Consider the generating functions

$$\begin{array}{ll} A=1+x+x^2, & B=1+2 x+4 x^4+x^5, \\ C=1-x^2+x^4, & D=1+x+x^2+x^3+\cdots, \\ E=1+x^3+x^6+x^9+\cdots, & F=1-x+x^2-x^3+x^4-\cdots . \end{array}$$

In exercise given below, write each indicated expression in the form

$$a_0+a_1 x+a_2 x^2+a_3 x^3+\cdots .$$

If the expression is a polynomial, then compute it completely; otherwise compute it through the $x^7$ term.

AD

If

$$\overrightarrow{\mathrm{AB}}=\left( \begin{array}{c}{-1} \\ {3} \\ {2}\end{array}\right), \overrightarrow{\mathrm{AC}}=\left( \begin{array}{c}{2} \\ {-1} \\ {4}\end{array}\right) \quad \text { and } \quad \overrightarrow{\mathrm{BD}}=\left( \begin{array}{c}{0} \\ {2} \\ {-3}\end{array}\right)$$

find: a. $\overrightarrow{\mathrm{AD}}$ b. $\overrightarrow{\mathrm{CB}}$ c. $\overrightarrow{\mathrm{CD}}$

Consider the system of equations AX = 0 where

$$A=\left[\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right]$$

is a $2 \times 2$ matrix over the field F. Prove the following. (a) If every entry of A is 0, then every pair $\left(x_{1}, x_{2}\right)$ is a solution of AX = 0. (b) If $a d-b c \neq 0$, the system AX = 0 has only the trivial solution $x_{1}=x_{2}=0$. (c) If ad - bc = 0 and some entry of A is different from 0, then there is a solution $\left(x_{1}^{0}, x_{2}^{0}\right)$ such that $\left(x_{1}, x_{2}\right)$ is a solution if and only if there is a scalar y such that $x_{1}=y x_{1}^{0}, x_{2}=y x_{2}^{0}$.

Correlative conjunctions are paired conjunctions that connect two words or groups of words of equal significance . Correlative conjunctions\ include either... or; both...and; neither... nor; whether... or; and not only...but also.
Example : For he was lord and king of all the earth, Before poor Eve h ad either life or breath,... Rewrite below item, using correlative conjunctions to combine the sentence.
Lanier believed that men had to take responsibility for sin. She believed that women also had to take responsibility.

In the steel bars $BE$ and $AD$ each have a $6 \times 18-\mathrm{mm}$ cross section. Knowing that $E=200 \mathrm{~GPa}$, determine the deflections of points $A,B$, and $C$ of the rigid bar $ABC$., the $3.2-\mathrm{kN}$ force caused point $C$ to deflect to the right. Using $\alpha=11.7 \times 10^{-6} /{ }^{\circ} \mathrm{C}$, determine (a) the overall change in temperature that causes point $C$ to return to its original position, (b) the corresponding total deflection of points $A$ and $B$.

Proofread the following paragraphs. Write the words that are incorrectly capitalized, changing capital letters to lowercase letters and lowercase letters to capital letters where necessary. If a sentence is already correct, write $C$.

EXAMPLE: Chattanooga, Tennessee, is the seat of Hamilton county.

• County*

The city has been welcoming tourists since at least 1866, when an ad in the Chattanooga Times invited people from the north to visit with the assurance that the Ku Klux Klan had no power in Chattanooga.

$$\begin{aligned} &\text { Match the following word elements with the definitions. }\\ &\begin{array}{llll} \text { Combining Forms } & \text { } & \text { Suffixes } & \text { Prefixes } \\ \text { caud/o } & \text { kary/o } & \text {-genesis } & \text { ad- } \\ \text { dist/o } & \text { leuk/o } & \text {-gnosis } & \text { infra- } \\ \text { dors/o } & \text { morph/o } & \text {-graphy } & \text { ultra- } \\ \text { eti/o } & \text { poli/o } & & \\ \text { hist/o } & \text { somat/o } & & \\ \text { idi/o } & \text { viscer/o } & & \\ \text { jaund/o } & \text { xer/o } & & \end{array} \end{aligned}$$

unknown, peculiar _________.

The table refers to a survey of senior high school students in Dayton, Ohio. It cross-tabulates whether a student had ever smoked cigarettes and whether a student h ad ever drunk alcohol and shows counts and the conditional distributions of alcohol use.

	Alcohol
Cigarettes	Yes	No	Total
Yes	1449 (97%)	46 (3%)	1495 (100%)
No	500 (64%)	281 (36%)	781 (100%)

Find the odds of having used alcohol for users and non- users of cigarettes. Interpret each. Then, describe the strength of association, using the odds ratio.

Look over your chart, and choose one invention for your advertisement . Then answer these question to help you plan your advertisement: Who is your audience? Who will buy this invention?How will the invention benefit this audience? What words or phrased will best persuade this audience? Once you have answered these questions, design your advertisement. To draw readers' attention to your ad, include an illustration, a catchy heading , and a few lines of text.

Calculate the quantities required in the problem, where

$$\begin{aligned} &\mathbf{A}=\left[\begin{array}{rrr} {-1} & {0} & {3} \\ {2} & {1} & {2} \\ {-1} & {0} & {1} \end{array}\right], \quad \mathbf{B}=\left[\begin{array}{lll} {1} & {3} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1} \end{array}\right]\\ &\mathbf{C}=\left[\begin{array}{ll} {1} & {0} \\ {2} & {1} \\ {1} & {3} \end{array}\right], \quad \text { and } \quad \mathbf{D}=\left[\begin{array}{rrr} {3} & {-1} & {0} \\ {2} & {1} & {2} \end{array}\right] \end{aligned}$$

or explain why they are undefined. AD

Diogenes statim ad armarium contendit. in armadio erant quinque fustes, quod Diogenes extraxit et nobis tradidit. Diogenes immediately hurried to the closet. In the closet there were five clubs, which Diogenes drew out and gave to us. Diogenes had prepared for an eventual attack by stacking his closet with weapons. To be prepared for battle means to expect a battle. The situation could not have been good and peaceful in their part of town. Diogenes was expecting an attack, sooner or later.

Rewrite the following sentences, adding, deleting, or changing periods, question marks, exclamation points, and commas as necessary.

EXAMPLE: My best friend has moved to 9782 , Revere Avenue, New York NY 10465-2879

My best frieñ has moved to 9782 Revere Avenue, New York, NY 10465-2879.

Using hyperbole the store claimed in a colorful full-page newspaper ad that it would be having the "World's Most Spectacular Labor Day Sale."

$$\begin{aligned} &\text { Match the following word elements with the definitions. }\\ &\begin{array}{llll} \text { Combining Forms } & \text { } & \text { Suffixes } & \text { Prefixes } \\ \text { caud/o } & \text { kary/o } & \text {-genesis } & \text { ad- } \\ \text { dist/o } & \text { leuk/o } & \text {-gnosis } & \text { infra- } \\ \text { dors/o } & \text { morph/o } & \text {-graphy } & \text { ultra- } \\ \text { eti/o } & \text { poli/o } & & \\ \text { hist/o } & \text { somat/o } & & \\ \text { idi/o } & \text { viscer/o } & & \\ \text { jaund/o } & \text { xer/o } & & \end{array} \end{aligned}$$

yellow _________.

Use the following matrices. $$ A=\left[\begin{array}13&1\\-1&0\end{array}\right] $$ $$ B=\left[\begin{array}14&2\end{array}\right] $$ $$ C=\left[\begin{array}10\\-2\end{array}\right] $$ $$ D=\left[\begin{array}15&0\\1&2-\end{array}\right] $$ For each part, either do the calculation or explain why it does not make sense. a. $3D$ b. $A+B$ c. $A-D$ d. $B\cdot C$ e. $A\cdot C$ f. $AB$ g. $AC$ h. $AD$