Use the following expression to answer the questions below:

$(frac{1}{x+3} + frac{6}{x^2+4x+3})bulletfrac{x+3}{x+1}$

a. Simplify the expression down to one rational expression.
b. Explain the steps you used to do the simplification.
c. Are there any values $footnotesize{x}$ cannot be? Explain.
d. Find the values of $footnotesize{x}$ that make the expression equal to 0. Explain the steps you used to find these values.

To calculate the volume of a chemical produced in a day a chemical manufacturing company uses the following formula below:
$V(x) = [C_{1}(x) + C_{2}(x)](H(x))$
where $footnotesize{x}$ represents the number of units produced. This means two chemicals are added together to make a new chemical and the resulting chemical is multiplied by the expression for the holding container with respect to the number of units produced. The equations for the two chemicals added together with respect to the number of unit produced are given below:
$C_{1}(x) = frac{x}{x + 1},,,C_{2}(x) = frac{2}{x - 3}$
The equation for the holding container with respect to the number of unit produced is given below:
$H(x) = frac{x^{3} - 9x}{x}$

a. What rational expression $footnotesize{C(x)}$ do you get when you combine the two chemicals?
b. What is the simplified equation of $footnotesize{V(x)}$?
c. What would the volume be if 50, 100, or 1000 units are produced in a day?
d. The company needs a volume of 3000 $footnotesize{mbox{c}^{3}}$ How many units would need to be produced in a day?

The following diagram represents all of the stages of a shrinking iceberg. The iceberg is circular, and when it started out, it had a radius of 100 feet. The iceberg shrank to a radius that was 75% of its original size every hour. Each circle in the drawing represents the size of the iceberg in each of the successive hours it was shrinking. Using this information, answer the questions below the image:

a. Is this a geometric series or an arithmetic series? Explain why you chose this answer.
b. Describe the steps you would use to calculate the radius of the 6th circle.
c. Assuming this pattern continued forever, what would be the total length if you added all of the circles’ radii* together? Explain your answer.
*The word $footnotesize{radii}$ is the plural of $footnotesize{radius}$.

A Venn diagram is a common way for teachers and students to organize information. Have you ever seen a three-circle Venn diagram?

a) Create a Venn diagram by writing equations for three circles. Your equations do not have to create a perfect replica of this image, but your circles must overlap in a similar way. It may help you to visualize your equations if you graph them! What are the equations to your circles?
b) What are coordinates of the centers of your circles? Assign each center a name.
c)What is the distance between each pair of centers? Show how you arrived at your answer.
d) If you choose two of your circle equations and solve them, how many solutions will you find?
e)Now consider all three circles together. How many solutions would you find if you were to solve them all as one system? Explain how you know your answer is correct

Scientists have collected a sample of the bacteria responsible for an illness and determined the equation below:
$B = 100e^{(0.592t)}$
$footnotesize{B}$ is the number of bacteria after $footnotesize{t}$ hours.

a. Describe the graph of the exponential equation. Is it growth or decay? Does it have any asymptotes?
b. How much bacteria will there be in 1, 5, 10, and 24 hours?
c. We need to be able to find how long the bacteria have been present given the amount of bacteria we find. To do this we need to solve the equation for time using logs. What is the log equation to calculate time?
d. There is found to be 3500 bacteria; how long has the bacteria been growing?

You are visiting a Redwood tree forest and want to verify the height of one of the trees. You measure its shadow along the ground and use trig to calculate the height.
The shadow measures 500 feet and you calculate the angle of elevation to be $footnotesize{35^{circ}}$. This forms a right triangle.

a. What is the measure of the other acute angle?
b. What is the height of the tree?
c. You are standing at the end of the tree’s shadow and want to take a picture of the tree but your camera can only focus at distance less than 500 feet. When you hold the camera to take the picture it is 5 feet above the ground. What is the distance from the end of the shadow to the top of the tree?
d. Can you take a clear picture of the top of the tree from where you are standing?

Let’s assume the following statements are true: Historically, 75% of the five-star football recruits in the nation go to universities in the three most competitive athletic conferences. Historically, five-star recruits get full football scholarships 93% of the time, regardless of which conference they go to. If this pattern holds true for this year’s recruiting class, answer the following:

a. Based on these numbers, what is the probability that a randomly selected five-star recruit who chooses one of the best three conferences will be offered a full football scholarship?
b. What are the odds a randomly selected five-star recruit will not select a university from one of the three best conferences? Explain.
c. Explain whether these are independent or dependent events. Are they Inclusive or exclusive? Explain.

The math teacher and cheerleading coach have teamed up to help the students do better on their math test. The cheer coach, using dance move names for the positioning of their arms, yells out polynomial functions with different degrees.
For each position the coach yells out, write the shape by describing the position of your left and right arm.

a1. Constant Function:
a2. Positive Linear Function:
a3. Negative Linear Function:
a4. Positive Quadratic Function:
a5. Negative Quadratic Function:
a6. Positive Cubic Function:
a7. Negative Cubic Function:
a8. Positive Quartic Function:
a9. Negative Quartic Function:

When it comes time to take the test not only do the students have to describe the shape of the polynomial function, you have to find the number of positive and negative real zeros, including complex. Use the equation below:
$f(x) = x^{5} -  3x^{4} - 5x^{3} + 5x^{2} -  6x + 8$

b. Identify all possible rational zeros.
c. How many possible positive real zeros are there? How many possible negative real zeros? How many possible complex zeros?
d. Graph the polynomial to approximate the zeros. What are the rational zeros? Use synthetic division to verify these are correct.
e. Write the polynomial in factor form.
f. What are the complex zeros?