x²−4)/(x−2) and keep its original domain. Candidate answers: (A) x+2, for every real x (B) x+2, with x≠2 (C) x+4, with x≠2 (D) x−2, with x≠2.
• Independent check inputs: Simplify (x²−9)/(x−3) and keep its original domain. Candidate answers: (A) x+3, for every real x (B) x+3, with x≠3 (C) x+9, with x≠3 (D) x−3, with x≠3.

Learn the idea

Rational expressions divide polynomials, so denominator zeros are excluded from their domains. Factoring can reveal cancellation or help solve equations, but it does not erase the original restrictions or turn inverse variation into linear change. A shared polynomial factor can be canceled only where it is nonzero. The simpler formula agrees with the original expression on its original domain, which may retain a missing point.

Here is the method on the supplied teaching case. Factor the numerator and cancel the common nonzero factor x−2. The new formula is x+2, but the original exclusion x=2 remains; cancellation cannot fill that missing input.

Cancellation applies to factors, not terms of a sum. Retain every original denominator exclusion in the simplified answer; a new formula without that restriction would represent a different domain.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Simplify (x²−81)/(x−9) and keep its original domain. Candidate answers: (A) x+81, with x≠9 (B) x+9, for every real x (C) x+9, with x≠9 (D) x−9, with x≠9. Show the mathematical reason, not only the selected answer. Expected result: x+9, with x≠9. Factor the numerator and cancel the common nonzero factor x−9. The new formula is x+9, but the original exclusion x=9 remains; cancellation cannot fill that missing input.

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