2Rational Root Theorem give direction in testing possible zeros. Let a0xn + a1xn-1 + …an-1x + an = 0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of an and q is a factor of a0.
3Ex 1 List all the possible rational roots then determine the rational roots. 3x3 – 13x2 + 2x + 8 = 0
4Ex 2 Find ALL of the roots.x3 + 6x2 – 13x – 6 = 0
5Descartes’ Rule of Signs Used to determine the possible number of positive real zeros a polynomial has.P(x) is a polynomial in descending order. The # of positive real zeros is the same as the number of sign changes of the coefficients or is less than this by an even number.The # of negative real zeros is the same as the number of sign changes of the coefficients of P(-x), or less than by an even number.(Ignore zero coefficients.)
6Ex 3 find the number of possible positive and negative real zeros for then determine the rational zeros:P(x) = 2x4 – x3 – 2x2 + 5x + 1