Question

If the sum of two of the roots of $x^4-2x^3-3x^2+10x-10=0$ is zero then the roots are

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

Answer 1

The correct option is A $\pm \sqrt{5},1\pm i$

let roots are $\pm a,b,cb+c=2-a^{2}bc=-10a^{2}bc=10{-a}^2+ab+ac+bc-ab-ac=-3bc-a^2=-3a^2-bc=3$

let $bc=t$

from $\left(2\right)t=\frac{10}{a^2}\left(3\right)a^2-t=3a^2-\frac{10}{a^2}=3a^4-{3a}^2-10=0a^4-{5a}^2+{2a}^2-10=0a^2(a^2-5)+2(a^2-5)=0(a^2-5)(a^2+2)=0a=\pm \sqrt{5}bc=2c=\frac{2}{b}b+\frac{2}{b}=2b^2-2b+2=0$

$b=1\pm i$

$c=\frac{2}{b}=\frac{2}{1\pm i}=1\mp i$

$\therefore$ roots are $1\pm i,\pm \sqrt{5}$