Chapter 8.5, Problem 28WE

To determine

To find: the polynomial with the provided numbers as roots.

Answer to Problem 28WE

the polynomial obtained is $x^3-5x^2+11x-15$

Explanation of Solution

Given information: the numbers as roots provided in the question,

  $3,1+2i,1-2i$

Calculation:

Each root r of the polynomial has the factor x − r , -2 is a double root, thus, it needs to be squared,

  $\begin{array}{l}\left(x-3\right)\left(x-\left[1+2i\right]\right)\left(x-\left[1-2i\right]\right)\\ =\left(x-3\right)\left(x-1-2i\right)\left(x-1+2i\right)\\ =\left(x-3\right)\left[\left(x-1\right)-2i\right]\left[\left(x-1\right)+2i\right]\\ =\left(x-3\right)\left[{\left(x-1\right)}^2+4\right]\\ =\left(x-3\right)\left(x^2-2x+5\right)\\ =x^3-2x^2+5x-3x^2+6x-15\\ =x^3-5x^2+11x-15\end{array}$

Hence, the polynomial obtained is $x^3-5x^2+11x-15$