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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]]>Space figures exist in three dimensions. Some examples include spheres, cubes, tetrahedra, polyhedra, cylinders, cones, and other types of solids.

Plane figures such as polygons and curved figures are two-dimensional and flat. They have length and width. Space figures exist in three dimensions, adding depth. A sphere has a center point, and the myriad circles that exist around it form the sphere. A square in three dimensions is a cube.

A sphere has one fixed point at the center. When all points on the outside of a sphere are connected with its center, it becomes solid. Many real world objects are spherical or spheroid in shape, such as the earth and other planets, the sun and other stars, and bubbles. A round shape such as an oval can become a three-dimensional ovoid. Real-world examples include eggs or teardrops.

A tetrahedron is a solid with triangular faces. If the triangles that form a solid have any other polygon at the base, that solid is a pyramid. If other polygons are joined into a solid, that solid figure is called a polyhedron. If two faces of the polyhedron are congruent, such as a square at the top, and a congruent square at the bottom, the polyhedron is called a prism. Some polyhedra are regular, and others are irregular. Polyhedra have special properties; and their volume can be measured.

Cylinders are classic solids with two circular bases of the same size. If the bases are directly opposite each other, the figure is called a right cylinder. If the circular bases are not directly opposite, the figure is called an oblique cylinder. Cones are figures that rise from a circular base to a vertex. They may also be right or oblique. Classical solid figures include tetrahedra, cones, cubes, prisms, cylinders, and other polyhedra. However, other solid figures can be easily imagined as combinations of the classic shapes.

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]]>Plane figures in geometry include points, segments, lines, rays, and angles, polygons, curves, and circles. They appear on a flat plane.

Think of a pencil point, and make it smaller, until it is a no-dimensional, undefined, geometric concept that only exists in the minds of mathematicians. Segments are made up of points, and they consist of the beginning, the end and all points between the beginning and the end. They exist in two dimensions. Extend the segment to infinity in both directions, and it is a line. Start from only one point in any direction and extend it to infinity, and it is a ray. Rays are usually designated by the beginning point and the endpoint of a line segment. Two rays that meet at a common point, the vertex, are called an angle, and the angle can be measured with a protractor or duplicated using a drawing compass.

A three-sided polygon is called a triangle, because it consists of line segments arranged in three angles. Four-sided polygons may be squares, if all four sides are equal and each side meets at right angles. Other types of four-sided polygons include the rectangle, the rhombus, the parallelogram, and the trapezoid. Polygons are usually named for the number of sides they have; such as a pentagon with 5 sides, a hexagon with 6 sides, a heptagon with 7 sides, and octagon with 8 sides, and so on.

A set of points connected by a continuous line is called a curve. Curves can be open or closed. A circle is a type of closed curve that has a single center. Each point along the circumference of the circle is at the same distance from the center. The distance from the center to the circle itself is called the radius, and any circle measures 360^{o}.

Many other curved figures exist in geometry and in the real world, such as ellipses, arches, ovals, lenses, and crescents. Technically, polygons are also curves, because they can be formed by a continuous line, even if that line doesn’t appear curved.

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]]>If division were limited to single-digit divisors and math facts, it would be comparatively easy. Long division is more difficult to visualize and tie to real-world manipulative objects. Paradigms exist for long division, and there are tips and tricks for dealing with other problems.

Sometimes it is easier to tell if a dividend is evenly divisible by a divisor before working the problem out on paper. If a number in the units digit is divisible by 2 or 5, the entire number will be divisible by 2 or 5. For example, 385 is divisible by 5. Similarly, if the sum of the digits in a number is divisible by 3 or 9, the number is divisible by 3. If the sum of the last 2 digits is divisible by 4, the number will be divisible by 4. If a number is divisible by both 2 and 3, it will be evenly divisible by 6.

The steps for division include directions for repeated subtraction. This paradigm can be used for one-digit divisors as well as long division. It involves subtracting multiples of the divisor, adding the results up, and continuing until the leftover amount is less than the divisor. Suppose the number is 124/32. Then subtract 124 – 32 = 92 – 32 = 60 – 32 = 28. This method does not use place value, and makes it easier to see the concept.

One of the ways to look more carefully at division is to color-code both the dividend and the divisor using the color-coding that is available for chip-trading, such as red, thousands place, green, hundreds place, blue, tens place, and yellow, units place. Suppose the division problem is 4800/25. That would result in 4 red chips and 8 green chips. The 4 red chips cannot be evenly divided by 25, but they could be traded in for 40 green chips. The 40 green chips can be added to the 8 green chips that are already there to leave 48 green chips. The 48 green chips can then be divided by 25, which will leave 23 green chips. The 1 goes above the green. The 23 green chips become 230 blue chips. 225 of them (25*9) can be eliminated, and the 9 can go above the blue. The remaining 50 is evenly divisible by 25, so the solution is 100 + 90 + 2 = 192.

Normally, any number divided by itself is 1. What happens when zero is divided by itself? The answer is not 1, it is not determined. If any other number is divided by zero, the answer cannot be determined either. Dividing by zero is not allowed, because it doesn’t make any sense.

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]]>Division is the inverse of multiplication. If a times b equals c, and b is not equal to zero, then c divided by b equals a. Also, for any real number a not equal to zero, a times 1/a equals 1.

Division is one of the four basic arithmetic operations. It is usually the last one to be taught, because solving division problems can involve addition, subtraction, and multiplication. Suppose the problem is 36 divided by 4. The answer, 9, is a math fact, because 9 times 4 is 36. It can be solved by using the definition of division, as 36/4 =9. It can be solved by subtraction, as 36 – 4 = 32 – 4 = 28 – 4 = 24 – 4 = 20 – 4 = 16 – 4 = 12 – 4 = 8 – 4 = 4 – 4 = 0. The number 4 was subtracted 9 times from 36. To check division, students can multiply the quotient by the divisor, or add the divisor repeatedly.

Suppose you had 100 movie tickets and you wanted to give 4 free tickets to as many people as possible who were waiting in line outside a theatre. One way that could be done is to measure sets of 4 tickets and give them to the first people in line until there were no more left. In this case, 100 is the dividend, 4 is the divisor, and dividing 100/4, gives 25 groups of 4 tickets each. The first 25 people in line get 4 free movie tickets.

Similarly, suppose the 100 movie tickets were divided into 4 equal parts. How many tickets would be in each set? The quotient would be the same, 25, but this time the divisor 4 represents the number of equal parts. Even though the numbers are the same in both models, the concepts are a little different.

Many hands-on models for division exist, such as dividing beans or cubes into groups, or using the abacus to illustrate division by measurement. When students number off in groups of 4 to take part in activities or register for one of 5 home rooms, they illustrate the partitive model.

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]]>If two linear equations are solved, and their solution is graphed on a Cartesian coordinate plane, there are three possibilities. The lines may be the same, with all solutions in common. They may never intersect, with no solutions in common. They may intersect at only one point, with one solution in common.

The coordinate plane is also called the Cartesian plane, in honor of the great mathematician Rene Descartes. It is one of the most familiar graphs in mathematics, with an x-axis and a y-axis, stretching out to infinity in all directions. The point where the x-axis and y-axis cross is called the origin. Its coordinates are (0, 0). It is divided into four quadrants. They are usually called I, II, III, and IV. In Quadrant I, values of x and values of y are positive. In Quadrant II, values of x are negative and values of y are positive. In Quadrant III, values of x and values of y are negative. In Quadrant IV, values of x are positive and values of y are negative.

Figure 1: The coordinate plane.

If two or more equations have all solutions in common, the graph of their solutions will contain all solutions along the same line. In geometric terms, the points are collinear. It algebraic terms, the systems are dependent. Suppose the equations were x +y = 5 and 2x +2y =10. If x equals 1, then y equals 4; x equals 2, y equals 3, and so on. If the line for the equation x +y =5 is blue, and the graph of 2x + 2y = 10 is red, the lines will be superimposed, and the graphed collinear line is purple.

If two or more equations have no solutions in common, the graph of their solutions will contain all solutions along parallel lines. Suppose one equation is y = 3x + 2 and the other equation is y = 3x -1. The slope of each line is 2, so they have the same slope. The y-intercept of the first can be graphed vertical or perfectly horizontal have the same slope and different y-intercepts. Horizontal lines have a slope of zero by definition, but they each cross the y-axis at different points. Vertical lines have the same undefined slope, but they each cross the x-axis at different points.

Figure 2: When two linear equations have no solutions in common, their graphed lines are parallel.

Many systems of linear equations have a single solution that satisfies both equations. When those linear equations are graphed, they intersect at a single point. The linear equations may meet at many different angles, either acute or obtuse. When they meet at a 90 degree perpendicular angle, the values that solve each equation are in definite relationship to one another. Suppose that the equation for Line 1 has a slope of m_{1} and the equation for Line 2 has a slope of m_{2. }The product of m_{1} and m_{2} is -1. Similarly, if one equation can be graphed in a horizontal line parallel to the x axis, with slope 0, and the other equation in a vertical line parallel to the y axis, with undefined slope, the lines meet at one perpendicular point, a singular solution.

Figure 3: If a single solution satisfies both equations in a system, their graphed lines will meet at a single point.

Figure 4: The graphed lines will be perpendicular if the equations meet at a single point and the product of their slopes is -1.

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]]>Operations with sets include the union of sets, the intersection of sets, and complements of sets. These operations can be applied to solve survey problems using sets.

Some special types of sets include the null or empty set, the universal set, and proper and improper subsets. The null or empty set ⦰ has no elements. Its set notation is { }. In contrast, the universal set **U **contains all the members of every set in an operation. While the empty set is always the same, a set with no elements, the universal set changes with each problem. It is defined before any operations take place. Suppose that set F is the universal set for a problem. It contains the elements {2, 4, 6, 8, 10, 12}. Therefore, all operations on set F will contain only those elements. Proper subsets of Set F will include some of the elements of Set F, such as Set G {2, 4, 6} or Set H {8, 10}. In set language, the symbol shorthand is G ⊂ F. Improper subsets of Set F will contain all the members of Set F. Set I {12, 10, 8, 6, 4, 2} is an improper subset of Set F.

Figure 1: Set F is the universal set for this problem; Sets G, H, and M are proper subsets.

The union of sets is designated by the symbol ⋃, so if Set G {2,4,6} and Set H {8, 10} G ⋃ H = {2, 4, 6, 8, 10}. The intersection of sets consists of the elements that two or more sets have in common. Suppose that Set J {1, 2, 3,4, 5} and Set K {a, 3, d, 4, q}. The intersection of sets J⋂K is {3, 4}. Those are the only points common to sets J and K.

Figure 2: Venn diagrams as operational definitions of sets and operations.

The complement of a set is in relation to the universal set for that problem. Remember the universal set F with the elements {2, 4, 6, 8, 10, 12}? If Set O {6, 8, 10}, the complement of O (Ō), is {2, 4, 12}. If the universal set for a problem is** I**, the set of integers, and set L is {1, 3, 5, 7 …} the complement of L will be {0, 2, 4, 6 …}.

Figure 3: Some symbols and terminology of sets.

Suppose that 100 people are asked what types of shows they like to watch on TV. Suppose that 15 watch sports, 15 watch mysteries and true crime, 12 watch reality TV, 20 watch reality TV and mysteries, 28 watch sports and mysteries, and 3 watch sports, mysteries, and reality TV. How many people watch neither sports, mysteries, nor reality TV? This can be solved by using a Venn diagram and logic. Let Set S equal sports, Set M equal mysteries, and Set R equal reality TV. The intersection of S⋂M⋂R is 3, so the complement will be 15 -3 or 12. In other words, there are 12 others of the people who watch sports. Some of them watch only sports, some of them watch sports and mysteries, and some of them watch sports and reality TV. We already know that there are 15 people who watch mysteries, but some of them also watch sports, and some of them also watch reality TV. Similarly, the intersection of S⋂M⋂R is 3, so the complement will be 15 -3 or 12. There are 12 people who watch reality TV, but 3 of them also watch sports and mysteries, so the complement of 12-3 equals 9. The part of Set M that is not part of Set S is (15 – 3) or 12, and the part of Set M that is not part of set R is 12 -3 or 9. Finally, the part of Set R that is not part of Set S is 15 -9 or 6. 15 + 9 + 3 = 27. 100 – 27 = 73, or 73 people watch neither sports, mysteries, nor reality TV.

Figure 4: Venn diagram showing regions and relationships of 3 different sets.

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The dimensions of a matrix are important to its definition. In order for the multiplication of two matrices to be meaningful, each matrix must have certain dimensions. Multiplication of matrices is not commutative.

In order to multiply a matrix by a constant, every member of the matrix is multiplied by the constant. Suppose that matrix A consists of the members [wxyz]. Multiplying each member by a constant e means that the new matrix will consist of [ew ex ey ez]. Suppose Matrix B consists of [9 3 7 6]. Multiplying each member by 3 would lead to the new matrix [27 9 21 18].

Figure 1: Matrix A

Figure 2: In Matrix 1A, every element of Matrix A is multiplied by the constant e.

Figure 3: Matrix B is a 2 X 2 Matrix, in the same form as Matrix A, but the variables are replaced with real numbers.

Figure 4: In Matrix 1B, every element of Matrix B is multiplied by 3

The first step in multiplying matrices is to determine the dimensions of each matrix. If the number of columns of the first matrix is equal to the number of rows in the second matrix, multiplying the matrices is possible. Suppose Matrix D consists of the elements [1 3 5 7] and Matrix E consists of the elements [ 2 4 6 8 10 12]. Matrix D has the dimensions 2 X 2 and Matrix E has the dimensions 2 X 3. Matrix DE would also have the dimensions 2 X 3. It would be equal to multiplying the elements of Matrix D by the elements of Matrix E.

Figure 5: Matrix D

Figure 6: Matrix E

In the next step, multiply each element of the row in the first matrix by each element of the column of the second matrix, and add those products together to form the new elements of the product matrix. Using the example of Matrix DE, the first row of the new matrix would equal (1∙2 + 3∙8) (1∙4 + 3∙10) (1∙6 + 3∙12), or (2 + 24) (4 + 30) (6 + 36). The second row of the new matrix DE would equal each element of the second row of the first matrix multiplied by each element of the column of the second matrix or (5∙2 + 7∙8) (5∙4 + 7∙10) (5∙6 + 7∙12) or (10 + 56) (20 + 70) (30 + 84). The new matrix would consist of the elements [26 34 42 66 90 114] in a 2 x 3 matrix. .

Figure 7: Matrix DE

Suppose Matrix F consisted of the elements [1 2 3 4] in a 2 x 2 matrix and Matrix G consisted of the elements [ 0 1 3 5 7 9] in a 2 x 3 matrix.. The new Matrix FG would consist of the elements [10 15 21 20 31 44] in a 2 x 3 matrix. Matrix GF would be a 2 X3 matrix X a 2 X2 matrix, which would be undefined, as the number of columns in G (3) is not equal to the number of rows in F(2).

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]]>Thinking about sets of objects and numbers is as simple as counting and as complex as infinity and transfinite numbers. Sets are an essential underlying concept of mathematics and logic.

Mathematicians use the language of set theory to describe sets. For example, small, finite sets can be defined using the roster method, which is exactly like a team roster. Each player on the team roster can be listed as a member of a set, such as {Russell Wilson, Marshawn Lynch, Derrick Coleman, Percy Harvin, Luke Wilson …Jimmy Staten, Julius Warmsley}. The other term for sets is the description method, such as {players on the active list for the Seattle Seahawks}. Using math language, Set A = {3, 4, 5, 6} using the roster method, or Set A = {x |x is an integer ≥3 and ≤6} using the description method. In symbol form {x| x** I ≥**3** **≤6}.

Sets are equal if they contain exactly the same elements. They are equivalent if they have the same number of elements. If each member of one set can be paired with only one member of another set, they are in one-to-one correspondence. The members do not have to equal one another, but the sets have to be equivalent, the same size. This is the basic idea behind counting, as one and only one number is paired with exactly one object. The sets have the same cardinality. Some sets are finite, such as Set A or the players on the active list of the Seattle Seahawks. Other sets are infinite, such as the set of integers** I** or **Z**, the set of whole numbers **W**, the set of rational numbers **Q**, real numbers** R**, and complex numbers **C**. A mathematician named Georg Cantor showed using one-to-one correspondence that the cardinality of infinite sets differed, so that there are levels of infinity. He designated the lowest level of infinity as א_{0}, a level that is shared by the sets of integers, the sets of odd numbers, and the sets of even numbers. There are even more real numbers, and even more complex numbers, in higher levels of infinity, א_{1} and beyond.

Figure 1: Georg Cantor theorized that there were different levels of infinity.

If an element is a member of a set, the symbol for that relationship is ∈. For example, {Marshawn Lynch ∈ Seattle Seahawks}. If an element is not a member of a set, the symbol for that relationship is ∉, so that {Peyton Manning ∉ Seattle Seahawks}. Suppose set B consists of the numbers {1, 2, 3, 4, 5, 6}. Set C consists of the numbers {2, 4, 6}. In this case, C is a subset of B, because all the numbers in Set C are in set B. In symbol form, C⊂ B. If Set D contains the number {7}, it is nowhere in set B, so D⊄B. The empty set has no elements in it. By definition, the empty set is a subset of all sets.

Figure 2: Using set language to describe a relationship between an element of a set and the entire set.

Relationships between different sets are shown with Venn Diagrams. If one set is a complete subset of another, so that all the members of one set are also members of another, the circle that represents the subset will be inside the circle that represents the larger set. If circles intersect, that means that both sets have some elements in common. If the sets have no common elements, the circles will be independent from each other.

Figure 3: The Venn diagram of the types of real numbers shows their relationship.

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]]>Infinity in math means endlessness, limitlessness, and unboundedness. Different levels and symbols exist, which depend on how it is defined. There are many applications in science for the endlessly fascinating concept of infinity.

This is the most familiar concept of infinity. The number line is endless in both directions, stretching from its origin at zero, with negative numbers to the left and positive numbers to the right. A line in geometry also is endless in both directions. One of the symbols used for infinity is ∞, which looks like a figure eight turned on its side. It is often compared to the endlessness of the Mobius strip, and also is called the lemniscate.

Figure 1: The infinity symbol against infinite space.

The concept of limitlessness is related to endlessness, but is not quite the same. A familiar mathematical series such as the set of odd numbers or the set of even numbers are limitless, because the series never reaches the end. In more advanced mathematics and statistics, the idea of limitlessness is used in summations and in function analysis.

Unboundedness is related to endlessness and limitlessness. A geometric plane is unbounded, because it extends infinitely in all directions. The asymptotes of the normal curve never touch the horizontal axis completely, which means that the space under the normal curve is technically unbounded.

Figure 3: The asymptotes of the normal curve never touch the horizontal axis.

Theoretically, the infinity represented by the infinity symbol is only the beginning of infinity. A mathematician named Cantor showed that there are more real numbers than natural numbers comparing the two sets of numbers, and designated another symbol for infinity, using the first letter of the Hebrew alphabet, or א (aleph). The natural numbers can be represented by א_{0}, and sets that are even more infinite are designated as א_{1},א_{2, }and so on, in advanced mathematics. Infinity has many applications in science, such as the infinite universe, black holes and other singularities, and infinite values used in computer programs. Chemists, physicists, and other scientists apply calculations of infinite series and limits to data from experiments.

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