A polynomial is an essential mathematical expression that consists of a sum of n-number of terms. Each expression in a polynomial contains a variable term allocated to positive power and raised by a coefficient. We at Sample Assignment, are one of the most trusted academic help that affords the best mathematical experts for Polynomial assignment help to students across the world.

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A polynomial is an algebraic expression of following formula:

P (x) = a_{n} x ^{n} + a_{n-1} x ^{n-1}+ a_{n-2} x ^{n-2}+...+ a_{n-1} x ^{n-1}+a_{0}

Being:

a_{n}, a _{n âˆ’ 1}... a_{1}, a_{0}_{ are }numbers, called coefficients, where,

n is a natural number

x is the variable or indeterminate

a_{n} is the principal coefficient

a_{0} is the independent term

The degree of a polynomial P (x) is the highest exponent to which the variable x is elevated.

According to their degree, the polynomials can be of:

TYPE |
EXAMPLE |

1^{st} DEGREE |
P (x) = 3x + 2 |

2^{nd} DEGREE |
P (x) = 2x ^{2}+ 3x + 2 |

3^{rd} DEGREE |
P (x) = x ^{3}-2x ^{2} + 3x + 2 |

**1. Null Polynomial**- It is that polynomial that has all its zero coefficients.

**P (x) = 0x ^{2}+ 0x + 0**

**2. Homogeneous Polynomial-** It is that polynomial in which all its terms or monomials are of the same degree.

**P (x) = 2x ^{2} + 3xy**

**3. Heterogeneous Polynomial**- It is that polynomial in which not all its terms are not of the same degree.

**P (x) = 2x ^{3} + 3x^{2} - 3**

**4. Complete Polynomial**- It is that polynomial that has all the terms from the independent term to the highest degree term.

**P (x) = 2x ^{3}+ 3x^{2} + 5x - 3**

**5. Incomplete Polynomial**- It is that polynomial that does not have all the terms from the independent term to the highest degree term.

**P (x) = 2x ^{3} + 5x - 3**

**6 .Ordered Polynomial**- A polynomial is ordered if the monomials that form it are written from highest to lowest degree.

**P (x) = 2x ^{3}+ 5x - 3**

**7. Equal Polynomials** -Two polynomials are equal if they verify:

- The two polynomials have the same degree.
- The coefficients of the terms of the same degree are equal.

**P (x) = 2x ^{3} + 5x - 3**

**Q (x) = 5x - 3 + 2x ^{3}**

**8. Similar Polynomials**- Two polynomials are similar if they verify that they have the same literal part.

**P (x) = 2x ^{3} + 5x - 3**

**Q (x) = 3x ^{3} + 7x - 2**

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Factoring is the topic that has been discussed by many mathematicians, making a journey through the history of mathematics, specifically with the solution of polynomial equations with rational coefficients.

Factoring is one of the most used tools in mathematical work to "transform" an algebraic expression in a convenient way, to solve a problem. It has appreciable importance throughout history, it is the solution of algebraic equations, at first, factorization arises from the need to solve second-degree equations.

The Babylonians were the first to solve quadratic equations. In some tablets deciphered by Neugebaveren 1930, which are about 4000 years old, solutions to several of these equations were found, using the method known today as "completing the square". About 4,000 years ago, the Babylonians knew how to find the positive solution of certain types of quadratic equations. They had a very precise "recipe" for solving equations of the type

**x ^{2} âˆ’ bx = c**

The work of the Babylonians was a remarkable achievement, considering that they did not have the modern notation and because of their high level of abstraction, considering the quartic equations as quadratic equations "in disguise" and solving them as such.

Later, Greek, Hindu, Arab, and European mathematicians studied these equations and managed to advance through time until they found the formula to solve any quadratic equation, that is, an equation of the form:

**ax ^{2} âˆ’ bx +c= 0**

Where a, b, c can be any number in the development of which the Babylonians used simple factorizations they already knew. Later, the Greeks and the Arabs managed to solve equations of the second degree using, also, the method of completing the square with the application of areas; both civilizations used geometric representations to show algebraic facts, as evidenced in the II book of Euclid's Elements. The formula that allows finding the solutions of any third-degree equation (or cubic equation) was not found until the 16th century in Italy. A cubic equation is of the form

**ax ^{2} - bx^{2} +cx +d= 0**

Where a, b, c, and d are any numbers, and a â‰ 0.

What all these equations have in particular, and what makes them be of the third degree or cubic, is that the unknown appears raised to the exponent 3, and that is the highest exponent of the unknown. For many centuries, before the 16th century, mathematicians tried to find the formula that would serve to determine the solutions of any cubic equation, without succeeding.

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