Polynomial Functions and Complex Zeros
A 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. 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.