Chapter 5.6, Problem 27PPSE

Solution

To calculate : The number of complex roots, the possible number of real roots and the possible rational roots for the given equation.

The given equation has five complex roots. The possible number of real roots is one, three or five and the required possible rational roots are $\pm 1,\pm 2,\pm 3,\pm 6,\pm 9,\pm 18$ .

Given information :

The equation is: $x^5-x^3-11x^2+9x+18=0$

Calculation :

Given equation $x^5-x^3-11x^2+9x+18=0$ has five complex roots.

So, the possible number of real roots is one, three or five.

The standard form of the only possible roots of an equation is

  $\frac{\text{factor of constant term}}{\text{factor of leading coefficient}}$

Here, the constant term factors of the given equation are $\pm 1,\pm 2,\pm 3,\pm 6,\pm 9,\pm 18$ and the leading coefficient factors is $\pm 1$ .

Therefore, the required possible rational roots of the given equation are $\pm 1,\pm 2,\pm 3,\pm 6,\pm 9,\pm 18$ .