Rewrite each expression in standard polynomial form.

a. 3x(4 - 2x) + 8(x$^2$ - 5x)

b. (x - 5)(4x$^2$ + 7x - 9)

c. (5x$^3$ - 7x$^2$ + 5) - (x$^3$ +10x - 12)

d. $\frac{4 x^3-6 x^2+2 x}{2 x}$

Fifty randomly selected families in a neighborhood in a large city were surveyed and each family was asked how many vehicles were registered at that address. The results of the survey are summarized in the table below.

$$\begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Number of } \\ \text { Vehicles } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Families } \end{array} & \begin{array}{c} \text { Percentage of } \\ \text { Families } \end{array} \\ \hline 0 & 17 & \\ \hline 1 & 23 & \\ \hline 2 & 7 & \\ \hline 3 & 2 & \\ \hline 4 & 1 & \\ \hline \end{array}$$

a. Explain what the 23 in the table indicates.

b. Use the data in the table to compute the average number of vehicles registered per address in this neighborhood.

c. Determine the percentage of families in the neighborhood that had no vehicles registered at their address. Then fill in the rest of that column of the table with percentages. Enter each percentage as a decimal number between 0 and 1. The percentage of times that a data value occurs is also called the relative frequency.

d. You can also find the average number of vehicles per address using the relative frequencies. In the formula below, f(x) is the relative frequency of the data value x. Use this formula to find the average number of vehicles registered per address.

Mean $=\sum(f(x) \cdot x)$, where x takes on all possible data values.

Your answer here should match the one you got in Part b.

e. What is the median number of vehicles registered per address in this neighborhood?

f. Which measure of center is most appropriate for these data? Explain your reasoning.

Consider the following numbers:

$$\begin{array}{lllllllll} 123 & \frac{2}{3} & \pi & -\frac{8}{4} & \sqrt[3]{8} & 31.72 & -53 & 6 \frac{3}{8} & \sqrt{7} \end{array}$$

a. Which of the numbers are whole numbers?

b. Which of the numbers are integers?

c. Which of the numbers are rational numbers?

d. Which of the numbers is irrational numbers?

e. Order the numbers from smallest to greatest.

Write each expression in a simpler equivalent exponential form using only positive exponents.

a. (3x$^6$)$^2$

b. (3$x^6$)(5x$^2$)

c. $\frac{10 x^6}{4 x^2}$

d. $\frac{-2 x^2}{8 x^6}$

e. 14(7x)$^{-1}$

f. (3x$^{-6}$)(2x$^4$)

A survey of 648 residents of Summit found that 480 of them thought the town should have a 4th of July parade.

a. What is the parameter that this survey was trying to estimate?

b. What is the point estimate of the proportion of residents of Summit who think the town should have a 4th of July parade?

c. Compute the margin of error for this survey. Then write a sentence explaining what it means in this context.

In recent years, the number of students enrolled in colleges and universities in the United States has been increasing linearly. In 2001, there were approximately 15.9 million students enrolled in U.S. colleges and universities (including community colleges). In 2012, total U.S. college and university enrollment had increased to approximately 19.8 million students.

a. Using the information above, write a function rule that can be used to estimate student enrollment in U.S. colleges and universities. Let t represent the number of years since 2000 and E(t) represents the number of students in millions.

b. What is the slope of the graph of this function? Be sure to include units of measure in your answer. What does it tell you about student enrollment in U.S. colleges and universities?

c. What is the y-intercept of the graph of this function? What does it tell you about student enrollment in U.S. colleges and universities?

d. Use your function to estimate college and university enrollment in 2015.

e. Use your function to determine the year during which the enrollment will first be greater than 23 million.

At the beginning of the school year, the 186 seniors at Sterling High School were surveyed and asked about whether they plan to go to college and if they had a summer job. Some of the results are shown in the chart below.

$$\begin{array}{|l|c|c|c|} \hline& \text { Had Summer Job } & \text { No Summer Job } & \text { Total } \\ \hline \text { College Plans } & & & \\ \hline \text { No College Plans } & 24 & & 72 \\ \hline \text { Total } & & 56 & 186 \\ \hline \end{array}$$

a. Fill in the remaining cells on a copy of the chart.

Suppose one of the seniors at Sterling High School is randomly chosen. Determine each of the following probabilities.

b. P(college plans)

c. P(had summer job)

d. P(college plans and had summer job)

e. P(college plans or had summer job)

f. What is the probability that someone who had college plans also had a summer job? (This can be written in symbols as P(had summer job | college plans). The symbol | means "given that" and restricts the group under consideration. So in this case, rather than determining the probability based on all of the students, we are only considering the 114 students who planned to go to college.)

g. P(college plans | had summer job)

Factor each of the following expressions into linear factors.

a. y$^2$ - 9

b. 3q$^2$ - 20q - 7

c. ax$^2$ - 5ax + 4a

d. 4(p - 1) + k(p - 1)

e. 6x$^2$ + 3x - 63

Often it is valuable to rewrite numbers in different but equivalent forms. Doing so may provide insight into properties of the number and it can also allow you to identify relationships between numbers.

a. Rewrite each number as a product of prime numbers.

i. 18

ii. 64

iii. 140

iv. 196

v. 320

b. Determine the greatest common divisor for each pair of numbers.

i. 18 and 6

ii. 196 and 64

iii. 64 and 320

iv. 140 and 320

v. 320 and 18

c. How can you use the prime factorization of a pair of numbers to help you determine the greatest common divisor of the numbers?

Fill in each blank with always, sometimes, or never in order to make the statement true. Be prepared to explain your reasoning.

a. A parallelogram is _____________ a quadrilateral.

b. A rectangle is _____________ a square.

c. A square is _____________ a rhombus.

d. A parallelogram is _____________ a rectangle.

e. A quadrilateral with exactly two lines of symmetry is _____________ a kite.

Each bank in the U.S. has an identification number. This ID number is usually printed at the bottom-left of all checks from the bank. The first eight digits of the ID number are information digits, which identify the particular bank. The ninth digit is a check digit. Suppose the eight information digits are $a_1, a_2, a_3, a_4, a_5, a_6, a_7$, and $a_8$. The check digit is chosen so that it is the last digit of $7a_1 + 3a_2 + 9a_3 + 7a_4+ 3a_5+ 9a_6+ 7a_7+ 3a_8$.

a. A bank in Iowa has the following ID number: 073901877. Verify that this number satisfies the check-digit system described above.

b. Explain why choosing the check digit as described above is the same as choosing the check digit so that it is equivalent to the indicated sum mod 10.

c. Examine the bottom of a check from a U.S. bank to find the nine-digit bank ID number, and verify that the ID number satisfies the check-digit system described in this task.

The elements in a set often share a common property. In such cases, you can define the set in terms of that property. There is a specific notation, called set-builder notation, that is used to do this. For example, the set of all positive numbers can be written as on the right.

a. Suppose a set A is described in words as follows:

• A is the set containing all numbers x such that x is greater than 2 and less than 10.

Write A in set-builder notation. (Use the diagram above as a model. The vertical line segment | is read "such that.")

b. List the elements in each of the sets below.

i. S = {x | x is an integer and -2 $\leq$ x $\leq$ 5}

ii. P = {x | x = 2$^n$, for n=0, 1, 2, or 3}

iii. M = {x | x is the name of a month that has exactly 30 days }

c. Finish writing each of the following sets using set-builder notation.

i. S = {1, 4, 9, 16, 25} = {x | x = n$^2$, for n = $\ldots$}

ii. E = {2, 4, 6, 8, 10, $\ldots$} = {x | $\ldots$}

iii. P = {triangle, quadrilateral, pentagon, hexagon } = {x | $\ldots$}

Look closely at the series of long and short black bars in the POSTNET barcode at the bottom of the business reply card. Each long bar represents a 1 and each short bar represents a 0. The first and last long bars are not used in the code; they are only used to define the beginning and end.

a. What five-bit string is represented by the first five bars in the code? Is this a 2-out-of-5 string?

b. Write all the five-bit strings represented by the bars. Separate each five-bit string with a space so that you can easily distinguish them.

c. Use the POSTNET Binary Code Dictionary in the earlier problem to translate the five-bit strings into decimal digits. Compare the result to the ZIP code printed in the address on the card. Does it match? If not, how is it different?

To translate numbers into characters in Problem 1, you first needed to convert any given integer into an integer between 0 and 26. This is an example of what is called modular arithmetic. In particular, you are using a mod 27 system, since you are limited to the 27 integers from 0 to 26. It is possible to do arithmetic in this system. For example, 2 + 3 = 5 in mod 2, just as in regular arithmetic. But some computations in mod 27 give results that are quite different than those in integer arithmetic.

Perform the following computations in mod 27. Compare your reasoning and results with other classmates. Resolve any differences.

a. Explain why 25 + 9 = 7 in in mod 27.

b. 18 + 14 = ? in mod 27

c. 3 $\times$ 25 = $\underline{\ \ ?\ \ }$ in mod 27

d. 7$^2$ = $\underline{\ \ ?\ \ }$ in mod 27

e. 4 $\times$ 5 = $\underline{\ \ ?\ \ }$ in mod 27

f. -5 = $\underline{\ \ ?\ \ }$ in mod 27