Question

The equation $a_8{x}^8+a_7{x}^7+a_6{x}^6+...+a_0=0$ has all its roots positive and real $(where{a}_8=1,a_7=-4,a_0={\frac{1}{2}}^8)$, then

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let the roots be ${\alpha }_1,{\alpha }_2,....,{\alpha }_8$

$\Rightarrow {\alpha }_1+{\alpha }_2+....+{\alpha }_8=4{\alpha }_1{\alpha }_2....{\alpha }_8=\frac{1}{2^8}\Rightarrow ({\alpha }_1{\alpha }_2....{\alpha }_8{)}^{\frac{1}{8}}=\frac{1}{2}=\frac{{\alpha }_1+{\alpha }_2+...+{\alpha }_8}{8}$

$\Rightarrow$ AM = GM $\Rightarrow$ all the roots are equal to $\frac{1}{2}$

$\Rightarrow a_1={-}^8{C}_7{\left(\frac{1}{2}\right)}^7=-\frac{1}{2^4}{a}_2{=}^8{C}_6{\left(\frac{1}{2}\right)}^6=-\frac{7}{2^4}{a}_3={-}^8{C}_5{\left(\frac{1}{2}\right)}^5$