Chapter 8.6, Problem 21WE

To determine

To write a fourth-degree polynomial equation with integral coefficients.

Answer to Problem 21WE

The required fourth-degree polynomial equation is

  $x^4-2x^3+6x^2-8x+8=0$

Explanation of Solution

Given information:

The given roots are $2i,1-i$

Concept used:

Conjugate Root Theorem

If a polynomial equation with real coefficient has $a+bi$ as root ( a and b real, $b\ne 0$ ), then $a-bi$ is also a root.

Calculation :

If $2i,1-i$ are the roots of the equation then according to conjugate root theorem $-2i,1+i$ are also the roots of the equation.

Therefore the factors of the equation are

  $\left(x-2i\right),\left(x+2i\right),\left(x-1+i\right)\text{ and }\left(x-1-i\right)$

The required equation can be formed as

  $\begin{array}{l}\left(x-2i\right)\left(x+2i\right)\left(x-1+i\right)\left(x-1-i\right)=0\\ \left(x^2+4\right)\left(x^2-2x+1+1\right)=0\left[i^2=-1\right]\\ x^2\left(x^2-2x+2\right)+4\left(x^2-2x+2\right)=0\\ x^4-2x^3+2x^2+4x^2-8x+8=0\\ x^4-2x^3+6x^2-8x+8=0\end{array}$

Therefore,

The required fourth-degree polynomial equation is $x^4-2x^3+6x^2-8x+8=0$