5. The algebra of polynomials and cubic equations

Abstract

Polynomials are expressions of the form p(x) = a0 + a1x + a2x2 + ... + anxn; the number ai is the coefficient of xi, a 0 is the constant term of p(x), and an is the leading coefficient. The underlying algebra of polynomials mirrors that of the integers. Polynomials can be added, subtracted, and multiplied, and the laws of associativity and commutativity and the distributive law of addition over multiplication all hold. Division is more complicated. ‘The algebra of polynomials and cubic equations’ outlines the Remainder and Factor Theorems along with complex numbers in the Argand plane. The factorization of polynomials, the Rational Root Theorem, the Conjugate Root Theorem, and solution of cubic equations are also discussed.