Root bounds, sign rules and polynomial proofs

Extend polynomial reasoning with Descartes’ rule of signs, synthetic upper and lower root bounds, and short algebraic proofs. Work from sign patterns and synthetic rows to justified conclusions, keeping positive roots, negative roots, zero roots and nonreal conjugate pairs distinct.

This optional precalculus practice set includes single-select and select-all problems, theorem-hypothesis checks, proof-step analysis and exact real factorization. Detailed solutions explain multiplicity, why a failed sufficient test proves less than it appears to, and how rational-root arguments can establish irrationality. No calculus is required.