Polynomials

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) which may be multiplied by a finite number of variables (usually represented by letters), also called indeterminates.[5] Each variable may have an exponent that is a non-negative integer, i.e., a natural number. The exponent on a variable in a term is called the degree of that variable in that term, the degree of the term is the sum of the degrees of the variables in that term, and the degree of a polynomial is the largest degree of any one term. Since x = x1, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a (nonzero) constant term is 0. The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers modulo of some prime number p. In most of the examples in this section, the coefficients are integers.
For example:

is a term. The coefficient is –5, the variables are x and y, the degree of x is in the term two, while the degree of y is one.
The degree of the entire term is the sum of the degrees of each variable in it, so in this example the degree is 2 + 1 = 3.
Forming a sum of several terms produces a polynomial. For example, the following is a polynomial:

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero.
The commutative law of addition can be used to freely permute terms into any preferred order. In polynomials with one variable, the terms are usually ordered according to degree, either in "descending powers of x", with the term of largest degree first, or in "ascending powers of x". The polynomial in the example above is written in descending powers of x. The first term has coefficient 3, variable x, and exponent 2. In the second term, the coefficient is –5. The third term is a constant. Since the degree of a non-zero polynomial is the largest degree of any one term, this polynomial has degree two.
Two terms with the same variables raised to the same powers are called "like terms", and they can be combined (after having been made adjacent) using the distributive law into a single term, whose coefficient is the sum of the coefficients of the terms that were combined. It may happen that this makes the coefficient 0, in which case their combination just cancels out the terms. Polynomials can be added using the associative law of addition (which simply groups all their terms together into a single sum), possibly followed by reordering, and combining of like terms. For example, if



then

which can be simplified to

To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other. For example, if


then

which can be simplified to


The sum or product of two polynomials is always a polynomial.
Alternative forms
In general any expression can be considered to be a polynomial if it is built up from variables and constants using only addition, subtraction, multiplication, and raising expressions to constant positive whole number powers. Such an expression can always be rewritten as a sum of terms. For example, (x + 1)3 is a polynomial; its standard form is x3 + 3x2 + 3x + 1.

Division of one polynomial by another does not, in general, produce a polynomial, but rather produces a quotient and a remainder.[6] A formal quotient of polynomials, that is, an algebraic fraction where the numerator and denominator are polynomials, is called a "rational expression" or "rational fraction" and is not, in general, a polynomial. Division of a polynomial by a number, however, does yield another polynomial. For example,

is considered a valid term in a polynomial (and a polynomial by itself) because it is equivalent to and is just a constant. When this expression is used as a term, its coefficient is therefore . For similar reasons, if complex coefficients are allowed, one may have a single term like ; even though it looks like it should be expanded to two terms, the complex number 2 + 3i is one complex number, and is the coefficient of that term.

is not a polynomial because it includes division by a non-constant polynomial.

is not a polynomial, because it contains a variable used as exponent.
Since subtraction can be replaced by addition of the opposite quantity, and since positive whole number exponents can be replaced by repeated multiplication, all polynomials can be constructed from constants and variables using only addition and multiplication.
Polynomial functions
A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies

for all arguments x, where n is a non-negative integer and a0, a1,a2, ..., an are constant coefficients.
For example, the function ƒ, taking real numbers to real numbers, defined by

is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in

An example is also the function which, although it doesn't look like a polynomial, is a polynomial function since for every x it is true that (see Chebyshev polynomials).
Polynomial functions are a class of functions having many important properties. They are all continuous, smooth, entire, computable, etc.
Polynomial equations
Main article: Algebraic equation
A polynomial equation, also called algebraic equation, is an equation in which a polynomial is set equal to another polynomial.

is a polynomial equation. In case of a univariate polynomial equation, the variable is considered an unknown, and one seeks to find the possible values for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x2 – y2, where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality. This means that a polynomial identity is a polynomial equation for which all possible values of the unknowns are solutions.

Polynomials

Quadratic Equations
This is what a "Standard" Quadratic Equation looks like:



The letters a, b and c are coefficients (you know those values). They can have any value, except that a can't be 0.
The letter "x" is the variable or unknown (you don't know it yet)
Here is an example of one:



The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).
It is also called an "Equation of Degree 2" (because of the "2" on the x)
More Examples of Quadratic Equations:
In this one a=2, b=5 and c=3

This one is a little more tricky:
Where is a? In fact a=1, as we don't usually write "1x2"
b = -3
And where is c? Well, c=0, so is not shown.
Oops! This one is not a quadratic equation, because it is missing x2 (in other words a=0, and that means it can't be quadratic)
Hidden Quadratic Equations!
So far we have seen the "Standard Form" of a Quadratic Equation:

ax² + bx + c = 0

But sometimes a quadratic equation doesn't look like that!

Here are some examples of different forms for you:

In disguise → In Standard Form a, b and c
x2 = 3x -1 Move all terms to left hand side x2 - 3x + 1 = 0 a=1, b=-3, c=1
2(w2 - 2w) = 5 Expand (undo the brackets), and move 5 to left 2w2 - 4w - 5 = 0 a=2, b=-4, c=-5
z(z-1) = 3 Expand, and move 3 to left z2 - z - 3 = 0 a=1, b=-1, c=-3
5 + 1/x - 1/x2 = 0 Multiply by x2
5x2 + x - 1 = 0 a=5, b=1, c=-1



Now Play With It
I have a "Quadratic Equation Explorer" so you can see:

the graph it makes, and
the solutions (called "roots").
How To Solve It?
The "solutions" to the Quadratic Equation are where it is equal to zero. There are usually 2 solutions (as shown in the graph above).
They are also called "roots", or sometimes "zeros"
There are 3 ways to find the solutions:

1. You can Factor the Quadratic (find what to multiply to make the Quadratic Equation)
2. You can Complete the Square, or
3. You can use the special Quadratic Formula:


Just plug in the values of a, b and c, and do the calculations.

We will look at this method in more detail now.

About the Quadratic Formula
Plus/Minus

First of all what is that plus/minus thing that looks like ± ?



The ± means there are TWO answers:



Why two answers? Just look at this typical graph:


Discriminant

Do you see b2 - 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

when b2 - 4ac is positive, you get two real solutions
when it is zero you get just ONE real solution (both answers are the same)
when it is negative you get two Complex solutions
I will explain about Complex solutions later.

Using the Quadratic Formula
Just put the values of a, b and c into the Quadratic Formula, and do the calculations.

Example: Solve 5x² + 6x + 1 = 0
Coefficients are: a = 5, b = 6, c = 1

Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a

Put in a, b and c: x = [ -6 ± √(62-4×5×1) ] / (2×5)

Solve: x = [ -6 ± √(36-20) ]/10
x = [ -6 ± √(16) ]/10
x = ( -6 ± 4 )/10
x = -0.2 or -1


Answer: x = -0.2 or x = -1

And you can see them on this graph.

Check -0.2: 5×(-0.2)² + 6×(-0.2) + 1
= 5×(0.04) + 6×(-0.2) + 1
= 0.2 -1.2 + 1
= 0
Check -1: 5×(-1)² + 6×(-1) + 1
= 5×(1) + 6×(-1) + 1
= 5 - 6 + 1
= 0
Remembering The Formula
I don't know of an easy way to remember the Quadratic Formula, but a kind reader suggested singing it to "Pop Goes the Weasel":

♫ "x equals minus b ♫ "All around the mulberry bush
plus or minus the square root The monkey chased the weasel
of b-squared minus four a c The monkey thought 'twas all in fun
all over two a" Pop! goes the weasel"
Try singing it a few times and it will get stuck in your head!

Complex Solutions?
When the Discriminant (the value b2 - 4ac) is negative you get Complex solutions ... what does that mean?

It means your answer will include Imaginary Numbers. Wow!

Example: Solve 5x² + 2x + 1 = 0
Coefficients are: a = 5, b = 2, c = 1

The Discriminant is negative: b2 - 4ac = 22 - 4×5×1 = -16

Use the Quadratic Formula: x = [ -2 ± √(-16) ] / 10

The square root of -16 is 4i, where i is √-1
(Read Imaginary Numbers to find out why)

So: x = ( -2 ± 4i )/10


Answer: x = -0.2 ± 0.4i

The graph does not cross the x-axis. That is why we ended up with complex numbers.
In some ways it is actually easier ... you don't have to calculate the two solutions, just leave it as -0.2 ± 0.4i.

Summary
Quadratic Equation in Standard Form: ax² + bx + c = 0
Quadratic Equations can be factored
Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a
When the Discriminant (b2-4ac) is:
positive, there are 2 real solutions
zero, there is one real solution
negative, there are 2 complex solutions

Polygons

Polygons Property

Oslob,Cebu

Oslob is located 117km south of Cebu City

Oslob is a 4th class municipality in the province of Cebu, Philippines.

Long time ago, there was a popular place in town called “Bolocboloc”. The place was called Bolocbol because of its existing spring located beneath the shoreline at the foot of the barangay. The flow of the water was so strong as if the water was boiling which can still be seen at this present time. At present, the place is now known as Nigad (a name of a tree). The place is named Nigad because of the existing tree that grew in the place which is seldom seen to grow in the shoreline. From the name Nigad the word “Oslob” was born due to the misunderstandings between the native couple and the two guardia civil (civil guards) in the year 1785. While the said couple were taking a rest under the tree and were eating their brought boiled bananas soaking it with vinegar and salt, the two guardia civil suddenly appeared with the words, as if they were asking: “Como se llama esto pueblo?” – which if translated in English would mean:”What is the name of this town?”. The couple were astonished for they were not able to comprehend what the civil guards were saying. Since, the couple, at that time, were then soaking bananas with the vinegar and salt, they thought that the civil guards were asking them as to what they were doing, and thus, the couple answered in unison saying ”Toslob”, which means “soaking”. After hearing the word “Toslob”, the civil guards kept on repeating the word “Toslob” in the thought that the said word was the name of the town. This has been the start of the word “Toslob” which was later changed to “Oslob” due to the passes of time. Until now, the flowing of the water at Nigad was still there quenching the thirst of the many people of the place including the nearby inhabitants specially when there is a shortage of water.

The Heritage of Oslob

Out of extreme excitement to discover the heritage trail of southern Cebu which usually begins in the farthest of the five heritage towns, we missed out doing a research about the town of Oslob.  Calle del Aragones Heritage structures in Oslob are easy to identify. For one thing, the focal point of Aragones Street (the town’s oldest) from the national highway is the unfinished Spanish cuartel. The street was constructed readable  -1879. The street was named after the first parish priest of Oslob –Fray Juan Jose Aragones.   The Unfinished Spanish Cuartel We continued walking towards the cuartel. No historical marker has been installed yet and no document has been retrieved about its history but this is what we’ve found out –it was built by El Gran Maestro Don Marcus Sabandal to serve as barracks for the Spanish armies. Local historians also believed that it served as the first line of defense for the naval infantry due to the town’s strategic location. However, its construction was put on hold when the Americans arrived in 1899. The Spanish cuartel with its double rows of arches and its 91 centimeter-thick walls of coral stones were left to stand unfinished for more than a century. It is said that the stones used for its construction came from the remnants of the collapsed floor of the once five-level bell tower of the nearby church of Nuestra Señora de la Inmaculada Concepcion . Church of Nuestra Señora de la Inmaculada Concepcion Fronting the coast of Oslob is the massive church of Nuestra Señora de la Inmaculada Concepcion. Constructed in1830, the plan for the church was designed by Bishop Santos Gomez Marañon. He is the same prelate who had built the kiosk of Magellan’s Cross in Cebu City. However, the construction of the current structure is attributed to Fray Julian Bermejo, the warrior-priest who organized a military defense system composed of bantay sa hari or watchtowers and fortress churches in the southern coastal region of Cebu. Unfortunately, the church together with its centuries-old rectory is undergoing major reconstruction after both structures were razed by an 8-hour fire in 2008. Nothing was left of the old rectory except for the ruins of what seems to be an oversized bahay na bato. Flanking on the left side of the main church is its four-storey massive belfry. It is said that the Oslob belfry used to be five-levels high however the topmost floor was destroyed by a typhoon and was never rebuilt. Another account describes the belfry to be a seven-storey high with 10 bells which have collapsed in 1871. The walls and gates surrounding the church, called the paril were constructed in 1875 to act as defense frontline of the complex against the moors that enjoyed invading the pueblos along the coast. The thick antique coral stonewalls are topped by a series of inverted cone shaped stones. This unique feature completes the medieval fortress. In front of the church is a prayer room. Built in 1847, it is also known as the waiting chapel during the Spanish period since it has been used as an isolation chamber for those inflicted with leprosy. The panteon is the focal point of the cemetery complex. According to tradition, it was in this site where the Spanish priests have their penitential rituals every 8 in the evening over a platform that once stood at the center. The Oslob Watchtower Ruins From Calle Eternidad, we slipped through a street leading to the beach to view one of the seven watchtowers built along the coastline of Oslob. Constructed in 1788 by Fray Bermejo, one can only imagine how it looked like from the ruins which can still be traced based from the hexagonal plan, the rear portion of the massive wall, gun-slits and the small entrance.                          Oslob Town Plaza Once the site of the Ayutamiento, the town plaza has played host to political rallies that were attended by Philippine presidents like Ramon Magsaysay, Diosdado Macapagal and Carlos Garcia along with famous political figures.

elow its me jhocel