Question

If the equation $x^4-4x^3+a{x}^2+bx+1=0$ has four positive roots, then the value of $(a+b)$ is:

• A. $-4$
• B. $2$
• C. $6$
• D. cannot be determined

Answer 1

The correct option is B $2$

Given, $x^4-4x^3+a{x}^2+bx+1=0$

let the root of equation be $\alpha ,\beta ,\gamma ,\sigma$

$\alpha +\beta +\gamma +\sigma =4$ ...(i)

$\alpha \beta \gamma \sigma =1$ ... (ii)

$\frac{1}{4}(\alpha +\beta +\gamma +\sigma )=1$

$\Rightarrow \frac{1}{4}(\alpha +\beta +\gamma +\sigma )=\left(\alpha \beta \gamma \sigma \right)\frac{1}{4}$

$\therefore A.M.=a.m.$

$\therefore \alpha =\beta =\gamma =\sigma$

$4\alpha =4$

$\therefore \alpha =1$

$1-4+a+b+1=0$

$\therefore a+b=2$