Chapter 4.4, Problem 19E

To determine

To find: The number of possible positive real zeros and the number of possible negative real zeros for function $f\left(x\right)=x^3+3x^2-10x-24$ . Then determine the rational zeros.

Answer to Problem 19E

Possible zeros: $\pm 1\text{, }\pm \text{2, }\pm 3\text{, }\pm 4,\pm \text{6, }\pm \text{8, }\pm \text{12, }\pm \text{24}$

  $\text{Zeros: }x=-3,-2,4$

Explanation of Solution

Given information: $f\left(x\right)=x^3+3x^2-10x-24$

Calculation:

  $f\left(x\right)=x^3+3x^2-10x-24$

Possible zeros: $\pm 1\text{, }\pm \text{2, }\pm 3\text{, }\pm 4,\pm \text{6, }\pm \text{8, }\pm \text{12, }\pm \text{24}$

Descartes Rule of Signs gives us:

1 positive root and 0 or 2 negative roots.

Using synthetic division:

  $\left(x+2\right)\left(x^2-x-12\right)\to$ Factoring the quadratic

  $\begin{array}{l}\left(x+2\right)\left(x-4\right)\left(x+3\right)\\ \text{Zeros: }x=-3,-2,4\end{array}$