Chapter 4.4, Problem 11E

To determine

To list: The possible rational roots of equation $x^3-2x^2+x+18=0$ and then determine the rational roots.

Answer to Problem 11E

0 or 2 positive roots, 1 negative root

Explanation of Solution

Given information: $x^3-2x^2+x+18=0$

Calculation:

  $x^3-2x^2+x+18=0$

Starting simple, you can find the first root at $x=-2$

  $\frac{x^3-2x^2+x+18}{x+2}=x^2-4x+9$

  $x^2-4x+9\text{ has}$ 2 irrational roots

The possible roots are all factors of 18.

  $\pm 1,\pm 2,\pm \text{3, }\pm \text{6, }\pm \text{9, }\pm \text{18}$

Using Descartes Rule of signs there are:

0 or 2 positive roots, 1 negative root