Chapter 11, Problem 3CLR

To determine

To Calculate: The equation $x^4-2x^3+4x^2+30x-13=0$ roots completely

Answer to Problem 3CLR

The required roots are $x=2\pm 3i,-1\pm \sqrt{2}$ .

Explanation of Solution

Given Information:

One of the given roots is $2+3i$ .

Calculation:

If one of the roots is $2+3i$ then the other root will be its conjugate $2-3i$ .

Writing two factors with the two roots and multiplying them gives us,

  $\begin{array}{c}(x-(2+3i))(x-(2-3i))=x^2-x(2-3i)-x(2+3i)+(4+9)\\ =x^2-2x+3ix-2x-3ix+13\\ =x^2-4x+13\end{array}$

Now dividing the equation by the result of the result of the imaginary roots we get,

  $\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}{x}^2+2x-1$

  $x^2-4x+13)\overline{x^4-2x^3+4x^2+30x-13}$

  $\underset{\_}{-x^4+4x^3-13x^2}$

  $2x^3-9x^2+30x$

  $\underset{\_}{-2x^3+8x^2-26x}$

  $-x^2+4x-13$

  $\underset{\_}{x^2-4x+13}$

  $0$

Now solving the remaining we get,

  $\begin{array}{c}0=x^2+2x-1\\ 1=(x^2+2x+1)-1\\ $ 2={(x+1)}^2$\pm \sqrt{2}=x+1\\ x=-1\pm \sqrt{2}\end{array}$

Thus, the roots are $x=2\pm 3i,-1\pm \sqrt{2}$ .

Hence, the required roots are $x=2\pm 3i,-1\pm \sqrt{2}$ .