Question

Consider the cubic $f\left(x\right)=8x^3+4a{x}^2+2bx+a$, where $a,b,\in R$.

If the sum of the base $2$ logarithms of the roots of the cubic $f\left(x\right)=0$ is $5$ then the value of '$a$' is

• A. $-64$
• B. $-8$
• C. $-128$
• D. $-256$

Answer 1

The correct option is D $-256$

Let $x_1,x_2,x_3$ be the roots of cubic equation $8x^3+4a{x}^2+2bx+a=0$

Thus by theory of equation, $x_1\cdot x_2\cdot x_3=\frac{-a}{8}$......(1)

It is also given that, ${\log }_2{x}_1+{\log }_2{x}_2+{\log }_2{x}_3=5$

$\Rightarrow {\log }_2{x}_1{x}_2{x}_3=5,[\because \log a+\log b+\log c=\log (abc\left)\right]$

$\Rightarrow x_1{x}_2{x}_3=2^5=32$

$\Rightarrow -\frac{a}{8}=32$, ....using (1)

$\Rightarrow a=-256$