Question

If there is a multiple root of order $3$ for the equation $x^4-2x^3+2x-a=0$, then the other root is

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is A $-1$

$\to$ Let the roots be $\alpha ,\alpha ,\alpha ,\beta$

Sum of roots = $\frac{-b}{a}=2=3\alpha +\beta$

sum of roots taken two at a

time = ${\alpha }^2+{\alpha }^2+{\alpha }^2+\alpha \beta +\alpha \beta +\alpha \beta =0$

$\Rightarrow 3{\alpha }^2+3\alpha \beta =0$

$\Rightarrow 3\alpha (\alpha +\beta )=0$

either $\alpha +\beta =0$

$\Rightarrow \alpha =-\beta$

$\Rightarrow -3\beta +\beta =2\Rightarrow -2\beta =2\Rightarrow \beta =-1$