Students factor higher-degree real-coefficient polynomials using common factors, grouping, identities, substitution, the Rational Root Theorem, the Factor Theorem, and polynomial division, and solve equations for real and nonreal complex roots, including repeated roots and multiplicities. They perform long division and synthetic division by linear divisors, using placeholders for missing powers, and apply the Remainder Theorem to evaluate polynomials and connect remainders, factors, and zeros. They determine rational-function restrictions, holes, vertical, horizontal, and slant asymptotes, solve rational equations and inequalities through domain and sign analysis, and sketch polynomial functions using zeros, multiplicities, degree, leading coefficient, intercepts, and end behavior.