Rational Expressions and Algebraic Fractions

Master polynomial operations, algebraic long division, the Remainder and Factor Theorems, and factoring techniques for higher-degree polynomials.

Simplify, add, subtract, multiply, and divide complex rational expressions and simplify compound algebraic fractions.

Decompose proper and improper rational expressions into partial fractions with distinct linear, repeated linear, and irreducible quadratic denominators.

Analyze quadratic forms using discriminant analysis, solve equations reducible to quadratics, and optimize objective functions via completing the square.

Solve simultaneous linear and nonlinear systems in two variables algebraically and interpret solutions geometrically.

Solve single-variable linear and quadratic inequalities, graph two-variable inequality regions, and express solution sets using interval and set notation.

Expand binomial expressions with positive integer powers using Pascal's triangle, combinatorics, and sigma notation.

Expand (1 + x)ⁿ and (a + bx)ⁿ for any rational exponent n, determine radius of convergence, and perform numerical approximations.