General Form of Polynomial

Polynomial expression in mathematics can be defined as an expression that consists of intermediates also termed variables and coefficients. These constituents are used to carry out operations such as addition, subtraction or multiplication. It can be also used to solve non-negative integer exponentiation. A polynomial function can also be defined as a function of a cubic, quadratic or quartic including nonnegative powers of integer x. The general form of a polynomial function with degree n is:

“f(x) = anxn + an-1xn-1 +…..+ a2x2 + a1x + a0 ”

Here, real numbers are denoted by a and are also known as coefficients of polynomials and x is intermediate.

Main Body

What is a polynomial? 

A polynomial expression can be bulky and well defined from symbols and constants which are known as intermediates or variables through multiplication, addition or exponentiation to non-negative integers. Constants are denoted by numbers but it can also be an expression where intermediates are not involved. Two expressions of a polynomial can be said to define a single polynomial if it can be transformed by application of properties such as associativity, commutativity and distributivity of multiplication and addition. If it is considered that (x-1)(x-3) and x2 – 4x + 5 are polynomial expressions representing same polynomial, then it can be written as (x-1)(x-3) = x2 – 4x + 5. 

x has been defined as an intermediate which indicates that there is no particular value for x and any value can be substituted against it. The substituted result is the mapping of associated substituted value and is a function known as a polynomial function. Thus it can be stated that value of a polynomial function can be zero and can be noted as sum of finite numbers of non-zero terms. The non-zero term will consist of a number product which is signified as a coefficient of term and intermediates that are raised to non-negative power of integers. The intermediate x with polynomial P is also denoted as P(x). This functional notation is in general used for specifying intermediate in a single phrase. 

Classification of a polynomial expression

The degree of intermediate can be determined from exponent value of that intermediate. The highest degree of a term having a non-zero coefficient has to be degree of a polynomial. The sum of intermediate degrees is signified to be the degree of a term. A term and polynomial having no intermediate can be defined as a constant polynomial or content term. Polynomials having a small degree are given specific names. As it is previously mentioned, a constant polynomial is a polynomial with zero degrees. Likely polynomials having degrees one, two or three are respectively termed as linear, quadratic and cubic polynomials.

Higher polynomial degrees are not used as usual. The term quartic and quintic polynomials are used for intermediate degrees of four and five respectively. Polynomial zero is considered as zero polynomial. Degree of such a polynomial is not considered zero rather it is explicit or undefined. It is also noted to have a finite number of roots. Zero polynomial graph is in x-axis having f(x)=0. Polynomial consisting of integer coefficients is known as an integer polynomial and a polynomial of complex coefficients is known as a complex polynomial. A single intermediate polynomial is known as a univariate polynomial whereas multiple intermediate polynomials are known as a multivariate polynomial. Two intermediate polynomials are known as bivariate polynomials. 

Application of polynomial 

Polynomial equations are applied in several fields. It is applicable in projectile motion. Projectile motion is formed when something is thrown in air and comes down due to gravitational force. Projectile motion follows a curved trajectory having predictable behaviour. A polynomial function can be modelled considering height from ground at time t. The function with respect to h can be noted as h(t) = at2 + bt + c. here, t is time and h is object’s height. Projectile motion is also termed parabolic trajectory because of its curved path of motion. 

Polynomial equation had its importance in the Pythagoras theorem. The theorem states regarding sides of a right triangle. One angle of the triangle is 90- degrees. Two sides of these triangles next to the 90-degree angle are called legs and the third side is called the hypotenuse. The Theorem is used for finding the length of the triangle. If it is considered that hypotenuse is labelled as c and the other two sides as a and b, then the equation stands as follows: a2 + b2 = c2. This theorem has found its application in civil engineering and celestial navigation. Linear equations are also in use to find out a company’s revenue and cost. The area between the two lines will determine the profit region of the company. 

Conclusion

Several streams of mathematics have demonstrated the implementation of polynomial equations. Polynomial equations have encoded a wider range of problems that include elementary word problems and complicated scientific problems. Polynomials are also applicable in physics, chemistry, social science and economics. Numerical analysis and calculus have their base in the equations. The central concept of algebraic geometry and algebraic varieties are all based on polynomial equations. Polynomials are also used for carrying out operations such as addition, subtraction, multiplication, division and composition. In addition, associative law has been followed.