Chapter 8.6, Problem 5WE

To determine

To solve: The equation.

Answer to Problem 5WE

The remaining root is $-2i$

Explanation of Solution

Given information:

The equation is given as:

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

Concept used:

Conjugate theorem,

If a polynomial equation with real coefficient has $a-bi$ as a root $\left(a\text{ and }b\text{ real, }b\ne 0\right)$ , then $a-bi$ is also a root.

Calculation:

Let $p\left(x\right)=x^3-3x^2+4x-12=0$

If $2i$ is a root of $p\left(x\right)=0$ , then $-2i$ is also a root

The answer can be checked by:

  $\begin{array}{l}{\left(-2i\right)}^3-3{\left(-2i\right)}^2+4\left(-2i\right)-12\\ 8i+12-8i-12\\ 0\end{array}$

Hence,

The answer is verified.