Chapter 8.6, Problem 7ST

To determine

To find: The polynomial equation.

Answer to Problem 7ST

The equation is $x^3+6x+20=0$

Explanation of Solution

Given information:

The integral coefficient is given as:

  $-2\text{ and }3+i$

Concept used:

Factor theorem:

The polynomial p( x ) has $r-x$ as a factor if r is a root of equation p( x )

Calculation:

The factor can determine as:

By the factor theorem, the required polynomial must have factors $\left[x-\left(-2\right)\right],\left[x-\left(1+3i\right)\right]\text{ and }\left[x-\left(1-3i\right)\right]$ .

Therefore a solution is,

  $\begin{array}{l}\left(x+2\right)\left(x-1-3i\right)\left(x-1+3i\right)=0\\ \left(x+2\right)\left[{\left(x-1\right)}^2-{\left(3i\right)}^2\right]=0\\ \left(x+2\right)\left[\left(x^2-2x+1\right)+9\right]=0\text{ As }{i}^2\text{ is equal to }-1\text{ and use }{\left(a-b\right)}^2\text{ formula}\text{.}\\ \left(x+2\right)\left(x^2-2x+10\right)=0\\ x^3-2x^2+10x+2x^2-4x+20=0\\ x^3+6x+20=0\end{array}$

Therefore,

The equation is $x^3+6x+20=0$