## Precalculus (6th Edition) Blitzer

The simplified partial fraction expansion is $\frac{\left( -2 \right)}{3\left( x-1 \right)}+\frac{2x+13}{3\left( {{x}^{2}}+x+1 \right)}$.
The provided partial fraction expression is as follows: $\frac{3x-5}{{{x}^{3}}-1}$ Simplify this as given below: \begin{align} & \frac{3x-5}{{{x}^{3}}-1}=\frac{3x-5}{{{x}^{3}}-{{1}^{3}}} \\ & =\frac{3x-5}{\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)} \end{align} Step 1: Set up the partial fraction expansion with unknown constant coefficients and then write a constant coefficient over each of the two distinct algebraic linear factors in the denominator of the expression. $\frac{3x-5}{\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)}=\frac{A}{x-1}+\frac{Bx+C}{{{x}^{2}}+x+1}$ Step 2: Multiply both sides of the equation by the expression $\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)$ and consider the least common denominator. $\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)\times \left( \frac{3x-5}{\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)} \right)=\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)\times \left( \frac{A}{x-1}+\frac{Bx+C}{{{x}^{2}}+x+1} \right)$ Then, multiply and simplify as follows: \begin{align} & 3x-5=\left( {{x}^{2}}+x+1 \right)A+\left( Bx+C \right)\left( x-1 \right) \\ & 3x-5=\left( {{x}^{2}}+x+1 \right)A+\left( Bx+C \right)\left( x-1 \right) \\ & =A\left( {{x}^{2}}+x+1 \right)+B\left( {{x}^{2}}-x \right)+C\left( x-1 \right) \\ & ={{x}^{2}}\left( A+B \right)+x\left( A-B+C \right)+A-C \end{align} Step 3: And equating the coefficients of like powers of $x$ and of the constant terms. Then, we get the system of linear equations with the unknown values of $A,B$ and $C$. $A+B=0$ (I) $A-B+C=3$ (II) $A-C=-5$ (III) Step 5: Now solve the system for $A$, $B$, and $C$, Eliminate $C$ by adding equations (II) and (III) as given below: $\left( A-B+C \right)+A-C=3-5$ $2A-B=-2$ (IV) Eliminate $B$ by adding equations (I) and (IV) as given below: \begin{align} & \left( A+B \right)+2A-B=0-2 \\ & 3A=-2 \\ & A=-\frac{2}{3} \end{align} Substituting the values of $A$ in equation (I) and simplify as given below: \begin{align} & A+B=0 \\ & -\frac{2}{3}+B=0 \\ & B=\frac{2}{3} \end{align} Similarly, put the values of $A$ and $B$ in equation (II), and find the value of $C$ as given below: \begin{align} & A-B+C=3 \\ & -\frac{2}{3}-\frac{2}{3}+C=3 \\ & C=3+\frac{4}{3} \\ & C=\frac{13}{3} \end{align} Step 5: By replacing the values of $A$, $B$, and $C$, write the partial function expression as given below: \begin{align} & \frac{3x-5}{\left( x-1 \right)\left( {{x}^{2}}+x+1 \right)}=\frac{A}{x-1}+\frac{Bx+C}{{{x}^{2}}+x+1} \\ & =\frac{-\frac{2}{3}}{x-1}+\frac{\frac{2}{3}x+\frac{13}{3}}{{{x}^{2}}+x+1} \\ & =-\frac{2}{3\left( x-1 \right)}+\frac{2x+13}{3\left( {{x}^{2}}+x+1 \right)} \end{align} Thus, $-\frac{2}{3\left( x-1 \right)}+\frac{2x+13}{3\left( {{x}^{2}}+x+1 \right)}$ is the required partial fraction expansion of the rational expression $\frac{3x-5}{{{x}^{3}}-1}$.