Question

If $\sqrt{5}-i\sqrt{7}$ is a root of $a{x}^4+b{x}^3+c{x}^2+dx+e=0$ ,where $a,b,c,d,e$ are rational numbers, then the other roots are:

• A. $\sqrt{5}+i\sqrt{7},\frac{2}{3},\frac{1}{3}$
• B. $\sqrt{5}+i\sqrt{7},-\frac{2}{3},\frac{1}{3}$
• C. $\sqrt{5}+i\sqrt{7},-\sqrt{5}+i\sqrt{7},-\sqrt{5}-i\sqrt{7}$
• D. $\sqrt{5}+i\sqrt{7},\frac{2}{3},-\frac{1}{3}$

Answer 1

The correct option is C $\sqrt{5}+i\sqrt{7},-\sqrt{5}+i\sqrt{7},-\sqrt{5}-i\sqrt{7}$

Coefficients are rational, hence all the irrationalities and imaginary parts must be cancelled.

Hence the roots are $\sqrt{5}+i\sqrt{7},\sqrt{5}-i\sqrt{7},-\sqrt{5}+i\sqrt{7},-\sqrt{5}-i\sqrt{7}$