Some special factoring formulas include the difference of two squares, the sum of two cubes, and the difference of two cubes. If there are three terms or more in the polynomial, students can use strategies such as finding common factors and factoring by grouping.

The difference of two squares [(a + b)(a - b)] is a common pattern with binomials involving variables to the second power. However, the concept can also be applied to exponents higher than x^{2}. Any even power (such as x^{2}, x^{4}, x^{6}, and so on) can be factored into squares evenly. For example, 16x^{4} can be rewritten as (4x^{2})^{2} and 49y^{6} can be rewritten as (7y^{3})^{2}. The expression 16x^{4} – 49y^{6} is then factored as (4x^{2} – 7y^{3})(4x^{2} + 7y^{3}).

The product of (a + b)(a^{2} – ab + b^{2}) can be evaluated using FOIL as a^{3} –a^{2}b + ab^{2} + a^{2}b –ab^{2} + b^{3}. That simplifies to a^{3} + b^{3}. Suppose that the binomial that needs to be factored is 27x^{3} + 8. That expression will factor as (3x + 2) (9x^{2} – 6x + 4).

Figure 1: Factoring the sum of two cubes and the difference of two cubes.

The product of (a – b)(a^{2} + ab + b^{2}) can also be evaluated and simplified to a^{3} – b^{3}. The easiest way to remember the direction of the signs when factoring the sum or difference between two cubes is to use the acronym SOAP. The sign between the terms of the binomial factor is in the same direction in both the sum of the cubes and the (a + b) factor. (If the difference of cubes is the issue, the sign in a^{3} – b^{3} and a – b is negative.) The sign is opposite between the a^{2} term and the ab term, such that if it is the sum of cubes the sign between a^{2} and ab is negative, and if it is the difference in cubes, the sign between the a^{2} and ab term is positive. The sign between the ab term and the constant is always positive.

Figure 2: Using the acronym SOAP to remember the direction of the signs.

The first step in factoring a polynomial is always to factor out anything that is common to every term in the polynomial. Suppose that the polynomial to be factored is 3x^{2} + 6x + 9. The first step in factoring would be to remove the common factor of 3 from all the terms as 3(x^{2} + 2x + 3). Next, check to see if it follows any of the special factoring forms. It can be factored by grouping or another method.

Figure 3: Following the general steps to factor a polynomial.

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]]>The AC-method, also known as factoring by grouping, is often used to factor polynomials when the a coefficient is equal to 1. It can also be used when the a coefficient is not equal to 1, similar to the FOIL method.

The first step to defining the polynomial expression, before finding factors or a numerical solution, is to see whether or not it takes the form ax^{2} + bx + c. Many polynomials used in high school math will follow that form, or one that is similar. If it is an expression, the variable x can take a number of different values. If they are in an equation or an inequality, the solution set for the variable will either consist of many solutions, one solution, or no solutions.

Figure 1: Check the form the polynomial takes before finding factors or a numerical solution.

If the polynomial expression has common factors other than 1 or -1, the common factors can be factored out, before looking at the rest of the equation. This is a very important step, because the polynomial that is left may be an ax^{2} + bx + c expression. Suppose the expression is 9x^{3} + 27x^{2} + 24x. The common factor to each element of the expression is 3x. Dividing 9x^{3} by 3xis 3x^{2}; 27x^{2} by 3x is 9x, and 24x by 3x is 8. Therefore one factor is 3x, and the other factor is in the form ax^{2} + bx + c.

Figure 2: An example of reviewing the polynomial for common factors.

The next step is to multiply the leading coefficient a by the constant c. In this example 3∙8 is 24. In the polynomial 14x^{2} + 67x + 81, 14 ∙ 81 is 1134. In the polynomial x^{2} – 3x + 7, 1∙7 is 7. In symbol terms, find integers p and q such that pq =ac. In symbol terms, integers p and q are such that pq =ac AND p + q =the coefficient b. Then the expression can be factored as (x + p) (x + q). This is also called splitting the middle term, and it is a method of trial and error. Suppose the expression is 3x^{2} -10x -8. The expression is in the form ax^{2} + bx + c. That a coefficient is 3, b is -10, and c is 8, so there are no common factors. In this example ac is equal to -24, so the task is to find two integers p and q, so that pq = -24 and p + q = -10. The factors that are equal to -24 are (-1, 24) with a sum of 23; (1, -24) with a sum of -23, (-2, 12) with a sum of 10, and (2,-12) with a sum of -10; (-3, 8) with a sum of 5; and (3, -8) with a sum of -5; (-4, 6) with a sum of 2, and (4, -6) with a sum of -2. The integers that follow both conditions are 2 and -12.

Figure 3: The process of defining AC and splitting the middle term.

Next, write the expression out and group the factors. The polynomial 3x^{2} – 10x – 8 = 3x^{2} – 12x + 2x – 8 = 3x(x – 4), because 3x ∙ x is 3x^{2} and 3x∙ (-4) is -12x. Also 2x -8 can be factored as 2(x – 4), because the expression 2x – 8 is equal to 2(x – 4). Putting the expression together 3x(x – 4) + 2(x – 4) equals (3x + 2) (x – 4), which can be checked by multiplying using FOIL, so that 3x∙x = 3x^{2}, 2x – 12x = -10x, and 2∙4 is 8.

Figure 4: An example of factoring by grouping.

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]]>The relationship of the length of the legs of the right triangle to the hypotenuse is expressed by one of the most famous equations in geometry, the Pythagorean Theorem. Deriving the measurement of the sides of the right triangle was one of the earliest uses of irrational numbers.

Right triangles have one 90-degree angle. The legs of the right triangle form the 90 degree angle, and the side opposite the 90-degree angle is the hypotenuse. The relationship between the lengths of the sides is given by the Pythagorean Theorem, a^{2} + b^{2} = c^{2}. That means that if the length of the hypotenuse and one leg is known, the length of the other leg can be derived. Suppose that the length of the hypotenuse is 5 cm and one leg is 3 cm. Using the Pythagorean Theorem, 3^{2} + b^{2} = 5^{2}, or 9 + b^{2} = 25. So 25 – 9 = b^{2}, or 16 = b^{2}. Taking the square root of both sides, b = 4 cm.

Figure 1: The Pythagorean Theorem.

The followers of Pythagoras soon found that there was a big scar in the perfect geometric world. This was such a shock to them that they took out one Hippasus of Metapontum (who used irrational numbers) and drowned him at sea, or so the story goes. They couldn’t drown the idea so quietly, because the measurement of the side is not always a rational number. Suppose one leg of the triangle measures 10 feet and the other leg measures 15 feet. What is the length of the hypotenuse? 10^{2} or 100 + 15^{2} or 225 equals 325. The square root of 325 (in symbol form √325) is not a perfect square. It is a little more than 18, 18.03 to two decimal places.

Right triangles are found all over, in the angle that a ladder makes with a building, the height of a tree, baseball diamonds and soccer fields. The applications of the Pythagorean Theorem and right triangles can be used to approximate and measure the lengths of the sides of these triangles. (Trigonometry is an entire branch of mathematics that gives more detail.) For example, suppose a ladder is 28 feet long. When the ladder is used against a building, it forms a right triangle. If the ladder is placed so the distance between the wall of the building and the base of the ladder is 2 feet, how high up will the ladder reach? One leg of the triangle is 2 feet, and the hypotenuse is 28 feet long. Using the Pythagorean Theorem, 2^{2} + b^{2} = 28^{2}, or 4 + b^{2} = 784. If b^{2} = 784 – 4 = 780, then b = √780, or 27.93.

Figure 2: A ladder against the side of a burning building forms a real-life right triangle.

Sometimes the right triangle is actually an “air triangle”. The applications of the Pythagorean Theorem and radical expressions still hold. Suppose a plane is at a height of 5000 feet when it approaches an airport. At 5000 feet, the line of sight between the plane and the terminal is 38,000 feet. How far is the horizontal distance between the plane and the airport? This is truly an “air triangle”, as one leg of the triangle is the distance between the plane and the ground, or 5000 feet. Let that be a^{2}. The line of sight between the plane and a point of ground at the terminal is the hypotenuse, or 38,000 feet. So 38000-5000 will be the other leg of the air triangle, or 33000. The square root of 33000 is 181.66.

Figure 3: Right triangles abound in the friendly skies.

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]]>In order to divide rational expressions accurately, special rules for radical expressions can be followed. Some of those rules include the quotient rule, rules for finding the square roots of quotients, and rationalizing the denominator.

Suppose the problem is to evaluate the square root of 36/49. The problem can be written in two different ways, either √ (36)/√ (49) or √ (36/49). The square root of 36/49 solved with the first method is √ (36)/√ (49) or 6/7. The square root of 36/49 solved with the second method is √ (36/49) or 6/7. This leads to the Quotient Rule, that for any non-negative number A and any positive number B, the quotient of two square roots is the same whether the square root is taken from the entire ratio or the square root is taken of the numerator and the denominator separately.

Figure 1: The Quotient Rule in Symbol Form

This rule is commutative, so that it means the same whether the square roots are taken together in a ratio or taken separately. Sometimes it is easier to solve the problem if the square roots of the numerator and the denominator are considered separately. That way, they can be simplified using perfect square roots if they are available. Suppose the problem is to find the square root of 18/50. Neither 18 nor 50 are perfect squares, but the ratio 18/50 can be multiplied by 2/2, to give 36/100. The square root of 36 is 6 and the square root of 100 is 10, to result in a fraction of 6/10, which can be further simplified to 3/5. Alternatively, 18 could be factored as 2 times 9 and 50 as 2 times 25, so the ratio could be divided by 2/2 to be further simplified as 3/5.

One of the ways to move a radical out of the denominator is to multiply the number by a form of 1, so that the denominator becomes a perfect square. Suppose the expression is √ (3/5). While the entire fraction is still under the radical sign, multiply each number by 5/5 so that the new fraction is 15/25 or √ (15)/5.

Figure 2: The process of rationalizing the denominator under the radical sign.

The other way to move a radical out of the denominator is to multiply both the numerator and the denominator for a form of 1 using radicals. Suppose that the ratio is 5/√ (3). In order to move the radical out of the denominator, both the numerator and the denominator can be multiplied by a form of 1 with the radical expression √3/√3 to result in (5√3)/3.

Figure 3: Rationalizing the denominator by using radicals, in symbol form.

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]]>Cellular respiration is the process that transforms food to energy within living organisms, requiring oxygen for the complete cycle. If the cell is in the presence of oxygen, the processes are aerobic and include glycolysis, the Krebs cycle, and electron transport. Processes such as fermentation occur after glycolysis. They release energy and do not require oxygen.

The first stage of cellular respiration is called glycolysis. Think of the terms glucose, as a type of sugar, and lysis, or loosening. In glycolysis, the bonds that hold together glucose are broken very quickly. When the bonds are broken, enough energy is released to jump-start the system. The part of the cell where glycolysis takes place is in the cytoplasm, the cell body itself.

The process of glycolysis may or may not take place in an oxygen-rich environment. If the cells are types that don’t need oxygen, anaerobic processes take place to form glycolysis. The two main types are alcoholic fermentation and lactic acid fermentation. Alcoholic fermentation is used by yeasts and some other types of organisms to produce energy, ethyl alcohol and carbon dioxide bubbles. When yeast ferments, carbon dioxide bubbles form. The alcohol that is produced by the bubbling yeast in bread burns away as it bakes, and the carbon dioxide bubbles leave behind air pockets. Lactic acid fermentation is familiar to athletes. During intense exercise, the muscles of the body run out of oxygen, and convert the glucose from glycolysis to lactic acid as a byproduct. When lactic acid builds up in the muscles, they cramp and burn.

The Krebs cycle requires oxygen, and begins after glycolysis. It takes place in a series of reactions in the mitochondria of the cell. The acids produced during glycolysis produce citric acid, and then the process starts over again several times. Carbon dioxide is released at every step, and more energy is produced. When people or animals breathe, releasing carbon dioxide into the atmosphere, the energy in cells that take part in the Krebs cycle is being released into the atmosphere.

Besides carbon dioxide, very high-energy electrons are released. Those high-energy electrons are responsible for the chemical reactions and enzymes that produce energy. The cellular respiration system converts food to energy efficiently. Muscles store energy, which can be released and steadily replenished during the processes within all phases, from glycolysis, the Krebs cycle, and electron transport.

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]]>Substances occur in elements or compounds, while mixtures can be any combination of one or more elements or compounds. Mixtures may occur as solutions or heterogeneous combinations.

Elements and compounds are known as pure substances. A molecule of one of the elements or compounds only contains that substance. For example, a molecule of oxygen, O_{2}, only contains oxygen. A molecule of water, H_{2}O, only contains 2 atoms of hydrogen to one atom of oxygen.

Pure substances are always homogeneous. The ideal pure substance contains molecules of only that substance, and nothing else. Mixtures may be homogeneous (if they are solutions) or heterogeneous. If they are heterogeneous, that means that different substances are throughout the mixture. For example, sand is a mixture that contains many different silicates, other types of rocks, organic debris such as particles of shells, and miscellaneous other items.

Solutions are mixtures that are homogenous. Dissolve a teaspoon of sugar in a cup of hot water. Both the sugar molecules and the water molecules exist separately, as can be demonstrated if the water is allowed to evaporate. Homogenized milk is a suspension of fat globules in liquid. By definition, the milkfat globules are evenly distributed throughout the liquid rather than being allowed to rise to the top as cream.

Suppose there is a mixture of white sugar and white sand. Both substances are solids and difficult to separate. It is also very hard to tell them apart, unlike a package of different-colored M &M’s. In some mixtures, all types of matter are solid. In others, such as soda pop, some types are solid, some are liquid, and some are bubbles of gas. Rocks and minerals are also heterogeneous mixtures. Different types of chemical compounds, water, and other solids combine to form them. A diamond is formed of crystalline carbon under conditions of extreme heat and pressure. However, diamonds in the real world contain impurities that affect their color and clarity. Boron is responsible for those that are blue; nitrogen, yellow and brown; and deformation or irradiation for other colors.

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]]>Over the centuries, the number of elements has changed as new elements have been discovered. Some unstable, heavy elements have only been synthesized in atomic laboratories, so very little is known about them.

The chemist Lavoisier listed 23 elements in 1789. Mendeleev arranged 63 elements in the first periodic table in 1869. In the 1930s, 93 elements were listed. In the 1960s, over 100 elements were listed, but not all of them had been isolated. The 2014 most recent version of the periodic table has 118 elements, but not all of them have been confirmed.

The periodic theory of the elements has been used to predict several gaps, to be filled in with later discoveries. For example, the Russian chemist Mendeleev predicted 8 elements that would have specific properties, which were later synthesized. The periodic table was modified to allow for the noble gases. In the 1940’s, physicist Glenn Seaborg theorized that elements should be grouped in a different transition series, and elements are synthesized to fill in those gaps. Many of those elements are radioactive, and have short half-lives.

The periodic table of the elements is arranged in 18 groups of elements with similar properties. Group 1 elements are the most reactive, and Group 18 elements, called the noble gases, are the least reactive. Most of the elements are metals. Many periodic tables show a huge jump from Element 57 (La) to Element 72 (Hf), and another huge jump in the row below it, from Element 89 (Ac) to Element 104. The first jump is called the Lanthanide series, and the second jump is called the Actinide series. In many periodic tables, those series are shown right below the main periodic table. They have similar properties, (even the ones that are on the verge of being discovered), as rare-earth elements, sometimes radioactive. Some of the elements with higher atomic numbers than uranium (U) have only been observed in the laboratory.

Most elements in the periodic table have one or two-letter abbreviations, even those radioactive elements like Fermium (Fm)) and Lawrencium (Lr). Those abbreviations mean that the element has been accepted by the International Union of Pure and Applied Chemistry (IUPAC). Elements that have a three-letter designation (Un* or Uu*) are those elements that are unconfirmed or theoretical. Element 104 is called Unq in earlier periodic tables and Rf (Rutherfordium) in others. It has a half-life of 1.3 hours. Element 116 is called Uuh in earlier periodic tables, and was confirmed as Lv (Livermorium) in 2012. Its half-life is measured in milliseconds.

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]]>The part of chemistry that deals with matter is concerned with the world of objects that have mass and take up space. A material object may be in solid, liquid, or gaseous form; consist of pure substances or mixtures of pure substances; contain elements or compounds; and may be composed of metal, nonmetal, or metalloid.

A material object can be classified according to its state. If it is a solid, it has a definite shape and volume. The shape of a solid is not dictated by its container. Suppose you have a solid marble, hard, spherical, and smooth. It will be the same size and shape whether it is immersed in water, on the edge of a table, or lost in the bushes outside. By contrast, the molecules of a liquid move much more freely and take the shape of their container. They do have a definite volume. Gases have indefinite volume and indefinite shape. They expand and contract under pressure.

A material object also exists as a pure substance or a mixture of various substances. Suppose an ingot of gold is refined until it is 24-carat pure gold, and then it is refined more, until as many atoms as possible are gold. The more pure gold is, the softer it is, so jewelers alloy gold with other substances to make it more useful. Rose gold is a mixture of gold and copper, and white gold is a mixture of palladium, silver, or titanium with gold.

Matter may also exist as one of the 109^{+} elements in the periodic table. Some, such as carbon (C), are very common, while others are very rare. The heaviest elements have been produced in minute quantities in scientific laboratories, and decay rapidly into other elements. Compounds consist of elements joined with various types of chemical bonds. Water, H_{2}O, is an example of a compound. Each molecule contains 2 atoms of hydrogen and 1 atom of oxygen.

Elements can be classified further as metal, nonmetal, or metalloid, and have been arranged in the periodic table on the basis of their properties. Most of the elements are metals, which are usually solid at room temperature, lustrous, good conductors, malleable, and ductile. Nonmetals may be solids, liquid, or gaseous at room temperature, not lustrous, poor conductors, and brittle. Metalloids share some properties of metals and some of nonmetals. Some nonmetals are highly reactive with other elements.

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]]>Equations that involve multiplication and division of fractions can be solved in similar ways to multiplication and division of whole numbers. There are some special rules to follow and questions to ask before solving them.

Before solving math problems with fractions using multiplication or division, read the problem carefully and ask questions. What math operations need to be done in this problem other than multiplication or division? What relationships are important to the problem and which are not important? Will drawing a sketch of the problem help me to understand it? Be a math detective and use all the skills you have.

Think of the math fact 3 times 4. The product is 12, and it is the same thing as adding 4 + 4 + 4. What about 1/3 times 5? That is the same thing as saying 1/3 of 5, which is the same thing as dividing 5/3 or 1 2/3. It can also be illustrated by repeated addition. Suppose Rom has 5 bars of gold-pressed latinum and he is going to give Leeta 1/3 of them as an anniversary gift. One-third of each of the bars could be cut, so that 1/3 + 1/3 + 1/3 + 1/3 + 1/3 = 5/3 or 1 2/3. However, the method for multiplication is much quicker than repeated addition, either for whole numbers or for fractions. Suppose the equation is 2/3 times 5/8. A quick way to think of it is (2·3)/(5·8) or 6/40, which can be simplified still further to 3/20. In symbol form, if two fractions a/b and c/d are multiplied, a/b times c/d means the same thing as ac/bd.

When whole numbers are multiplied, the product is larger than either of the factors. When fractions are multiplied, the product is smaller. This puzzled mathematicians in the 15^{th} century, and is a twist to multiplying with fractions. It makes sense to math detectives, because a fraction of anything is going to be smaller than the whole.

If multiplication is repeated addition, division is its inverse; repeated subtraction. This is true for whole numbers. It’s also true for fractions, but with a twist, which will make sense to math detectives. The definition for division in symbol form is a/b divided by c/d is the same thing as multiplying a/b by d/c, or a/b ÷ c/d equals a/b times d/c or (ad)/(bc). Suppose the problem were 2/3 divided by 5/8. That would be the same thing as saying 2/3 times 8/5, turning the denominator upside down. By definition, 2/3 ÷ 5/8 = 2/3 ·8/5 = (2·8)/(3·5) or 16/15.

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]]>Before solving math problems with fractions using addition or subtraction, read the problem carefully and ask questions. What math operations need to be done in this problem? Do the fractions have like or unlike denominators? Can a mixed number be turned into an improper fraction? Which fraction is smaller and which fraction is larger after common denominators are found?

The rules for addition and subtraction of fractions are similar to the rules for math operations with whole numbers. One important change: finding common denominators if they are needed. In order to perform some operations, mixed numbers may have to be turned into improper fractions, where the numerator is larger than the denominator.

For both addition and subtraction of fractions, common denominators are needed. If the fractions in the problem, have common denominators, addition is exactly the same as for whole numbers, following the number line. To add 2/5 and 1/5, add (2 + 1)/5 for 3/5. Suppose the problem was 7/8 – 2/8. The problem would be solved as (7 – 2)/8, or 5/8. If the fractions do not have common denominators, they can be turned into fractions with common denominators. In symbol form, a/b ± c/d = ad/bd ± cd/bd, (ad + cd)/bd, with neither b nor d equal to zero. For example, adding 2/3 and ¼ equals 8/12 + 3/12, which equals (8 + 3)/12, or 11/12.

A mixed number, such as 1 ¼, can always be turned into an improper fraction by changing the whole number into a fraction, then adding or subtracting the fractions with common denominators. The number 1 is the same thing as 4/4 and then adding (4 + 1)/4 equals 5/4. Suppose the math problem is 1 ¾ – 7/8. There are two steps to solving this problem. First, the mixed number should be changed to an improper fraction, so (4 + 3)/4 is equal to 7/4. Then, the fractions need to be changed to equivalent fractions with common denominators. The fraction 7/4 is equal to 14/8 by multiplying 7/4 by 2/2. Then (14 – 7)/8 equals 7/8. Another method of finding the common denominator was used by finding common factors.

Fractions are also called rational numbers in math language. Rational numbers are part of the number line, so the rules for adding and subtracting extend to the negative part of the number line. Suppose the math problem is ¾ – 1 1/8. The mixed number can be changed to an improper fraction, so that the equation is ¾ – 9/8. The fraction ¾ can be changed to its equivalent 6/8. Then the problem becomes (6-9)/8 or -3/8.

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