Calculate: (x^2+3x+2)/(x^3)

Expression: $\frac{ {x}^{2}+3x+2 }{ {x}^{3} }$

For each factor in the denominator, write a new fraction using the factors as new denominators. The numerators are unknown values

$\frac{ ? }{ {x}^{3} }$

Since the factor $x$ is raised to the power of $3$, it is a repeating factor, so include each power from $1$ to $3$ as denominators of separate fractions

$\frac{ ? }{ x }+\frac{ ? }{ {x}^{2} }+\frac{ ? }{ {x}^{3} }$

Since the factor in the denominator is linear, the numerator is an unknown constant $A$

$\frac{ A }{ x }+\frac{ ? }{ {x}^{2} }+\frac{ ? }{ {x}^{3} }$

Since the denominator is a linear factor raised to a power, the numerator is an unknown constant $B$

$\frac{ A }{ x }+\frac{ B }{ {x}^{2} }+\frac{ ? }{ {x}^{3} }$

Since the denominator is a linear factor raised to a power, the numerator is an unknown constant $C$

$\frac{ A }{ x }+\frac{ B }{ {x}^{2} }+\frac{ C }{ {x}^{3} }$

To get the unknown values, set the sum of fractions equal to the original fraction

$\frac{ {x}^{2}+3x+2 }{ {x}^{3} }=\frac{ A }{ x }+\frac{ B }{ {x}^{2} }+\frac{ C }{ {x}^{3} }$

Multiply both sides of the equation by ${x}^{3}$

${x}^{2}+3x+2=A{x}^{2}+Bx+C$

When two polynomials are equal, their corresponding coefficients must be equal

$\left\{\begin{array} { l } 2=C \\ 3=B \\ 1=A\end{array} \right.$

Solve the system of equations

$\left( A, B, C\right)=\left( 1, 3, 2\right)$

Substitute the given values into the formed partial-fraction decomposition

$\frac{ 1 }{ x }+\frac{ 3 }{ {x}^{2} }+\frac{ 2 }{ {x}^{3} }$