Equivalent Representations of Polynomial and Rational Expressions
One function, several forms, each with its own gift. Factored form hands over the zeros; standard form hands over the leading term (end behavior) and the constant (y-intercept); dividing out a rational expression hands over the quotient, which is the end-behavior model: the polynomial the graph imitates far from the origin. Equal degrees give a horizontal asymptote at the ratio of leading coefficients, a top degree one higher gives a slant asymptote, and a smaller top gives y = 0.
Two disciplines police the rewrites. Only factors cancel, never terms joined by plus signs, and every reduction carries its domain: (x squared minus 1)/(x minus 1) is the line x + 1 with a hole, not the whole line. The classic traps, dividing leading coefficients no matter the degrees, banning graphs from crossing their asymptotes, and slashing terms, all get named and drilled in the lesson.
The work
3 ways in · any order
Lesson
Equivalent Representations of Polynomial and Rational Expressions
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Factored, standard, and divided forms each show one feature at sight. The lesson derives the asymptote rules from long division so they cannot be misremembered, enforces factors-only cancellation with domain notes, then closes with ten scenarios on picking and trusting the right form.
Diagnostic
10-item topic check
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Ten items spanning the Topic 1.11 misconceptions: term-by-term cancellation, leading-coefficient division applied at the wrong degrees, and asymptotes treated as uncrossable fences. The bank ships with the Unit 1 data drop.
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 the misconception and moves you to the next.