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Our education staff publish regular articles, tips and tutorials to help students with their homeworkMon, 08 Feb 2016 00:52:21 +0000en-UShourly1http://wordpress.org/?v=4.3.2Math Review of Adding and Subtracting Polynomials
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Adding and subtracting polynomials is similar to adding and subtracting other expressions. Terms can be combined using the Commutative and Associative properties for addition, then like terms can be added or subtracted as needed.

Find Like Terms

Recall that like terms have the same variables and are at the same degree. If they have the same degree, the terms are raised to the same power. For example, 4x and 2x can be added or subtracted because they both have the variable x and no other. Addition of 4x + 2x equals one term, the monomial 6x. They are also at the same degree, because both 4x and 2x have the same exponent 1. The expressions 3x^{2} and 4x both have the variable x, but they are not at the same degree, as 3x^{2} has degree 2, and 4x has degree 1. Combining them will result in the polynomial expression 3x^{2} +4x.

Use Properties

Polynomials can be rearranged using the Commutative Property without changing their value. For example, 4x + 2x has the same meaning as 2x +4x. This was already shown in rearranging polynomials in standard form, so that 5x + 6x^{3} +109 +8x^{2} can be rearranged to 6x^{3} +8x^{2} + 5x + 109. The Associative Property also applies, so that (6x^{3} +8x^{2}) + (5x + 109) equals 6x^{3 }+ (8x^{2} + 5x + 109). The one thing to remember when using either property with polynomials is that the sign in front of each term must stay with that therm. Therefore, 8x^{2} + 5x -109 + 6x^{3} can be rearranged as 6x^{3} +8x^{2} +5x -109.

Adding Polynomials

In order to add polynomials, first find like terms, rearrange them using the Commutative Property, and combine like terms using the Associative Property. For example, (2x^{2}-x) +(x^{2} +3x -1) have the like terms of 2x^{2 } +x^{2}, -x +3x, and the constant -1. Notice that the negative signs stay with the terms –x and -1. The new expression, using all the terms, is 2x^{2} +x^{2} –x +3x -1. Combining terms, (2x^{2} +x^{2}) +(-x + 3x) -1 results in a new expression where (2x^{2 } +x^{2}) can be replaced by 3x^{2}, and (-x +3x ) can be replaced by 2x , so the simplified expression is 3x^{2} +2x -1.

Subtracting Polynomials

Subtracting is the inverse of addition, so the difference between adding and subtracting polynomials is that each term subtracted has the opposite sign. For example, -(2x^{3} + 4x^{2} +12x +42) is -2x^{3}-4x^{2}-12x -42 and –(2x^{2}-12x – 36) is -2x^{2} +12x +36. Therefore, (a^{4}-2a) –(3a^{4} -3a -1) is the same thing as adding a^{4} -2a -3a^{4} +3a +1. Like terms can be arranged, so that a^{4}-3a^{4} -2a +3a +1is the same expression. Then like terms can be combined, as -2a^{4} +a +1.

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Polynomials contain variables, constants, and exponents. The simplest polynomial is called a monomial, as a monomial consists of a single expression. Other polynomials are a sum of monomials.

Monomials

A monomial expression can consist of a constant, variable, and exponent, all multiplied together. The variable is not in the denominator of the expression, such as 1/x, and the exponent must be a whole number, neither a fractional exponent, nor a negative exponent. Therefore, 2x^{2} is a monomial, but 3x^{-5} is not, nor is 5y^{2/3}.

Polynomials

A polynomial is a monomial or the sum of monomials. For example, 2x^{2} +3x is a polynomial that is formed by the sum of the monomials 2x^{2} and 3x. Similarly, a polynomial such as x^{2}-4x^{3} +32x +16, is formed by 4 separate monomials, x^{2}, -4x^{3}, 32x, and 16. An expression such as 3y^{2} + 9y, 10x – 3, or 3a^{6}-9a^{3}, is also called a binomial because it contains two terms. Similarly, an expression such as 2x^{2}-5x-16 is also called a trinomial because it contains three terms. A constant such as 16 can be thought to contain x^{0}, because any variable with a zero exponent equals 1.

Degree of a Polynomial

The degree of a polynomial is the degree of the term in that polynomial with the highest exponent. In an expression such as x^{2} – 7x^{3} +94x +124, the degree is 3, because the highest exponent is in the term -7x^{3}. Similarly, in the expression 18 – 12b^{9}, the degree is 9, because the highest exponent is 9. In an expression such as 16x-5, the degree is 1, because 16x has an exponent of 1. By convention, x^{1} is the same thing as x. We generally leave out an exponent of 1.

Standard Form

When writing polynomial expressions in standard form, the term with the highest degree is written first, then the next highest degree, then the next highest to that, all the way to the constants. Therefore, an expression such as 20x-9x^{3} + 2 +18x^{2} can be rearranged in standard form as -9x^{3} + 18x^{2} +20x +2, as -9x^{3} has degree 3 , 18x^{2} has degree 2, 20x has degree 1, and 2 has degree 0. The coefficient -9 in the term -9x^{3} is often called the leading coefficient, as it is the coefficient of the highest degree term and “leads” the polynomial when it is rearranged in standard form. If the polynomial were x^{2} +2x +3, the leading coefficient would be 1. When there is no constant beside a variable in a monomial it is understood to be 1, as 1 times x^{2} is still x^{2}, using the multiplicative identity.

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One way to build good writing and studying habits is to find a good place to write and do homework. Not all writing and homework can be done at school, so it is important to find a place without distractions, where students can go to the same place each day, keep necessary tools together, and notice what works best to help work go together smoothly.

A Place without Distractions

Most students have a hard time writing if the place they choose is too distracting. An interesting movie or TV program, unfamiliar music, or friends who want to talk can break concentration. Some people need silence, especially if it is difficult to get started. Others need background music to keep the process going.

Choose the Time and Place

One way to build good writing and study habits is to choose the time and place, and try to make it the same time every day, or close to every day. That way, there will be enough time to develop a writing idea, find examples, and write and polish the draft enough times to have a finished product. Most students and writers cannot write a polished paper on the first try, and need time to revise their work.

Gather Tools

Gather writing tools together in the place you choose for writing and studying, so you won’t have to waste valuable time hunting for a pen that has enough ink, your English textbook, a dictionary, your writing notebook, or the charger for your laptop. If the place you choose is not at home (such as the public library), have a place for your tools that you can take with you, such as a briefcase or backpack. You can also have a smaller notebook with you at other times to jot down ideas or descriptions.

Be Prepared to Change

During the time you are writing or studying, if you find something that is not working, be prepared to change things to make them work better for you. Maybe the music you are listening to while you write is too loud, so it is distracting, or you’ve forgotten your textbook in your tool kit. Turn down the volume on the radio or change the station, find your textbook, and keep writing. It is always better to revise what you have written than stare at a blank page.

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Many selective colleges and universities ask prospective students to write an essay as part of their application for admissions. Usually, there are guidelines, such as word length, a quotation, or a common question. The goals of the essay are for the admissions officer to see how well students can express themselves in writing, and reveal themselves as interesting potential students.

Choosing a Topic

Most of the time, college applications contain essay questions that are open-ended, but specific. Some common application questions ask students to describe an experience, achievement or event that stood out to them; a person who has influenced their life; a character in fiction, an historical figure, or a creative work and how it has impacted personal philosophy; or any other topic.

Write About Yourself

Admissions officers are looking for student essays that reflect individuality. Maybe you haven’t spent a summer in a foreign country or received awards for outstanding community service or heroism. However, even mundane life experiences can be made interesting. Write about something that excites you, whether it be making art or music, playing sports, hobbies, or academic interests.

Show, Don’t Tell

Engage the senses to engage the reader. If your passion is for hiking in the mountains, describe the feel of the warm sun on your back, the smell of the pine forests, the taste of the icy water from a mountain stream, the sight of the glaciers and peaks; and the cries of hawks as they glide by. Draw the reader into the scene, whether it is onstage presenting a drama or traveling in a foreign land.

Proofread Carefully

Most writers cannot write a perfect essay on the first draft. Even with spell-check as part of the word processing program, words remain that are not the correct ones, such as “there” for “their”, typos that are still words, or when a word is left out of the sentence. Wordiness can create problems, as well as weak forms of the verb “to be”, strings of adjectives; run-on sentences; and mixing up tenses. Leave enough time to write and rewrite, ask someone else to read the draft, and feel free to go in a different direction if one attempt doesn’t say what you want it to say.

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Desert lands generally receive less than 10 inches of rainfall per year, and sometimes receive no rainfall in a year. Desert climates may be hot or cold, and the topography may be flat or mountainous.

Polar Desert

The continent of Antarctica is considered a polar desert. It receives less than 8 inches of rainfall on the average, and the Dry Valleys are not covered by snow and ice, but exist as barren rock. Few forms of vegetation can live in such extreme conditions. There are some lichen-forming fungi, snow algae, and various forms of hardy bacteria. Some parts of the Arctic Basin also receive less than 6 inches of precipitation, and are also technically considered a desert.

Sahara, Arabian, and Syrian Desert

The Sahara Desert is the third largest desert area in the world, and is the largest hot desert. It is located in northern Africa, and includes some of the hottest areas during the summer, with average temperatures over 100 degrees F (38° C). Most of the Sahara consists of barren, rocky areas where most of the sand has blown away, although large sand dunes cover other areas, as well as salt flats. The plants and animals that live on the desert conserve what little water there is. The Arabian Desert and the Syrian Desert cover most of the Middle East. Some of the land is rocky and some is covered with sand.

Deserts in Asia and Australia

The Gobi Desert is a large desert area in Mongolia and Northern China. Clouds carrying rain from the Indian Ocean cannot cross the Himalayas, so the rain shadow area has less than 8 inches of precipitation in a year. Sometimes snow falls on the sand dunes during the winter. By contrast, much of the desert land in Australia is hot, with the hardiest shrubs and grasses growing along sand dunes. Small marsupials and lizards survive the heat by burrowing into the sand.

Deserts in North and South America

Some of the desert area in North and Central America include regions that are parts of the states of Utah, Nevada, California, southern Idaho, and Arizona, as well as parts of Mexico. These desert lands contain many more familiar species of cactus, sagebrush, and other scrub vegetation, as well as lizards and many different varieties of snakes. The Patagonian Deserts in Chile and Argentina are in the rain shadow of the Andes Mountains.

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Glaciers form on land when the amount of snow falling in winter is greater than the amount that melts in the summer. The compact snow and ice form sheets that flow under their own weight. Those moving rivers of ice are especially found in mountainous areas and in the polar regions on Earth.

Types of Glaciers

Alpine glaciers form near mountain summits, especially in areas where the winter snows are deep and do not melt completely during relatively short summers. The largest glaciers flow great distances, down high elevations to the lowlands. Continental glaciers, or ice sheets, cover more than 50,000 square kilometers, and exist mostly in Greenland and Antarctica. By contrast, the Arctic area is more oceanic, and glaciers do not form on oceans.

Movement of Glaciers

Glaciers move at different rates depending upon landforms and weather conditions. Over steep terrain, glaciers can move several meters at a time, especially if their inner temperature is relatively high. However, when terrain is not as steep and the weather is cold and dry, glaciers only move at the rate of a few centimeters a day. For the most part, glaciers move by a combination of friction and low pressure. Crevasses open as brittle upper ice cracks, making it dangerous for mountaineers to travel along them.

Glacial Erosion

Glaciers pick up rocks and sediment and carry them along, scraping and gouging the bedrock as it freezes, melts, and moves. In addition, water seeps through cracks beneath the glacier, causing more weathering and erosion. U-shaped valleys are gouged out of mountain landscapes by glacial action, and mountain peaks are carved into steep formations. The Matterhorn in the Alps is an example of this type of erosion, as glaciers sculpted a steep peak from three different sides of the mountain.

Glaciers and Global Warming

Glaciers store fresh water, and are a very important reservoir for much of the fresh water on the earth. However, many areas that were covered in ice, such as the ice sheets in Greenland and glaciers throughout the world, are melting more rapidly than ever before. Even Antarctica is losing mass along its great ice shelves at a greater rate than they are growing.
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As division is the inverse of multiplication, so division of exponents is the inverse of multiplication of exponents. There are properties for the division of exponents similar to the properties for the multiplication of exponents.

Quotient of Powers

Recall that finding a product of powers with the same base is expressed by the rule in algebra language of x^{m}x^{n} = x^{m +n}. This concept was developed by using the definition of exponents as repeated multiplication. If x^{3} is x·x·x and x^{2 }is x·x, then x^{3}x^{2} is (x·x·x)(x·x). What about (x^{3})/(x^{2})? That also can be simplified by using the definition of exponents, as (x·x·x)/(x·x) is x^{3-2} or x. In the language of algebra, x^{m}/x^{n} equals x^{m-n}, when x is a real number not equal to zero and m and n are integers.

Positive Power of a Quotient

Recall also that any expression (ab)^{n} can be expanded to a^{n}b^{n}. For example, an expression such as (3x)^{3} means 3·3·3·x·x·x or 27x^{3}. Similarly, a quotient such as (2/5)^{4} means (2/5)(2/5)(2/5)(2/5) or (2^{4})/(5^{4}). The quotient is in simplest form with the exponents, or it can be multiplied out as 16/625. In the language of algebra, (a/b)^{n} equals a^{n}/b^{n}, if a and b are real numbers not equal to zero, and n is a positive integer.

Negative Power of a Quotient

Remember that x^{-n} is the reciprocal of x^{n} or 1/x^{n}. For example, 5^{-3} is the same thing as 1/5^{3} or 1/125. Similarly, the reciprocal of 5/6 is 1/(5/6) or 6/5. Suppose that the value of x is the fraction a/b. In that case, x^{-n} would equal 1/(a/b)^{n}, or 1/(a^{n}/b^{n}), or b^{n}/a^{n}, which is the reciprocal of a^{n}/b^{n}. In the language of algebra, if a and b are real numbers not equal to zero and n is a positive integer, then (a/b)^{-n} is equal to (b/a)^{n}, which is equal to b^{n}/a^{n}.

Applications of Exponents

Exponents are often used in scientific notation to express numbers that are very large or very small. For example, the distance from the Sun to the Earth is about 9.3 x 10^{7} miles. However, the size of an average dust particle is about 7.53 x 10^{-10} kg. Exponents are also used in other applications, including economics, measurement, engineering, and accounting.
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Exponents are used as a type of shorthand directing how many times that a base number is multiplied by itself. Exponential expressions are useful when working with very large or very small numbers, as in scientific notation. Like other numbers and variables, exponential expressions can be multiplied following certain rules.

Review of Simplifying Expressions

If there are negative exponents in an expression, it is not simplified. The expression x^{-3} can be defined as 1/x^{3}. Similarly, if the same base is used more than once in an expression, it is not simplified. Suppose the expression is x^{2}x. That actually means x·x·x, or x^{3}. Also, powers that are raised to powers are not simplified, such as [x^{5}]^{2}. If any coefficients have common factors, those common factors must be simplified. For example, (15a^{2})/ (10b^{3}) can be simplified to (3a^{2})/ (2b^{3}) because 15 and 10 have a common factor of 5.

Products of Powers

Since an exponent stands for how many times a base is multiplied by itself, the rule for multiplying exponents can be discovered by showing each power as repeated multiplication. If the expression is y^{3}y^{2}, that can be expanded as y·y·y·y·y, or y^{5}. The product of the powers can be found by adding the exponents, as 3 +2 = 5. However, the bases must be the same in order for the exponents to be added. In the language of algebra, for any real number a not equal to zero, a^{m}·a^{n} = a^{m+n}.

Powers of Powers

Expressions such as [x^{5}]^{2} actually mean x^{5}·x^{5}, or (x·x·x·x·x) (x·x·x·x·x), by using the meaning of exponents as repeated multiplication. The product of powers of powers can be found by multiplying the exponents, as 5·2 equals 10. In the language of algebra, for any real number a not equal to zero, (a^{m})^{n}=a^{mn}. The way to know whether to add or multiply the exponents is to expand the expression to repeated multiplication. For example, z^{3}z^{4} means z·z·z·z·z·z·z, or z^{7}, while (z^{3})^{4} means (z^{3})(z^{3})(z^{3})(z^{3}), or (z·z·z)(z·z·z)(z·z·z)(z·z·z) or z^{12}.

Powers of Products

An expression such as q^{5} only involves one component, the variable q. However, an expression such as (3z)^{3} has two components, the coefficient 3 and the variable z. It can be expanded as (3z)(3z)(3z), or 3·3·3·z·z·z, or 3^{3}z^{3}, or 27z^{3}. In the language of algebra, (ab)^{n}, for any real number a and b not equal to zero and n as an integer, (ab)^{n} equals a^{n}b^{n}.

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http://schooltutoring.com/help/english-review-of-ways-to-build-vocabulary/#commentsSun, 20 Dec 2015 05:41:59 +0000http://schooltutoring.com/help/?p=8801OverviewBuilding a good vocabulary is an important part of success in English. It helps students to more easily understand what is read, as well as having more words to choose from when writing an essay or a paper. A good vocabulary helps people communicate in all walks of life, both in school and in the workplace.

Read Widely

The most important way to build vocabulary is to read a variety of books and publications and note any unfamiliar or unusual words. The novels and stories read in English classes and other classes are only a starting point. Many authors have written other books than those read in class. Charles Dickens wrote many more books than A Christmas Carol and A Tale of TwoCities. Newspapers such as The New York Times or The Seattle Times can be read online. Periodicals such as Sports Illustrated, Time, or National Geographic have articles that are informative and interesting. Reader’s Digest has regular columns for building word power, introducing unfamiliar words and words in unusual contexts.

Make a List

Keep a running list of unusual or unfamiliar words, along with definitions that make sense. Students can make flash cards of those words and quiz themselves or quiz one another. If those vocabulary words are going to be part of a test, it is better to practice them over several sessions than all at once.

Watch for New Words

It is always surprising how many times a new word will appear in different contexts. Suppose you read about agents trying to avoid setting off the alarms of a proximity detector. The proximity detector is set off when someone or something comes too close to it. Next, you hear about a surfboard shop in close proximity to the beach. Finally, you see a news report about people protesting a new airport runway because it is to be built in proximity to their houses.

Imagine the Word

Whether you are learning vocabulary to expand your knowledge or for upcoming SAT or ACT tests, one good strategy is to imagine the word in a meaningful context. It will be easier to remember that your friend Larry is very talkative, even loquacious, than to write the word loquacious without a context. Similarly, try imagining an artist drawing a picture for the word depict. The roof of a mine is weakened and collapses when its support beams are sabotaged, undermining it. The word erroneous begins with an error.
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http://schooltutoring.com/help/english-review-of-plot/#commentsSun, 06 Dec 2015 03:41:11 +0000http://schooltutoring.com/help/?p=8795OverviewPlot is essential to any story. The sequence of events within the story are linked by cause and effect to produce a desired outcome, and keep readers engaged. Plots tend to follow a structure where the action rises to a climax, then falls as the situation is resolved for the characters.

Exposition and Inciting Incident

Many stories begin with introducing the main characters as exposition. For example, Dickens introduces the miserly Ebenezer Scrooge in A Christmas Carol by showing how he limits the heat in the office to three coals, as well as begrudging his clerk, Bob Cratchit, time off to spend the holiday with his family. Scrooge also turns down an invitation to join his nephew Fred’s family for dinner, rebuffs men seeking donations, allowing readers to see that he considers Christmas as a humbug. The ghost of Jacob Marley, his old business partner, visits him as an inciting incident. Marley’s chains were forged by greed and selfishness, and the ghost is condemned to trudge the earth, just as Scrooge will be, if he does not change his ways while he lives.

Rising Action

The visits from successive ghosts follow. The Ghost of Christmas Past shows Scrooge what Christmas was like for him as a lonely child and as an apprentice, as well as the breakup between Scrooge and his fiancée Belle. The Ghost of Christmas Present appears, to take Scrooge to many different celebrations, including the one he spurned with his nephew Fred and his family. Bob Cratchit’s family, including his young son Tiny Tim, enjoys a happy holiday, despite the child’s serious illness.

Climax

The Ghost of Christmas Yet to Come takes Scrooge to view the events surrounding the death of a man no one mourns, with his possessions stolen. He contrasts that man’s death with the death of the beloved child Tiny Tim. He then shows Scrooge that the neglected grave belongs to him. At that point, a contrite Ebenezer gives that ghost a pledge that he will change his ways if he is given a chance to make amends.

Falling Action and Resolution

On Christmas morning, Scrooge does make amends. He sends a huge turkey to the Cratchit family, and later gives Bob a raise and helps the child Tiny Tim. He is a truly changed man, who keeps the spirit of Christmas in his heart, and does good deeds for others. The conflicts are resolved for the former miser Scrooge and the Cratchit family.

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