Chapter 8.6, Problem 10WE

To determine

To solve: The equation.

Answer to Problem 10WE

So the solution set of the given equation is,

  $\left\{i\sqrt{5},-i\sqrt{5}\text{ and }\frac{1}{2}\right\}$

Explanation of Solution

Given information:

The equation is given as:

  $2x^3-x^2+10x-5=0$ and a root of the equation is also given $i\sqrt{5}$

Calculation:

Let $p\left(x\right)=2x^3-x^2+10x-5=0$

If $i\sqrt{5}$ is a root of $p\left(x\right)=0$ , then $-i\sqrt{5}$ is also a root.

Therefore, $x-\left(i\sqrt{5}\right)$ and $x-\left(-i\sqrt{5}\right)$ are factors of $p\left(x\right)$

Their product,

  $\left[x-\left(i\sqrt{5}\right)\right]\left[x-\left(-i\sqrt{5}\right)\right]=x^2+5$

is also a factor of $p\left(x\right)$ ; that is $p\left(x\right)=x^2+5\cdot Q\left(x\right)$

Use long division, $Q\left(x\right)=p\left(x\right)÷x^2+5=2x-1$

The resulting equation is $2x-1=0$ , and its root is $x=\frac{1}{2}$

So the solution set of the given equation is,

  $\left\{i\sqrt{5},-i\sqrt{5}\text{ and }\frac{1}{2}\right\}$