Polynomial Functions and Complex Zeros
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA degree-n polynomial owns exactly n complex zeros, counted with multiplicity, but the graph only displays the real ones. This topic is the dictionary that keeps those books balanced: real zeros are linear factors are x-intercepts, multiplicity decides whether the graph crosses or just touches at each intercept, and non-real zeros travel in conjugate pairs that leave no mark on the axis at all. It closes with the symmetry tests, f(−x) = f(x) for even and f(−x) = −f(x) for odd, that graphs and tables get quizzed on all year.
The traps are structural, not computational: counting x-intercepts as if they equaled the degree, expecting a bounce where the graph actually crosses, reversing the sign in the zero-to-factor dictionary, and promoting "degree 4" into "even function." The lesson names each one and drills it.
The work
3 ways in · any order
Lesson
Polynomial Functions and Complex Zeros
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Zeros, factors, and intercepts are one dictionary; multiplicity sets crossing versus touching; conjugate pairs keep the count at n without showing on the graph. The lesson builds each rule, then closes with ten scenarios reading factored forms, graphs, and symmetry claims.
Diagnostic
10-item topic check
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Ten items spanning the three Topic 1.5 misconceptions: multiplicity behavior misread at intercepts, complex zeros expected to appear on the graph, and even or odd symmetry claimed from the degree alone. 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.