Chapter 4, Problem 32SGA

To determine

Find the possible rational roots of the equation.

Answer to Problem 32SGA

  $-\frac{1}{4}$

Explanation of Solution

Given information:

  $4x^3+x^2+8x+2=0$

Calculation:

Here, we have:

  $4x^3+x^2+8x+2=0$

Now, according to the rational root theorem, if $\frac{p}{q}$ is a root of the equation, then $p$ is a factor of $2andq$ is a factor of $4$ .

Thus,

The possible values of p:

  $\pm 1,\pm 2$

The possible values of $q$ :

  $\pm 1,\pm 2,\pm 4$

The possible rational roots $\frac{p}{q}$ :

  $\pm 1,\pm 2,\pm \frac{1}{2},\pm \frac{1}{4}$

Now, we know that all the possible rational roots fall in the domain $-2\le x\le 2$ .

Thus, set the $x-axis$ viewing window at $[-3,3]$ .

Now, graph the function:

  

Now, by synthetic division method:

  $\begin{array}{lllll}-\frac{1}{4}\hfill & 4\hfill & 1\hfill & 8\hfill & 2\hfill \\ \hfill & \hfill & -1\hfill & 0\hfill & -2\hfill \\ \hfill & 4\hfill & 0\hfill & 8\hfill & 0\hfill \end{array}$

Therefore, $4x^3+x^2+8x+2=\left(x-\frac{1}{4}\right)\left(4x^2+8\right)$ .

Now, solving for $x^3+x^2+1$ :

The other roots are $4x^2+8$ .

  $\begin{array}{l}4x^2+8\\ =4\left(x^2+2\right)\\ =4\left(x^2-{\left(\sqrt{2}i\right)}^2\right)\\ =4\left(x+\sqrt{2}i\right)\left(x-\sqrt{2}i\right)\end{array}$

The roots of the equation are $-\sqrt{2}i,\sqrt{2}iand-\frac{1}{4}$ .

Hence, the rational root of the equation is $-\frac{1}{4}$ .