Question

Let $P\left(x\right)$ be a cubic polynomial with zeroes $\alpha ,\beta ,\gamma$ if $\frac{P\left(\frac{1}{2}\right)+P(-\frac{1}{2})}{P\left(0\right)}=100$find $\sqrt{\frac{1}{\alpha \beta }+\frac{1}{\beta \gamma }+\frac{1}{\gamma a}}$.

Answer 1

Let $P\left(x\right)=a{x}^3+b{x}^2+cx+d$
So $P\left(\frac{1}{2}\right)+P(-\frac{1}{2})=2b\left(\frac{1}{4}\right)+2d$
and $P\left(0\right)=d$
$\Rightarrow \frac{2\frac{b}{4}+2d}{d}=100$
$\Rightarrow \frac{2b}{4d}+2=100\Rightarrow \frac{b}{d}=196$
also $\alpha +\beta +\gamma =-\frac{b}{a},\alpha \beta \gamma =-\frac{d}{a}$
Hence $\sqrt{\frac{1}{\alpha \beta }+\frac{1}{\beta \gamma }+\frac{1}{\gamma a}}=\sqrt{\frac{\alpha +\beta +\gamma }{\alpha \beta \gamma }}=\sqrt{196}=14$