Chapter 4.3, Problem 41AYU

To determine

To find: To determine the polynomial where zeros are $4+i,4-i,2+i$

Answer to Problem 41AYU

The given facts are contradictory.

Explanation of Solution

Given information:

Zeros are $4+i,4-i,2+i$

Calculation:

Let, polynomial

  $\begin{array}{l}f\left(x\right)=\left\{x-\left(4+i\right)\right\}\left\{x-\left(4-i\right)\right\}\left\{x-\left(2+i\right)\right\}\\ =\left\{\left(x-4\right)-i\right\}\left\{\left(x-4\right)+i\right\}\left\{x-2-i\right\}\\ =\left\{{\left(x-4\right)}^2-i^2\right\}\left\{x-2-i\right\}\\ =\left\{{\left(x-4\right)}^2+1\right\}\left\{\left(x-2\right)-i\right\}\left[\text{Since, }{i}^2=-1\right]\end{array}$

Clearly here, the polynomial is degree $3$ but the co-efficient of the polynomial is not real.

Since, one complex number $\text{'}i\text{'}$

Present,

So, the given facts are contradictory.