Polynomial Division - The Factor and Remainder Theorems

Factor and Remainder Theorems, Long Division and Synthetic Division

In this module we will study two important results on polynomials - the Factor Theorem and the Remainder Theorem.

The Remainder Theorem asserts that the value P(c) of a polynomial at a point c is the remainder on dividing the polynomial P(x) by (x - c). The quantity (x - c) divides P(x) evenly if and only if P(c) = 0, in which case, (x - c) is a factor of P(x).

The Factor Theorem asserts that c is a zero of a polynomial if and only if (x - c) is a factor of the polynomial.

Learning Objectives:

  • Learn long division method to divide a polynomial by another polynomial
  • Learn synthetic division method to divide a polnomial by a linear factor (x - c)
  • Use the Remainder Theorem
  • Use the Factor Theorem to factor a polynomial into linear and quadratic factors
  • Find zeros of a polynomial