Polynomials

An expression with variables having non-negative integer exponents. p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀. Degree = highest power. Monomial: one term. Binomial: two terms. Trinomial: three terms. Constant polynomial: just a number (degree 0). Zero polynomial: p(x) = 0 (undefined degree).

Value of x that makes p(x) = 0. Also called roots. A polynomial of degree n has AT MOST n zeroes. For linear p(x) = ax+b: zero is x = -b/a. For quadratic: can have 0, 1, or 2 real zeroes. Geometrically: x-intercepts of the graph of y = p(x).

When p(x) is divided by (x-a), the remainder = p(a). Example: p(x) = x³-2x²+3x-5 divided by (x-2). Remainder = p(2) = 8-8+6-5 = 1. No need to do long division! Saves a lot of calculation time.

If p(a) = 0, then (x-a) is a factor of p(x). Converse: If (x-a) is a factor of p(x), then p(a) = 0. Extension: (ax-b) is a factor of p(x) if p(b/a) = 0. Use this to factorise polynomials by trial of rational factors.