Problem preview

N-CN.8 Warmup M3-045-A10-V02

Extend polynomial identities to complex numbers for higher-degree polynomial work.

List complex roots as repeated multiplicity records

Problem

For \((x~-~(2~+~3i))^{3}(x~-~(2~-~3i))^{3}\), list every root with multiplicity, give the total degree, and relate the conjugate multiplicities to real coefficients.

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Roots 2+3i and 2-3i each have multiplicity 1; total multiplicity and degree 2, ignoring both cubes.

Both roots have multiplicity 3, but report the degree as 3 instead of total multiplicity 6.

Root 2+3i has multiplicity 3 and 2-3i has multiplicity 1; total multiplicity and degree 4.

Roots 2+3i and 2-3i each have multiplicity 3; total multiplicity and degree 6. Equal conjugate multiplicities are consistent with real coefficients.

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Curriculum context

Course: Math III
Standard: N-CN.8
Category: Number and Quantity
Domain: The Complex Number System
Objective: Extend polynomial identities to complex numbers for higher-degree polynomial work.
Problem type: List complex roots as repeated multiplicity records