Chapter 8.5, Problem 32WE

To determine

To find:

The solution of given equation with the help of given two indicated roots

Answer to Problem 32WE

The solution of the given equation is $x=\pm i$

Explanation of Solution

Given information:

The equation is $2x^4-3x^3-3x-2=0$

The root of the equation is $2,-\frac{1}{2}$

Calculation:

To find the depressed equation, divide $P\left(x\right)=2x^4-3x^3-3x-2$ by $x-\left(-\frac{1}{2}\right)\text{ or }x+\frac{1}{2}$

  $\begin{array}{l}-\frac{1}{2}\overline{)\begin{array}{l}\text{2 }-3\text{ 0 }-3-2\\ -1\text{ 2 }-1\text{ 2}\end{array}}\\ \text{ 2 }-4\text{ 2 }-4\overline{)\text{0}}\end{array}$

So, $2x^4-3x^3-3x-2=\left(x+\frac{1}{2}\right)\left(2x^3-4x^2+2x-4\right)$ and the depressed equation is $2x^3-4x^2+2x-4$

Again, to find the depressed equation divide, $P\left(x\right)=2x^3-4x^2+2x-4$ by $x-2$

  $\begin{array}{l}2\overline{)\begin{array}{l}\text{2 }-4\text{ 2 }-4\\ \text{ 4 0 4}\end{array}}\\ \text{ 2 0 2 }\overline{)\text{0}}\end{array}$

So, the depressed equation is $2x^2-2$

This equation is in the form $a^2+b^2$ , hence it can be factored as $\left(a+bi\right)\left(a-bi\right)$

  $2x^2+2=2\left(x+i\right)\left(x-i\right)$

To find the remaining factor set the above factor equal to zero and solve for $x$

  $\begin{array}{l}2x^2=2\\ x=\pm i\end{array}$

Therefore, the solution of the given equation is $x=\pm i$