Chapter 2, Problem 104CR

To determine

To find a polynomial function with real co-efficient that has zeros $-6,-i$ .

Answer to Problem 104CR

The required polynomial function is $r(x)=x^3+9x^2+17x-27$ .

Explanation of Solution

Given information:

Given a polynomial function with real co-efficient that has zeros $1,-5+\sqrt{2}i$

Let $r(x)$ be a polynomial that has a zero $b$ of multiplicity $m$ then

  $r(x)={\left(x-b\right)}^m$

Since the given polynomial has zeros $1,-5+\sqrt{2}i$ then $\begin{array}{l}r(x)=a{\left(x-\left(1\right)\right)}^m{\left(x+5-\sqrt{2}i\right)}^n{\left(x+5+\sqrt{2}i\right)}^n\\ =a{\left(x-\left(1\right)\right)}^m{\left({\left(x+5\right)}^2-{\left(\sqrt{2}i\right)}^2\right)}^n\\ =a{\left(x-1\right)}^m{\left({\left(x+5\right)}^2+2\right)}^n\end{array}$

Where $a$ is a real number as polynomial contains real co-efficient.

For simplicity, we can take $a=1,m=n=1$

  $\begin{array}{l}r(x)=\left(x-1\right)\left({\left(x+5\right)}^2+2\right)\\ \Rightarrow r(x)=\left(x-1\right)\left(x^2+10x+25+2\right)\\ =\left(x-1\right)\left(x^2+10x+27\right)\\ =x^3+9x^2+17x-27\end{array}$

Hence, the required polynomial function is $r(x)=x^3+9x^2+17x-27$