Rational Expressions: Simplification, Domain, and Arithmetic

Analyze algebraic expressions by interpreting terms, coefficients, factors, and structural equivalence in real-world modeling contexts.

Perform addition, subtraction, and multiplication on polynomials of any degree and justify why polynomials form a system closed under these operations.

Derive, verify, and apply special product identities, Pascal's Triangle, and the Binomial Theorem to expand algebraic expressions efficiently.

Factor quadratic, cubic, and quartic polynomials completely using greatest common factors, grouping, quadratic form substitutions, and special identities.

Divide polynomials using long division and synthetic division, and apply the Remainder and Factor Theorems to determine roots and factors.

Analyze the relationship between zeros, multiplicities, end behavior, and graph shapes to sketch and interpret polynomial functions.

Identify domain restrictions, simplify rational expressions by factoring, and perform addition, subtraction, multiplication, and division.

Simplify complex fractions, rewrite improper rational expressions into quotient-remainder form, and solve applied rational modeling problems.