Chapter 4, Problem 24CR

To determine

To find: all real solutions of the given equation.

Answer to Problem 24CR

The real solutions are $-4,-3$ , and $2$ .

Explanation of Solution

Given information:

Given equation,

  $x^3+5x^2-2x-24=0$

Calculation:

Consider the equation.

  $x^3+5x^2-2x-24=0$

The above equation is of degree 3, so it has 3 solutions.

Here, the leading coefficient is $1$ and constant term is $-24$ .

The possible rational solutions are $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 8,\pm 12,\text{and}\pm 24$ .

Test possible solution using synthetic division until a solution is found.

For $x=2$

  $\begin{array}{ccccc}2& 1& 5& -2& -24\\ & & 2& 14& 24\\ & 1& 7& 12& 0\end{array}$

Therefore, $x-2$ is a factor.

Factor completely using the result of above synthetic division.

  $\begin{array}{c}{x}^3+5x^2-2x-24=0\\ \left(x-2\right)\left(x^2+7x+12\right)=0\\ \left(x-2\right)\left(x+4\right)\left(x+3\right)=0\end{array}$

Solve further,

  $\begin{array}{c}\left(x+3\right)\left(x+4\right)\left(x-2\right)=0\\ x=-4,-3,2\end{array}$

Therefore, all real solutions are $-4,-3$ , and $2$ .