Chapter 8 Notes

Theorems about Roots• Number of Roots Theorem: Every polynomial withcomplex coefficients and positive degree n has exactly nroots.• Conjugate Root Theorem: If a polynomial equation withreal coefficients has a + bi as a root (a & b ∈R, b ≠ 0),then a - bi is also a root.• Descartes’ Rule of Signs: Let P(x) be a simplifiedpolynomial with real coefficients and terms arranged indecreasing degree of x.– The number of positive real roots of P(x) = 0 equals thenumber of variations of sign of P(x) or is fewer thanthis number by an even integer.– The number of negative real roots of P(x) = 0 equals thenumber of variations of sign of P(- x) or is fewer thanthis number by an even integer.