The post Crush The GRE appeared first on Math Hacks.

]]>**The GRE Exam does not have any specific eligibility requirements**. Anyone may register for the GRE, and there are no age restrictions or prerequisite qualifications for GRE 2023. A few factors that need to be taken into consideration are:

**Education:**The GRE is often required or recommended for admission to various graduate-level programs, such as master’s and doctoral programs. While there is no specific degree requirement, GRE is intended for individuals who have completed or are about to complete their undergraduate education like Bachelor’s degree.- Test Taker’s Nationality: The GRE is an international test, and individuals from all countries are eligible to take it. There are no nationality-based restrictions.

**Registration**: To take the GRE, you need to create an account on the official GRE website (www.ets.org/gre) and register for the test. The test is typically computer-based and can be taken throughout the year at designated testing centers.**Validity**: The validity of GRE scores is typically five years from the date you take the test. This means that the GRE scores you obtain are valid for five years from the test date. After the five years, the scores are no longer considered current, and most graduate schools or business schools will not accept them for admission.**Cost**: GRE test fee is generally $220 while it is $228 in a few countries.

Maximum score that you can get in GRE is 340(170 each in Quantitative and verbal reasoning).

GRE is broken down into 6 sections:

**60-minute writing section:**Argumentative Writing Task and an Issue Writing Task**Verbal Reasoning**section 1 and section 2 – 30 minutes each**Quantitative Reasoning**section 1 and 2 – 35 minutes each**Research**– 30 or 35 minutes Verbal or Quantitative

The writing section remains first while the rest of the section comes in any order with a 1-minute break between each section and a 10-minute break after section 3. The computer test lasts about 3 hours and 45 minutes in total, the paper test lasts 3 hours 30 minutes.

The GRE test consists of three main sections – Verbal Reasoning, Quantitative Reasoning, and Analytical Writing. The test is administered via a computer, and the order of the sections may vary.

The total testing time for the GRE is about 3 hours and 45 minutes, including breaks. The Verbal and Quantitative Reasoning sections are each 1 hour and 10 minutes long, while the Analytical Writing section is 1 hour.

The GRE has three sections – Verbal Reasoning, Quantitative Reasoning, and Analytical Writing.

GRE scores are scaled differently for the Verbal and Quantitative sections, with a maximum of 170 points for each. A good GRE score depends on the specific graduate programs you are applying to, but generally, scores above 160 in each section are considered competitive.

To register for the GRE, visit the official ETS website (www.ets.org/gre), create an account, and follow the registration instructions. You can select a test date and location based on availability.

Effective GRE preparation involves creating a study plan, using study materials and practice tests, identifying weak areas, reviewing content, and practicing time management. Consider joining a GRE prep course or seeking study resources online.

Some popular GRE study materials and books include “The Official Guide to the GRE General Test” published by ETS

The GRE and GMAT are both accepted for admission to business schools. Some MBA programs accept both, while others may have a preference. Research the requirements of the schools you are interested in to decide which test to take.

GRE test dates are available throughout the year. You can check available test dates and schedule your exam on the ETS website

The GRE test fee varies by country. You can find the specific fee for your location on the ETS website. Payment methods typically include credit/debit cards or PayPal

GRE score requirements vary among graduate programs and institutions. Research the specific score expectations for the schools you are interested in

ETS provides accommodations for test-takers with disabilities. You need to submit the appropriate documentation to request accommodations

You can cancel your GRE scores at the test center immediately after completing the exam. The retake policy allows you to take the GRE once every 21 days, up to five times within any continuous 12-month period.

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]]>The post Names Of Polygons – 50 Easy Polygon Names appeared first on Math Hacks.

]]>The word “geometry” comes from the Greek geos which means “earth” and metron which means “measurement”. In this chapter we will learn the differences between polygons and polyhedrons and how to calculate various lengths (perimeters), areas, and volumes (in the case of the polyhedrons) of various geometric figures.

A polygon is a closed plane figure made up of segments (also called sides). Polygons are named according to the number of segments from which they are comprised. The term n-gon is used to denote a polygon with n sides. The n is substituted, according to the number of sides that

comprise the figure, by the Greek cardinal prefixes: penta (or 5), hexa (or 6), hepta (or 7), etc. The only two exceptions are the three- and four-sided polygons which are the triangles (never “trigons”) and quadrilaterals (never “tetragons”). The number of sides is equal to the number of

angles in all polygons.

Polygons can be classified in two different ways: internal angles and lengths of sides. Depending on the measure of the internal angles, polygons can be convex, with all internal angles less than 180° (see page 211), or concave (with at least one or more internal angles greater than 180°). You can distinguish a convex polygon from a concave polygon by noting that convex polygons have all their angles pointing out, while concave polygons have at least one angle that points inward. It is impossible to construct a 3-sided (triangular) concave polygon.

Convex polygons can also be classified as regular (when they are equilateral―that is, all sides have the same length; and equiangular―that is, all angles have the same measure) and irregular (when at least one angle or side differs from the rest).

The parts that make up a polygon are the following:

• **Sides**: the segments or straight lines that form the polygon.

• **Vertices**: the point where the sides meet. The sides always form an angle at each vertex.

• **Diagonals**: are the lines that join two non-consecutive angles.

Consider the following convex hexagon:

A “hexagon” is a 6-sided polygon having 6 vertices and 9 different diagonals. In general, the number of diagonals can be computed given the number of vertices, n, using this formula:

no. of diagonals = n(n – 3)/2

Diagonals that connect the vertices of a concave polygon, will sometimes lie partially or completely outside the boundaries of the object.

The polygons are named according to the number of sides they contain as given in the table below. In actual practice, instead of, for example, identifying a 40-sided polygon as a tetracontagon, most mathematicians will simply call it a 40-gon.

Name of Polygon | No. Of Sides | Name of Polygon | No. Of Sides |

triangle | 3 | enneadecagon | 19 |

quadrilateral | 4 | icosagon | 20 |

pentagon | 5 | triacontagon | 30 |

hexagon | 6 | tetracontagon | 40 |

heptagon | 7 | pentacontagon | 50 |

octagon | 8 | hexacontagon | 60 |

nonagon | 9 | heptacontagon | 70 |

decagon | 10 | octacontagon | 80 |

hendecagon | 11 | enneacontagon | 90 |

dodecagon | 12 | hectagon | 100 |

triskaidecagon | 13 | chiliagon | 1000 |

tetrakaidecagon | 14 | myriagon | 10,000 |

pentadecagon | 15 | Megagon | 1,000,000 |

hexakaidecagon | 16 | googolgon | 10^100 |

heptadecagon | 17 | n-gon | n |

octakaidecagon | 18 |

The sum of the interior angles of a general n-sided polygon is given by the following:

Sum of interior angles = (n – 2)180°

Given an interior angle, a, the exterior angle is 180 – a and the number of sides of a regular ngon having that interior angle is given by

n=360/180−a.

Knowing n, we can find a = 180 -(360/n).

The sum of the exterior angles of a polygon is always 360 degree.

All polygons by definition are two-dimensional (i.e., flat) objects. Let´s consider now threedimensional figures (i.e., solid figures that have height, width, and depth).

A polyhedron is a geometric solid in three dimensions with flat faces and straight edges. A polyhedron is a three-dimensional figure assembled with polygonal-shaped surfaces. In a manner similar to that of polygons. Polyhedrons are comprised of

• **Faces**: Plane polygon-shaped surfaces.

• **Edges**: Intersections between faces.

• **Vertices**: The vertices of the faces are the vertices of the polyhedron also. Three or more faces can meet at a vertex.

Interestingly, the number of vertices (V), edges (E), and faces (F) of a polyhedron are related according to this formula:

**V – E + F = 2**

Here are some examples of polyhedrons (or polyhedra):

Just as polygons are named based on the number of sides that comprise them, polyhedrons are named after the number of faces, that comprise them. The n-hedron names are generated by substituting the n with the Greek cardinal prefixes (tri—or 3, tetra—or 4, octa—or 8, dodeca—or 12, icosa—or 20) that match the number of faces that comprise the figure.

Name of polyhedron | Number of faces | Name of polyhedron | Number of faces |

trihedron | 3 | dodecahedron | 12 |

tetrahedron | 4 | tetradecahedron | 14 |

pentahedron | 5 | icosahedron | 20 |

hexahedron | 6 | icositetrahedron | 24 |

heptahedron | 7 | triacontrahedron | 30 |

octahedron | 8 | icosidodecahedron | 32 |

nonahedron | 9 | hexecontahedron | 60 |

decahedron | 10 | enneacontahedron | 90 |

undecahedron | 11 | n-hedron | n |

There are other types of non-polyhedral solids that we are going to consider in this chapter, known as prisms and pyramids. The main difference between these and polyhedrons consists in the different shaped-faces that constitute them.

**Prisms** are three-dimensional figures with two similar (or congruent—same shape and size, but in different positions) polygon bases that rest in parallel planes. Pyramids, on the other hand, have only one polygon base in a plane with the remainder of the faces connected to a point outside this plane.

**Regular** prisms and pyramids are named after the shape of the base they have. In the figures above, for example, since the base is a square, a regular square prism and a regular square pyramid are shown.

The post Names Of Polygons – 50 Easy Polygon Names appeared first on Math Hacks.

]]>The post Cramer’s Rule appeared first on Math Hacks.

]]>Let’s consider the system of linear equations that we solved earlier when we were graphing the solutions,

y = -2x + 3

y = x – 6

We are going to rewrite these equations in standard form, rather than work with the equations in slope and y-intercept format. The equivalent system of equations in standard form is

2x + y = 3

-x + y = -6

Rewriting these equations showing the implied coefficients, we have

2x + 1y = 3

-1x + 1y = -6

Next, we can form a matrix of the coefficients of x and y as shown,

$$\begin{bmatrix}2&1\-1&1\end{bmatrix}$$

Next, we compute what is called the determinant of the coefficient matrix by multiplying the number in the top left of the matrix times the number in the bottom right and then subtracting the product of the number in the bottom left of the matrix and the number in the upper right. Thus, we have the determinant computed as follows:

D= $$\begin{vmatrix}2&1\-1&1\end{vmatrix}$$ = (2)(1) – (-1)(1) = 2 + 1 = 3

We will also need to compute Dx and Dy. So based on the general system of equations:

a1x + b1y = c1

a2x + b2y = c2

The determinants are computed as follows:

D= $$\begin{bmatrix}a1&b1\a2&b2\end{bmatrix}$$ = a1b2 – a2b1

Dx = $$\begin{bmatrix}c1&b1\c2&b2\end{bmatrix}$$ = = c1b2 – c2b

Dy = $$\begin{bmatrix}a1&c1\a2&c2\end{bmatrix}$$ = = a1c2 – a2 c1

Then, the values of x and y that satisfy the equations are simply given by:

x = Dx/D

y = Dy/D

In summary, the x and y solutions can be computed for any system of two equations in two unknowns using these equivalent formulas (derived using Cramer’s Rule):

x = (c1b2 – c2b1)/(a1b2 – a2b1) or x = (c1b2 – c2b1)/D

y = (c2a1 – c1a2)/(a1b2 – a2b1) or y = (c2a1 – c1a2)/D

The solution to our system of linear equations is:

x = [(3)(1) – (-6)(1)]/3 = (3 + 6)/3 = 9/3 = 3

y = [(-6)(2) – (3)(-1)]/3 = (-12 + 3)/3 = -9/3 = -3

Thus, the solution is (3,-3).

It should be noted that if the determinant of the coefficient matrix is 0, then either the lines are parallel or they are identical. What is interesting about Cramer’s rule is that it can be used for systems of equations involving more than two unknowns.

Description | Formula |

Given the system of equations in two unknowns a1x + b1y = c1 a2x + b2y = c2 | |

The coefficient matrix determinant is | D = a1b2 – a2b1 |

The solution for x is | x = (c1b2 – c2b1)/D |

The solution for y is | y = (c2a1 – c1a2)/D |

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]]>The post Graphing Linear Equations appeared first on Math Hacks.

]]>If two lines are perpendicular to each other and positioned horizontally and vertically, then you have a rectangular coordinate system which is also called a Cartesian coordinate system. The horizontal line is called the X-axis and the vertical line is called the Y-axis. The other names for X-axis and Y-axis are **abscissa** and ordinate respectively. The figure shows X-axis and Y-axis. While there are four quadrants I, II, III, and IV as shown in the figure. Note: the name given to the quadrants helps to understand the system better and so should be remembered as shown in the figure.

From the figure, it is also clear that an infinite amount of values and points can be placed on both axes. The point where both lines intersect is called the origin. It divides each axis into positive and negative sections. Notice that the right-hand side of the X-axis or the horizontal axis has all the positive numbers which start from zero called the origin. The left part shows the negative numbers. Similarly, on the Y axis or the vertical axis the values above the origin are positive while below the origin are negative.

**Quadrant I:** In the first quadrant, the value of X and the value of Y are positive. If you want to plot a number (2,2) it will always lie in the first quadrant.

**Quadrant II:** In the second quadrant, the value of X is negative while the value of Y is positive. So a number(-2,2) will always lie in the second quadrant.

**Quadrant III**: (-2,-2) falls in the 3rd quadrant. This is because the value of x which is -2 and the value of Y which is again -2 are negative. So if the value of X is negative and Y is negative, the point lies in III Quadrant.

**Quadrant IV:** From the figure, we can see the value of X is positive and the value of Y is negative in the 4th quadrant. So, a point (2,- 2) lies in the IV Quadrant.

A rectangular coordinate system can be used to show the location of one or more points. A point is represented on a graph as a solid dot (or a very small black circle). The location of the point on the graph depends on its assigned x and y values that are often expressed inside parentheses like

this: (x, y). A point that is expressed in this manner is called an ordered pair (the order is always the x-value first, then a comma, followed by the y-value next). The x and y values are called the coordinates of the point. The x-value tells us where to go along the x-axis and the y-value tells us

where to go along the y-axis. Where the x and y values intersect (or meet) is the location of the point that is being specified.

To identify or plot (or graph) points on the Cartesian plane (or coordinate system) you need two numbers that comprise a coordinate pair (an x and y value) that corresponds to each point. By inspecting the signs of the coordinate pair or by the location of the point on the graph, it is easy to identify the quadrant that any given point lies in. Let´s plot the following four points on the graph below:

Notice, that a point (shown as a small black dot) has been placed for point A, given by the coordinates (2, 5) at the intersection of the vertical light-colored line going through 2 on the x-axis and the horizontal line going through 5 on the y-axis. The same is true for the positioning of the other

three points labeled B, C, and D

Name of point | Coordinates | Quadrant |

A | (2, 5) | I |

B | (2, -3) | IV |

C | (-1, 2) | II |

D | (-2, -3) | III |

Up until now, you have mostly seen expressions and equations with only one variable, such as:

x^2 + 17x – 5 or 19t + 5 = 12

However, now we are going to consider equations that have two different variables, such as:

y = 3x + 1 or y = -6x – 5

Notice in the examples above that for the equation y = 3x + 1 that if we set x equal to 1 (or x = 1), we can solve the equation for y by substituting 1 for x. Doing this substitution, we have

y = 3(1) + 1

or y = 4. When we have a pair of x and y values such as x = 1 and y = 4 that satisfy the equation (or make it true), this is called a solution that can be written as an ordered pair, (1, 4). Remember that the order of the x and y values in an ordered pair is always the x-value listed first, followed

by a comma, and then the y-value next, or (x-value, y-value). Sometimes an ordered pair is also referred to as a coordinate point or simply as a coordinate or as a point.

There are an infinite number of solutions to the equation y = 3x + 1. As an example of another solution, let us set x equal to 2 (or x = 2). Substituting the x with 2, we can solve for y,

y = 3(2) + 1

or y = 7. So that the ordered pair (2, 7) is another solution. Still, other solutions include (3, 10), or even (-2, -5) since when x = -2,

we obtain y = 3(-2) + 1 = -6 + 1 = -5.

These points four points (1, 4), (2,7), (3, 10), (-2,- 5) are shown on the graph. Notice that each of these coordinate pairs lie directly on the line since they are solutions to y = 3x + 1. What we have just graphed are points that are solutions to a linear equation. Linear equations are equations where if you were to graph all the ordered pairs that are solutions to the equation, they would lie along a straight line, and the line would extend into infinity

because there are an infinite number of solutions as x increases to infinity or as x decreases to minus infinity. A linear equation can always be represented in the general form y = mx + b; where m is called the “slope” or the coefficient (that is, the number that is multiplied by the variable x) and b is a constant (the y-coordinate value where the line intersects the y-axis). Notice that the variable x in a linear equation is raised to the power of 1 (in other words x as shown in the general form is the same as x^1). We learned earlier that this can be referred to as a first-degree polynomial. An equation that contains x raised to the 2nd power, or x^2, is not linear (or is non-linear) and is referred to as a quadratic equation (or 2nd-degree polynomial) which we will address at a later time.

It is straightforward to find solutions and graph linear equations. We simply substitute at least two different values for x into the equation and solve for the two corresponding values of y. Then, we plot the two points and draw a line through them.

Let’s demonstrate the procedure with the linear equation,

y = -2x – 1

Usually, x = 0 is an easy substitution to use so that we have

y = -2(0) – 1 = 0 – 1 = -1

So, the first coordinate point we have on the line is (0, -1). Now, let’s try substituting x = 1 into our equation so that we have

y = -2(1) – 1 = -2 – 1 = -3

So, our second coordinate point is (1, -3). Even though we just need two points to define a line, let’s go ahead and determine a third coordinate point using x = -1. Then we have

y = -2(-1) – 1 = 2 – 1 = 1

So, our third coordinate point is (-1, 1). Wasn’t that easy! Now we can simply plot our points,

(0, -1), (1, -3), and optionally (-1, 1)

and draw a line through them to graphically represent the linear equation y = -2x – 1.

The expression on the right side of the equation y = -2x – 1 contains the variable x, so we can say that y is a function of x (or y is dependent on the value that is assigned to x), where the function f(x) is equal to -2x – 1. This equivalent notation is sometimes used: f(x) = -2x – 1, where

f(x) is the same as y and indicates that the expression is a function of (or depends on) x. When x = 4, we can compute that y = -9. Equivalently, f(4) can be computed by making the assignment x = 4 and evaluating the expression on the right side—so that f(4) = -2•4 – 1, or f(4)= -9. Both of these conventions are commonly used so it is important to be familiar with this alternate notation.

When we are interested in two (or more) linear equations taken together at the same time, this is called a system of linear equations. Since each linear equation represents a line on the graph, two linear equations will usually intersect at some point that is common to both lines. This point of intersection is called the solution of the system of linear equations and it can be specified as a coordinate in the form (x, y). Let’s look at an example.

As the temperature increases in Chicago, ice cream sales increase, but hot dog sales decrease. This intuitively makes sense since something cold, such as ice cream, is a nice treat in the summer heat; yet, a hot snack, such as a hot dog, is not very popular in such heat.

After some research, it was discovered that the estimated number of ice cream sales per day, y, is given by the linear equation:

y = 3x – 25

where x is the temperature in degrees centigrade. Similarly, the estimated number of hot dog sales is given by the linear equation:

y = – 4x + 115

The charts and graph below show how ice cream sales and hot dog sales vary with temperature.

We have set the x-value to 0, then 10, then 20, 30, and 40 in both equations that describe ice cream and hot dog sales and show the corresponding y-values that we obtained using the equations. Notice that at degree C (and lower negative temperatures), ice cream sales are -25—and such

a negative number of sales is not practical; also, at 30 degrees C (and higher), a negative number of hot dog sales is not practical. Both of the lines that represent sales as a function of temperature (that is, sales are dependent on temperature) are plotted below. Notice the point of intersection

of the two lines below.

At what temperature are ice cream sales and hot dog sales the same?

The point of intersection occurs at a temperature of 20°C. Thus, if we substitute x = 20, into either the ice cream or hot dog sales equation we obtain the same number of sales in each case,

y = 3(20) – 25 = 60 – 25 = 35 or y = -4(20) + 115 = -80 + 115 = 35

The solution can be obtained by inspecting the graph above of the two equations. The two lines intersect at (20, 35), which is called the solution of the system of equations. Please note carefully that while point (10, 5) is located directly on the line representing ice cream sales, this point is not a solution to the system of equations—since the solution to the system of equations must consist of a point that lies on BOTH lines. In other words, the solution to the system of equations is usually only one point located at the intersection of both lines.

Now suppose we had chosen to substitute x = -1, x =0, and x =1, which represent temperatures of -1°C, 0°C, and 1°C, into the two linear equations, as we have done in earlier problems. Then for ice cream sales, using y =3x – 25 we would have the points (-1, -28), (0, -25), and (1, -22).

For hot dog sales, using y = -4x + 115, we would have the points (-1, 119), (0, 115), and (1,111). If we plotted and extended the lines we obtained from these points, this would yield the same solution (point of intersection) that we obtained, (20, 35).

Because of the large y-coordinates, however, our graphing of these lines would take a rather large sheet of paper, and it would take extra care to draw a rather lengthy y-axis and then extend the lines accurately. Notice that by labeling the axes 0, 5, 10, 15, 20, etc.―using 5-unit increments―we were able to plot the lines and find the intersection in a practical size on paper. Some problems may call for you to determine the scale to use on the x-and/or y-axes; it may not be practical to always increment the axes values by 1 unit when graphing a system of linear equations and attempting to find the point of intersection of the lines―especially if either the x- or y-coordinates of that point exceed perhaps 20.

At this point in your introduction to graphing, you may have noticed that some linear equations increase more rapidly than others as the x-value increases. The following linear equations are plotted on graphs with the same scale on the x-axes and the same scale on the y-axes so that the two

slopes or rates of increase may be compared.

The difference in the angles of the two lines is called the slope. A horizontal line has a slope of 0; however, the slope increases as the line gets steeper and approaches vertical (the slope of a vertical line is infinity). The slope of a line is defined as the ratio of the change in y to the change in x between any two points on the line. If we have any two points on a given line, such as (𝑥1,𝑦1) and (𝑥2,𝑦2), then the formula for the slope is given by slope = (𝑦2− 𝑦1)/(𝑥2− 𝑥1) = (change in y)/(change in x) = ∆𝑦/∆𝑥, where ∆ is the Greek letter “delta” meaning “change” (or difference). Let’s select two points on the line y = 0.5x, (2, 1) and (4, 2), notice that we compute the slope using the above formula we obtain

slope = (2 − 1)/(4 − 2) = 1/2 or 0.5

Please notice that the slope is the same as the coefficient of x in our equation of the line y = 0.5x. Also, if you happen to use the coordinates in the reverse order, the slope is unchanged:

Description | Definition | Comments |

Point or coordinate | (x, y) | Also called an ordered pair. |

Equation of a line in slope and y-intercept format | y = mx + b | Where m is the slope and b is the y-intercept |

Slope | m = (𝑦2−𝑦1)/(𝑥2−𝑥1) = ∆𝑦/∆x | Slope is undefined for a vertical line and 0 for a horizontal line. |

y-intercept | b = y – mx | y-value where the line intersects the y-axis (corresponding to x=0) |

Origin | (0, 0) | Point at which the x and y axes intersect |

The post Graphing Linear Equations appeared first on Math Hacks.

]]>The post Dividing Polynomials 4 Easy Ways To Remember appeared first on Math Hacks.

]]>- Numerator: The number above the line in a fraction is called a numerator. For example, 3/4 here 3 is called a numerator.
- Denominator: The number below the line is called denominator. For example, 3/4 here 4 is called denominator.
- Quotient: The result of the division is called Quotient. If you divide 12/4 you get the answer as 3, which is a quotient.
- Remainder: Some numbers are not completely divisible. There is left over after division which is called Remainder. Dividing 12/5 gives Quotient 2 and the remainder also as 2.

Let’s first consider how to divide a polynomial by a monomial. In this type of division, the only thing necessary is to divide each term of the polynomial by the monomial. This follows the basic rule in fractions, whereby such a fraction can be rewritten as the sum of fractions consisting of

each numerator term divided by the term in the denominator.

$$ \frac{a\;+\;b}c\;=\;\frac ac\;+\;\frac bc $$

Consider the following example word problem:

Mary Anne bought 6 identical shirts at the store and spent an additional $28 for a new pair of jeans. She had a coupon that gave half of her total purchase. If she ended up paying a total of $48, what was the original cost of each shirt (before the half-off coupon)?

Let x = the original cost of each shirt. Before using the coupon, Mary Anne spent 6x + 28 for the six shirts and pair of jeans. However, since she had a coupon that gave her half off, we must divide the total cost of her purchase by 2, thus we have the equation:

$$ \frac{6x\;+\;28}2\;=\;48 $$

This can be simplified by dividing each term on the left side of the equation by 2 so that we have

$$ \frac{6x\;+\;28}2\;=\;\frac{6x}2\;+\;\frac{28}2 $$

Performing the division, we have,

$$ 3x\;+\;14\;=\;48 $$

To solve for x, we must subtract 14 from each side of the equation:

$$ 3x\;=\;48\;-14 $$

Next, we divide by 3, to obtain

$$ x\;=\;\frac{34}3\;=\;11\frac13 $$

Thus, the original price for each shirt was $11.33.

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. The following property of exponents is useful when a polynomial is divided by a variable raised to some power:

$$ \frac{x^a}{x^b}\;=\;x^{a-b} $$

$$ \frac{20x^3\;-\;80x^2\;+\;16x}{4x} $$

First, we divide each term in the numerator by the term in the denominator,

$$ \frac{20x^3}{4x}\;-\;\frac{80x^2}{4x}\;+\;\frac{16x}{4x} $$

Then using the property of exponents (or simply canceling like terms), we can reduce the expression to,

$$ 5x^2\;-\;20x+\;4 $$

It is a formal method of division. Also know as bus stop method. We follow 4 steps in long division

- Divide: Take the dividend (Here – 24.60) and the divisor (Here – 12) and do normal division. Here you can take 24 and then divide it by 12.
- Quotient: Write the quotient above the line as shown in the above figure, which is 2 in this case.
- Subtraction: Do the multiplication (divisor x Quotient = 12 x 2 = 24) and then perform subtraction. Here 24 – 24 = 0
- Bring down: write down the remainder and bring the next number down which is 6 here.

Keep repeating the process until you are not able to bring down any number.

It is used to manually perform the division of polynomials. This method uses less number of steps compared to long division method. It is also as short hand method.

In synthetic division method, we follow these steps

- Write down the coefficients of the dividend. Here 2x^3 – 5x^2 -x + 3 is the dividend and has descending powers, meaning x has degrees starting from 3 up to 0.
**Note**: 3 can also be written as 3x^0. If the power is missing, then simply add that power with a coefficient as 0. Why add a coefficient of 0? This is because this doesn’t change the meaning of polynomials. coefficient 0 means a number is multiplied by 0 and any number when multiplied by 0 gives 0. So it doesn’t change the dividend in polynomials. - Use the root associated with the divisor. Divisor is x + 3 so if I considered x + 3 = 0 we get x = -3. Now place -3 and the coefficients of all the dividends as shown in the figure.
- Bring the first number down as it is ( Here 2). and then multiply it with -3, -3 x 2 = -6.
- Place the number below next number and add. Keep repeating the process till you reach last digit of the number to be multiped with.
- Number obtained at the end is the remainder. Keep it aside and starting from right place the variable with powers starting from 0. so 32 becomes 32x^0, -11 becomes -11x, 2 becomes 2x^2. So the quotient is 2x^2-11x+32.
- In case of remainder, it is always the number divided by the actual divisor. -93 here is remainder and x+3 is the number to be divided with. so remainder becomes -93/(x+3)

The splitting division method is another method to divide. In this method, you split the numbers. Say 144/8 can be split into the form 80 + something. You can use any number that is completely divisible by 8. Using 80 is simple as 8 x 10 = 80. so the number becomes 80 + 64. Dividing each number by 8 gives 10 and 8 respectively. As there is an addition between 80 and 64 we need to add 8 and 10 to get the final solution. This method is useful for dividing polynomials with a number. In this case, if I have 144x/8, we need to perform division similar way as we did without worrying about the variable that 144 has. In the end, you can just place the variables back so the solution would be 18x.

This can be applied to a polynomial as well say (144x^2 + 64x) / 8, divide 144/8 separately so answer is 18x^2 and 64x/8 which is 8x so the final answer is 18x^2 + 8x

As the name suggests breaking polynomial into factors and then dividing it is called factorization method.

In this method write down the polynomial in the form of factors. This can be done by various methods and by applying formulas.

In long division, we follow the steps divide, multiply, subtract, and bring down.

In short division, you do the multiplication and subtraction steps mentally (or with a calculator, or on scratch paper.

The difference obtained after division is brought down in long division while the difference is written next to the next digit instead of writing the difference below and bringing down the next digit to its level in short division. The calculations involved are the same.

Because more of the work is done mentally in the short division, the long division tends to be preferred over the short division for larger divisors.

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]]>The post 4 Easy Ways To Solve Recurrence Relation appeared first on Math Hacks.

]]>The number of terms arranged in a particular way is called a sequence. A recurrence relation occurs when some number in a sequence depends on the previous numbers

Example: 1,2,3,4… is a sequence of numbers whose difference is 1. By seeing this sequence you can say that the next number will be 5.

Common Examples of sequence are Arithmetic progression, Geometric Progression, and Fibonacci series.

What is a sequence?

The sequence is a set of terms where there’s a common role governing how we get from one term to the next term.

Say, for example, you have a sequence of numbers 6,13,27,55,… and we have to find the next number

So for instance, if we have six and double it, we’re going to have 12, and then having one gives 13. Likewise, if I’ve got 13 and double it, which will be 26 and have one. I get 27 and doubling 27 is 54 and adding one is 55. So you can see how we can build up the sequence. Now many authors use a particular type of notation for terms in sequences and the common notation tends to be a U. The next number that will come up in a sequence is double 55 and add 1 which is 2 * 55 + 1 = 111

what are we doing to any term to get the next term in the sequence? Well, we’re doubling it. So that would be twice U n. We’re adding one to that answer. This gives us the next term in the sequence. And the next term in the sequence would be you with a subscript N plus one… And this particular type of equation is called a recurrence relation.

If you have to find the 10th or 100th term, this is where things might get complex and so arises the need to set a formula to find it. To set the formula for it, identify what type of sequence it is and move ahead. You can find more details in arithmetic progression and geometric progression

Examples are shown in the video below

**Substitution Method**: The substitution method is**the algebraic method to solve simultaneous linear equations**. As the word says, in this method, the value of one variable from one equation is substituted in the other equation. Here we substitute the initial or given value in sequence to find the nth term. There are two ways to solve the substitution method- Forward Substitution Method: Using the below 4 steps we can find recurrence relation using this method. 1) Take the recurrence relation and initial condition 2)Put initial condition in equation and look for the pattern 3) Guess the pattern 4)Prove the pattern is correct. Example: We can calculate the running time for n=0,1,2,.. as follows n T(n) 1 1 2 T(2-1) + 1 = 1 + 1 = 2 3 T(3-1) + 1 = 2 + 1 = 3 Verifying the pattern here, T(n) = n
- Backward Substitution Method: In the forward substitution method, we put n=0,1,2… in the recurrence relation until we see a pattern. In backward substitution, we do the opposite i.e. we put n=n,n−1,n−2 until we see the pattern. After we see the pattern, we make guesswork for the running time and we verify the guesswork

**Iteration Method recurrence**: The iteration method is a**method of solving a recurrence relation**. The general idea is to iteratively substitute the value of the recurrent part of the equation until a pattern (usually a summation) is noticed, at which point the summation can be used to evaluate the recurrence.- Example: T(n) = 2T(n/2) + n , T(1) = 1. Expand and solve this recurrence relation. So what exactly are we doing here? We are going to find the pattern and reduce the given example till we reach the value of n as 1 1) Put n as n/2 so the given equation becomes T(n/2) = 2T((n/2)/2) + n/2 T(n/2) = 2T(n/2^2) + n/2 2)Substitute the value of T(n/2) in original equation T(n) = 2[2T(n/2^2) + n/2] + n T(n) = 2^2T(n/2^2) + n + n T(n) = 2^2T(n/2^2) + 2n 3) To find the pattern repeat step 2 by substituting n with n/2^2 we get, T(n/2^2) = 2T(n/2^3) + n/2^2 Substituting the value of t(n/2^2) in original equation from step 1 we get, T(n) = 2^3 T(n/2^3) + 3n 4) For ith term it will become, T(n) = 2^i T(n/2^i) + in But then we need to have value of i, for that lets compare n/2^i = 1 therefore n = 2^i Taking log on both sides, i=log n to base 2 5) Putting the value of i in equation derived in step 4 T(n) = nT(n/n) + (log n) n….. here 2^i = n from step 4 T(n) = nT(1) + n(log n) T(n) = n + nlogn So the time complexity here is O(n log n)

- Recursion Tree Method: In this method, a recurrence relation is converted into recursive trees. Each node represents the cost incurred at various levels of recursion. To find the total cost, the costs of all levels are summed up.
- Master Method: The master method is a formula for solving recurrence relations of the form:
**T(n) = aT(n/b) + f(n)**, where, n = size of input a = number of subproblems in the recursion n/b = size of each subproblem. All subproblems are assumed to have the same size.

Identify if the sequence is finite or infinite.

If the series is infinite then check the relation between numbers.

Based on the relation between numbers in sequence develop the formula.

- Linear
- Non Linear
- Homogenous Recurrence Relation
- Non Homogenous Recurrence Relation
- First Order
- Higher Order
- Divide and conquer: – general and binary

Linear recurrences are particular cases of sequences of numbers that, given initial values, the other elements can always be calculated as linear combinations of the previous

- Fibonacci Sequence
- Harmonic Numbers
- Partition of Integer
- Pell Numbers
- Pells Equation
- Diophantine Equation
- Modular forms

- Binomial Coefficient
- Pascal’s Triangle
- Dereangements
- Distribution of Identical Objects into Identical Bins
- Distribution of Distinct Objects into Identical Bins
- Permutations
- Combinations
- Partitions
- Catalian Numbers
- Markov Chains

- Arithmetic Progression
- Geometric Progression
- Euler’s Method
- Differential Equations

- Recursive Backtracking
- Dynamic Programming
- Memoization
- Computer Graphics
- Cryptography
- Numerical Analysis
- Huffman Coding
- Machine Learning
- Algorithmic Design

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]]>The post Algebra 2 Textbook With Solution – McGraw-Hills appeared first on Math Hacks.

]]>ISBN: 0-07-865980-9

Book Edition: California Edition

Pages: 1064

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]]>The post Adding and Subtracting Polynomials In Math 3 Days appeared first on Math Hacks.

]]>Adding and subtracting are two of the most basic operations you have. When solving equations with variables on each side, you have been able to

rearrange all variables to be on one side of the equation by adding or subtracting all the monomials on a selected side of the equation.

To simplify an equation, you have combined like variables by adding or subtracting terms with the same variable and exponents.

Polynomials are mathematical expressions that contain multiple terms. Basic operations such as addition, subtraction, multiplication, and division can be used to combine or simplify polynomials.

Adding and subtracting polynomials involves combining like terms, which are those with the same variables and exponents.

Step 1: Put the polynomials into standard form

To add or subtract polynomials, the first step is to put both polynomials in their standard form. In the following example, we have accomplished that step:

(-17x^3 + 8x^2 – 20x + 5) + (20x^3 – 4x^2 + 32x – 3)

Step 2: Arrange the like terms in columns and add as usual

**Step 1: Arrange Polynomials **

Both polynomials must be arranged in their standard form, which we have accomplished in this example,

(14z3 – 12z2 + 15z – 3) – (7z3 + 3z2 – 10z + 5)

Step 2: Change the sign of subtracting terms and arrange them and subtract

Since the entire second polynomial is subtracted, the sign of each term in the second polynomial can be changed so that all positive terms become negative terms and all negative terms become positive terms. Thus, we have

(14z^3 – 12z^2 + 15z – 3)- 7z^3 – 3z^2 + 10z – 5

Thus, the result is 7z^3 – 15z^2 + 25z – 8.

You can solve system of equations and cancel unknown variable in the process to find another unknown variable

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]]>The post Multiplying Binomials – 3 Ways To Solve[pdf] appeared first on Math Hacks.

]]>This stands for “First, Outside, Inside, Last.” This method involves multiplying the first term of one binomial by the first term of the other, the first term of one binomial by the second term of the other, the second term of one binomial by the first term of the other, and the second term of one binomial by the second term of the other. The results are then added together to give the final answer.

In general, this is of the form (a + b)(c + d) = ac + ad + bc + bd. A special mnemonic has been developed to describe this result: **F**OIL, or **F**irst + **O**uter + **I**nner + **L**ast which terms are defined below:

• **First**: Multiply the first terms of both binomials―ac

• **Outer**: Multiply the outer terms of both binomials, the first term of the first binomial and the last term of the last binomial―ad

• **Inner**: Multiply the inner term of both binomials, the last term of the first binomial and the first term of the last binomial―bc

• **Last**: Multiply the last term of both binomials―bd

This method involves distributing one term of one binomial to both terms of the other binomial, then adding the two products together.

The distributive property is used to derive the product of a monomial and binomials or trinomials. The distributive property is also used to derive the product of a binomial and another binomial or polynomial. Let’s consider the example of the product of two binomials as follows:

(15x + 8)(22x + 4)

Applying this general distributive property to the example product of two binomials above, we have,

**15x(22x) + 15x(4) + 8(22x) + 8 (4)**

where we have made the following substitutions: a = 15x, b = 8, c = 22x, and d = 4. Next we perform the multiplications to obtain,

**330x^2 + 60x + 176x + 32**

Finally, we simplify by combining like terms (the 60x and the 176x), to obtain the final result,

**330x^2 + 236x + 32**

Let’s do one more example together, this time using some negative terms:

**(-11x + 6)(5x – 18)**

Using the distributive property (FOIL method), we have,

**-11x(5x) + 11x(18) + 6(5x) – 6(18)**

**Step 1: **Make a box where each cell row and column is a term. Using the example (x+7)(3x + 5)

x | 7 | |

3x | ||

5 |

**Step 2**: Do the multiplication

x | 7 | |

3x | 3x^{2} | 21x |

5 | 5x | 21 |

**Step 3**: Add all the terms

3x^{2} + 21x + 5x + 21 = 3×2 + 26x + 21

There are 3 main methods explained above. Apart from this while solving sometimes you need to use various formulas like (a+b)(a+b), (a-b)(a-b), (a+b)(a-b)

Bi means two so binomial means two algebraic terms

FOIL, Distributive and BOX are three main methods for multiplying binomials

BOX method is the fastest method to do multiplication

FOIL, Distributive and BOX are three main methods for multiplying binomials

It depends on which method you are comfortable with. Using any method will lead to the same solution.

FOIL, Distributive, and BOX all these methods can be applied to multiply binomials and trinomials

This article provides free worksheet below that can be downloaded

(3x + 5)(2x-7) is one the example of binomial

The above example shows how to multiply binomial with exponent

Multiply each term with another and once done multiply with the next term.

To multiply expressions with exponents, first identify the base and exponent of each expression. Then multiply the bases together and add the exponents together to get your answer. For example, to multiply 8^3 * 7^2, you would take 8*7 = 56 and 3+2 = 5; so the answer is 56^5.

Yes the multiplication of variables with exponents is possible

To foil 4 terms, you would multiply the first term in each set of parentheses and then add that product to the product of the second term in each set of parentheses. Keep continuing the process till you reach final result where no terms can be added

Using binomial theorem helps to expand brackets to the power of 4

Binomials Notes provided below

Below is a quiz that will help you to check your knowledge of this topic

[contact-form-7]

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