Chapter 5, Problem 50E

Solution

To calculate : The additional roots of a polynomial $P\left(x\right)$ that has rational coefficients.

There are two additional root of the given polynomial and are given by $\text{3}i\text{ and }-7i$ .

Given information :

Some of theroots are $-3i\text{ and }7i$ .

Formula Used:

Conjugate Root Theorem: If the coefficients of the polynomial $P\left(x\right)$ are real numbers, then the roots of the equation $P\left(x\right)=0$ will be conjugate pairs of complex numbers in the form of $x+iy\text{ and }x-iy$ where $x\text{ and }y$ are real numbers and if the coefficients of the polynomial $P\left(x\right)$ are rational numbers, then the roots of the equation $P\left(x\right)=0$ will be conjugate pairs of complex numbers in the form of $x+\sqrt{y}\text{ and }x-\sqrt{y}$ where $x\text{ and }y$ arereal numbers.

Calculation :

Here $-3i\text{ and }7i$ are roots of the polynomial $P\left(x\right)$ .

It is known that the complex roots of a polynomial always come together in pairs. So, the additional root is given by the conjugate complex root of $-3i\text{ and }7i$ and the roots will be $\text{3}i\text{ and }-7i$ respectively.

Therefore, there are two additional root of the given polynomial and are given by $\text{3}i\text{ and }-7i$ .