Fundamental theorem of algebra and conjugate zeros

Fundamental theorem of algebra and conjugate zeros

Explore how degree accounts for every complex root of a polynomial, including repeated roots, and why nonreal zeros of a real-coefficient polynomial occur in conjugate pairs. This precalculus practice connects root lists with linear and quadratic factors.

Questions distinguish complex from nonreal numbers, count distinct roots separately from multiplicity, and reconstruct real polynomials with prescribed zeros. Examples include nonreal roots with nonzero real parts, repeated conjugate pairs and a fourth-degree polynomial that factors into real quadratics.

Prerequisites are complex-number arithmetic, factoring, polynomial division and exact root finding. Single-select and multiselect questions progress from theorem interpretation to construction and justification. Each stem states the coefficient assumptions needed for its conclusion.

Check a constructed polynomial by expanding its factors and accounting for the full degree. Worked explanations show why an omitted conjugate or repeated factor changes the problem. A counterexample with complex coefficients makes the theorem's boundary explicit. Use scratch work to develop the argument before viewing choices; recognizing the correct argument is useful preparation for later independent proof and polynomial construction.