A polynomial p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ where coefficients are real numbers and n is a non-negative integer. Degree = highest power of x. Types: monomial (1 term), binomial (2 terms), trinomial (3 terms). Linear (degree 1), quadratic (degree 2), cubic (degree 3).

A zero (root) of a polynomial p(x) is a value c such that p(c) = 0. A linear polynomial has exactly one zero. A polynomial of degree n can have at most n zeroes. Geometrically, zeroes are the x-intercepts of the graph y = p(x).

Remainder Theorem: When p(x) is divided by (x − a), the remainder is p(a). Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0. Used to find factors without long division.

Key identities: (a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², (a+b)(a−b) = a²−b², (x+a)(x+b) = x²+(a+b)x+ab, (a+b+c)² = a²+b²+c²+2ab+2bc+2ca, (a+b)³ = a³+3a²b+3ab²+b³, a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca).