Question

Use Descartes' rule of signs, Theorem 2.3, to determine the pos- sible numbers of positive, negative, and complex roots of each of the following real polynomials 8. f(x) = 3x4 +x2-2x-5 b. f(x) 4x3 - 6x2 x - 1

Answer 1

(a) F(x) = 3x^4 + x^2 - 2x -5 using Descartes rule of signs total number of sign changes = I possible numbers of positive roots = I f(-x) = 3(-x)^4 + (-x)^2 -2 (-x) -5 = 3x^4 + x^2 + 2x -5 total number of sign change = I possible number of negative roots = I possible number of complex roots = 4 - (1 + 1) = 4 - 2 = 2 (b) f(x) = 4x^3 -6 x^2 + x - 1 using Descartes rule of sign total number of sign changes = S total number of positive roots = 3 \r\n f(-x) = 4 (-x)^3 - 6 (-x)^2 + (-x) -1 = -4 x^3 -6x^2 -x -1 total number of sign changes = 0 possible number of negative roots = 0 possible numbers of complex roots = 3 - (3 + 0) = 3 - 3 = 0.