Rational Functions and Zeros
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA rational function is a polynomial over a polynomial, and each floor of the fraction has one job. The numerator owns the zeros: the function can only output 0 where its top is 0. The denominator owns the domain: its zeros are inputs where the function does not exist, which become asymptotes or holes, never x-intercepts. A candidate zero that also kills the denominator is struck from the list, because a function with no value there cannot equal zero there.
The same division of labor decides inequality endpoints: numerator zeros can be included when equality is allowed, denominator zeros never can. The classic mistakes are swapping the two jobs, and counting a shared zero as a crossing when it is really a hole. Both get drilled in the lesson.
The work
3 ways in · any order
Lesson
Rational Functions and Zeros
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Zeros come from the numerator; the denominator only decides where the function exists. The lesson works the two-step discipline (solve the top, check the bottom), the hole that steals a zero, and inequality endpoints, then closes with ten scenarios on telling the two jobs apart.
Diagnostic
10-item topic check
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Ten items spanning the Topic 1.8 misconceptions: reading denominator zeros as zeros of the function, counting a shared zero that is really a hole, and including forbidden endpoints in rational inequalities. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.