Finding Intercepts

Descartes Rules of Signs

Descartes Rules of Signs is used to determine the number of positive and/or negative real roots of a polynomial function. The rules are:

  • Maximum number of positive real roots of a polynomial equation f(x) is equal to the number of sign changes in f(x).
  • Maximum number of negative real roots of a polynomial equation f(x) is equal to the number of sign changes in f(-x).

Example Determine the number of real roots of

f (x) = x3 + 3x2 ? 12x + 3 = 0

    • Notice the signs are +, +, -, +. There are two sign changes as we move from left to right. Therefore, this cubic polynomial can have at most two positive real roots.

    • To find f(?x), replace every x with negative x.
      f(-x) = (?x)3+ 3(-x)2 -12(-x) + 3 = 0
      f(-x) = ?x3+ 3x2 +12x + 3 = 0

    • Notice the signs are -, +, +, +. There is one sign change as we move from left to right. Therefore, the polynomial equation can have at most one negative root.