Problem

Compare the factor \(x~-~3\) in a polynomial with \(x~-~3\) in a rational denominator. Include domain status, intercept behavior, and the rational cases with and without cancellation.

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What this problem is really about

A factor's graph role depends on where it appears. In a polynomial, its zero is an allowed input that can create an intercept because polynomials are defined everywhere. In an original rational denominator, the same zero is excluded; if the factor remains, it creates a vertical asymptote, and if it cancels, the preserved exclusion appears as a hole.

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Polynomial: x=3 is allowed and gives x-intercept (3,0). Rational denominator: x=3 is also allowed and gives x-intercept (3,0), whether or not cancellation occurs.

Polynomial: x=3 is allowed and gives x-intercept (3,0). Rational denominator: x=3 is excluded and always gives a vertical asymptote, even if the factor cancels.

Polynomial: x=3 is excluded and gives a hole. Rational denominator: x=3 is excluded and always gives a hole, even when the factor does not cancel.

Polynomial: x=3 is allowed and gives x-intercept (3,0). Rational denominator: x=3 is excluded; an uncanceled factor gives a vertical asymptote, while a canceled factor gives a hole.

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Compare the factor \(x~-~3\) in a polynomial with \(x~-~3\) in a rational denominator. Include domain status, intercept behavior, and the rational cases with and without cancellation.

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