Question

The coefficient of $x$ in the expansion of ${\left(1+x+x^2+x^3\right)}^{-3}$ is

• A. 6
• B. 9
• C. 5
• D. -3

Answer 1

Answer: (D)

${\left(1+x+x^2+x^3\right)}^{-3}={\left[\frac{\left(x^4-1\right)}{x-1}\right]}^{-3}$
$={\left(x^4-1\right)}^{-3}(x-1{)}^3$
$=(-1{)}^{-3}{\left(1-x^4\right)}^{-3}(x-1{)}^3$
$=(-1)\left(1+3x^4+\dots \right)\left(x^3-3x^2+3x-1\right)$
Coefficient of $x$ in ${\left(1+x+x^2+x^3\right)}^{-3}$ is $-3$