Question

Find all the roots of the equation $4x^4-24x^3+57x^2+18x-45=0$, if one of them is $3+i\sqrt{6}.$

• A. $3\pm i\sqrt{6};\pm \frac{\sqrt{3}}{2}$
• B. $\pm 3+i\sqrt{6};\pm \frac{\sqrt{3}}{2}$
• C. $3\pm i\sqrt{6};\frac{\sqrt{3}}{2}$
• D. $\pm 3+i\sqrt{6};-\frac{\sqrt{3}}{2}$

Answer 1

The correct option is A $3\pm i\sqrt{6};\pm \frac{\sqrt{3}}{2}$

Since the co-efficients of $4x^4-24x^3+57x^2+18x-45=0$ are real complex roots will occur in conjugate pairs.
So if $3+i\sqrt{6}$ is a root, then $3-i\sqrt{6}$ is also a root.
So a quadratic factor of $4x^4-24x^3+57x^2+18x-45$ is $x^2-6x+15$.
By dividing we get the other factor as $4x^2-3$ so the other roots are $\pm \frac{\sqrt{3}}{2}$