Question

lf $\alpha ,\beta ,\gamma$ are the roots of $x^3+2x^2-3x-1=0$, then the value of ${\alpha }^{-4}+{\beta }^{-4}+{\gamma }^{-4}$ is:

• A. $123$
• B. $36$
• C. $149$
• D. $795$

Answer 1

Answer: (C)

$149$

Given equation $x^3+2x^2-3x-1=0$
Since we need $\sum {\alpha }^{-4}$
${\alpha }^{-4}$ is the root of $x^3+3x^2-2x-1=0$
Multiply both sides by $x$, we get
$x^4+3x^3-2x^2-x=0$
$\sum {\alpha }^{-4}=\sum [-3x^3+2x^2+x]$
$\Rightarrow \sum {\alpha }^{-4}=-3\left(42\right)+2\left(13\right)-3$
$\Rightarrow \sum {\alpha }^{-4}=149$