Concept of Polynomial Functions

A polynomial function is the most basic, widely used, and fundamental mathematical function. These functions are used to represent algebraic expressions that meet particular constraints. They also serve a variety of purposes. Because of their wide range of applications, polynomial functions must be studied and comprehended. 

Definition of a polynomial function

Polynomial is composed of the term “poly” and “nomial”. Here “poly” means “many” and “nomial” means “term,” we may say that polynomials are “algebraic expressions containing many terms.”

Polynomial functions are sums of terms that are made up of a numerical coefficient multiplied by a unique power of the independent variable. A polynomial expression is one that can be constructed from constants and symbols known as variables or indeterminates by adding, multiplying, and exponentiating to a non-negative integer power. Constants are often integers, although they may be any expression that does not include indeterminates and represents mathematical objects that can be added and multiplied. Two polynomial expressions are regarded to define the same polynomial if they can be changed from one to the other using the normal addition and multiplication characteristics of commutativity, associativity, and distributivity.

Polynomial functions are expressions that can have various degrees, coefficients, positive exponents, and constants. Here are some polynomial function examples.

F(x) = 9x +y = 0

G(x) = 7x2 + 7y3 + 9z4 = 0

H(x) = x + y +z = 0

Standard form polynomial function

The standard form of polynomial function is 

F(x) = a0 + a1x + a2x 2 + a3x 3+…+ ar-1 xr-1+ arxr , this is the algebraic expression of the function of the variable “x”.

Here, a0, a1 ,a2,a3… ,ar-1, ,ar are real number constant, ar cant be zero and also it is called as leading coefficient, r is non- negative integer.

We usually write these degrees in decreasing order of variable power, from left to right.

Degree of the polynomial function

The polynomial function’s degree is the highest power of the variable to which it is raised. 

Let us take an example, ax3 + ax2 + a.  Here the highest power is 3. So, this implies that the degree of this polynomial is 3.

Type of polynomial function

On the basis of the number of terms, a polynomial function is classified into three types and they are:

• Monomial: It can be defined as that polynomial function that has only one term.

For example: 108x, 7y, 10z etc.

• Binomial: It can be defined as that polynomial function that has only two terms.

For example: 108x+ 8z, x + 7y, y2 + 10z etc.

• Trinomial: It can be defined as that polynomial function that has only three terms.

For example: 108x + 10y + z, 6x2 + 10y4 + z, etc.

Furthermore, polynomials are categorised according to their degrees. Zero polynomial function, linear polynomial function, quadratic polynomial function, and cubic polynomial function are the four most popular forms of polynomials used in precalculus and algebra.

• Zero polynomial function: A zero polynomial can include an unlimited number of terms as well as variables of varying powers, all of which have zero as their coefficient. It is of the form f(x)=0, this function is also known as the constant function.

For example y=108.

• Linear polynomial function: The linear polynomial function can be defined as the polynomial function which has degree 1. 

For example: y= 5x2

• Quadratic polynomial function: The quadratic polynomial function can be defined as the polynomial function which has degree 2. 

For example: y= x2+ 2x+ 9 = 0.

• Cubic polynomial function: The cubic polynomial function can be defined as the polynomial function which has degree 3. 

For example: y= x3+ 2x2+ x + 10 = 0.

How do you find a polynomial function?

To establish if a function is polynomial or not, the exponents of the variables must be tested against particular constraints. The following are the conditions:

• Every term’s exponent of the variable in the function must be a non-negative whole integer.

• In other words, the variable’s exponent should not be a fraction or a negative integer.

• The function’s variable should not be contained within a radical, i.e. It should not contain any square roots, cube roots, etc.

• The variable must not be included in the denominator.

Points to remember 

Here is a summary of some key points to know when studying polynomial functions:

• The greatest power of the variable increased to determine the degree of the polynomial function.

• Constant functions are zero-degree polynomial functions.

• Linear functions are degree 1 polynomial functions.

• Quadratic functions are degree 2 polynomial functions.

• Cubic functions are third-degree polynomial functions.

• Polynomial functions are functions of a single independent variable that can appear several times, increased to any integer power.

CONCLUSION

A polynomial function is the sum of terms, each of which consists of a converted power function with positive whole number power. A polynomial function can be thought of as the equivalent of a mathematical expression. The maximum power of a variable that appears in a polynomial is referred to as the degree of the polynomial function that it is used in.  The term containing the highest power of the variable is called the leading term.