91Ó°ÊÓ

For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor. $$\frac{4 x^{2}+4 x+12}{8 x^{3}-27}$$

For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor. $$\frac{3 x^{3}+2 x^{2}+14 x+15}{\left(x^{2}+4\right)^{2}}$$

For the following exercises, use the matrices below to perform the indicated operation if not possible, explain why the operation cannot be performed. (Hint: \(A^{2}=A \cdot A )\) $$ A=\left[\begin{array}{ll}{1} & {0} \\ {2} & {3}\end{array}\right], B=\left[\begin{array}{rrr}{-2} & {3} & {4} \\ {-1} & {1} & {-5}\end{array}\right], C=\left[\begin{array}{rr}{0.5} & {0.1} \\ {1} & {0.2} \\\ {-0.5} & {0.3}\end{array}\right], D=\left[\begin{array}{rrr}{1} & {0} & {-1} \\ {-6} & {7} & {5} \\ {4} & {2} & {1}\end{array}\right] $$ \(D^{3}\)

For the following exercises, write the augmented matrix for the linear system. $$\begin{aligned} 6 x+12 y+16 z &=4 \\ 19 x-5 y+3 z &=-9 \\ x+2 y &=-8 \end{aligned}$$

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. Every day, a cupcake store sells 5,000 cupcakes in chocolate and vanilla flavors. If the chocolate flavor is 3 times as popular as the vanilla flavor, how many of each cupcake sell per day?

For the following exercises, find the determinant. $$ \left|\begin{array}{rrr}{5} & {1} & {-1} \\ {2} & {3} & {1} \\ {3} & {-6} & {-3}\end{array}\right| $$

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution. Two numbers add up to \(56 .\) One number is 20 less than the other.

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule. At a women’s prison down the road, the total number of inmates aged 20–49 totaled 5,525. This year, the 20–29 age group increased by 10%, the 30–39 age group decreased by 20%, and the 40–49 age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the 30–39 age group than the 20–29 age group. Determine the prison population for each age group last year.

For the following exercises, write a system of equations to solve each problem. Solve the system of equations. A performer charges \(C(x)=50 x+10,000,\) where \(x\) is the total number of attendees at a show. The venue charges $75 per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?

For the following exercises, decompose into partial fractions. $$ \frac{-x^{2}+36 x+70}{x^{3}-125} $$