Polynomial Function

What is a Polynomial Function?

Polynomial functions are functions of single independent variables, in which variables can occur more than once, raised to an integer power, For example, the function given below is a polynomial.

Polynomial functions are the addition of terms consisting of a numerical coefficient multiplied by a unique power of the independent variables.

We generally represent polynomial functions in decreasing order of the power of the variables i.e. from left to right.

Polynomial functions are useful to model various phenomena. In Physics and Chemistry, unique groups of names such as Legendre, Laguerre and Hermite polynomials are the solutions of important issues.

In this article, we will discuss, what is a polynomial function, polynomial functions definition, polynomial functions examples, types of polynomial functions, graphs of polynomial functions etc.


Polynomial Function Definition

Polynomial functions are the most easiest and commonly used mathematical equation. It can be expressed in terms of a polynomial. The polynomial equation is used to represent the polynomial function. Generally, a polynomial is denoted as P(x). The greatest exponent of the variable P(x) is known as the degree of a polynomial. It is important to understand the degree of a polynomial as it describes the behavior of function P(x) when the value of x gets enlarged. The domain of polynomial functions is entirely real numbers (R).

Polynomial function is usually represented in the following way:

 an kn + an-1 kn-1+.…+a2k2 + a1k + a0, then for k ≫ 0 or k ≪ 0, P(k) ≈ an kn.

Hence, the polynomial functions reach power functions for the largest values of their variables.


Polynomial Function Examples

A polynomial function primarily includes positive integers as exponents. We can even carry out different types of mathematical operations such as addition, subtraction, multiplication and division for different polynomial functions.


Some of the examples of polynomial functions are given below:

  • 2x² + 3x +1 = 0

  • 4x -5 = 3

  • 6x³ + x² -1 = 0

All the three equations are polynomial functions as all the variables of the above equation have positive integer exponents. Buch some expressions given below are not considered as polynomial equations, as the polynomial includes does not have  negative integer exponents or fraction exponent or division.

  • 4x-1 + 1= 0

  • 5x1/2 +2x + 1

  • (8x +1) /1


Types of Polynomial Function

Some of the different types of polynomial functions on the basis of its degrees are given below :

  • Constant Polynomial Function -  A constant polynomial function is a function whose value  does not change. It remains the same and also it does not include any variables.

  • Zero Polynomial Function - Polynomial functions with a degree of 1 are known as Linear Polynomial functions.

  • Linear Polynomial Function - Polynomial functions with a degree of 1 are known as Linear Polynomial functions.

  • Quadratic Polynomial Function - Polynomial functions with a degree of 2 are known as Quadratic Polynomial functions.

  • Cubic Polynomial Function - Polynomial functions with a degree of 3 are known as Cubic Polynomial functions. 

  • Quartic Polynomial Function - Polynomial functions with a degree of 4 are known as Quartic Polynomial functions.


The General form of different types of polynomial functions are given below:


General Form of Different Types of Polynomial Function

Degree

Type

General Form

0

Constant

P(x) = p 

1

Linear

P(x) = px + q

2

Quadratic

P(x) = px² + qx +r

3

Cubic

P(x) = px³ + qx² + rx + s

4

Quadratic

P(x) = px⁴ + qx³ + rx² + sx¹ +t


The standard form of different types of polynomial functions are given below:


Standard Form of Different Types of Polynomial Function

Degree

Type

Standard Form

0

Constant

f(x) = a₀

1

Linear

f(x) = a₁x + a₀

2

Quadratic

f(x) = a₂x² + a₁x + a₀

3

Cubic

f(x) = a₃x³ + a₂x² + a₁x + a₀

4

Quadratic

f(x) = a₄x⁴ + a₃x³ + a₂x² + a₁x + a₀


Graphs of Polynomial Function

The graph of polynomial functions depends on its degrees. A polynomial possessing a single  variable that  has the greatest exponent is known as the degree of the polynomial.

Let us look at the graph of polynomial functions with different degrees.


Zero Polynomial Functions Graph

  • Standard form: P(x)= a₀ where a is a constant.

  • Graph: A horizontal line in the graph given below represents that the output of the function is constant. It doesn’t rely on the input.

Example, y = 4 in the below figure


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Linear Polynomial Function Graph

  • Standard form: P(x) = ax + b, where  variables a and b are constants. It draws  a straight line in the graph.

  • Graph: Linear functions include one dependent variable  i.e. x and one independent i.e y.

In the standard formula for degree 1, ‘a’ indicates the slope of a line where the constant b indicates the y-intercept of a line.

E.g., y = 2x + 3 ( In below figure)


Here, the values of variables  a and b are  2 and  3 respectively.


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Quadratic Polynomial Function

Quadratic polynomial functions have degree 2.

  • Standard form: P(x) = ax² +bx + c , where a, b and c are constant.

  • Graph: A parabola is a curve with a single endpoint known as the vertex. A parabola is a mirror-symmetric curve where each point is placed at an equal distance from a fixed point called the  focus.

In the standard form, the constant ‘a’ indicates the wideness of the parabola. The wideness of the parabola increases as ‘a’ diminishes. This can be seen by examining  the boundary case when a =0, the parabola becomes a straight line. The constant c indicates the y-intercept of the parabola. The vertex of the parabola is derived  by

(h,k) = (-b/2a, -D/4a)

where D indicates the discriminant derived by (b²-4ac).


Depends on the nature of constant ‘a’, the parabola either faces upwards or downwards

  • The parabola faces upward when  a > 0

  • The parabola faces downwards when a < 0

E.g. y = x²+2x-3 (represented  in black color in graph)

y = -x²-2x+3 ( represented  in blue color in graph)


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Graph of High Degree Polynomial Function

  • Standard form-  an kn + an-1 kn-1+.…+a0 ,a1….. an, all are constant.

  • Graph: Relies on the degree, If polynomial function degree n, then any straight line can intersect it at a maximum of n points. The constant term in the polynomial expression i.e .a₀ in the graph indicates the y-intercept.

Example: y = x⁴ -2x² + x -2, any straight line can intersect it at a maximum of 4 points ( see below graph).


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Solved Examples

1. Examine whether the following function is a polynomial function. If it is, express the function in standard form and mention its degree, type and leading coefficient.

\[f(x) = - 0.5y + \pi y^{2} - \sqrt{2}\].


Solution: Yes, the function given above is a polynomial function.

It standard from is \[f(x) = - 0.5y + \pi y^{2} - \sqrt{2}\].

The function given above is a quadratic function as it has a degree 2.

The leading coefficient of the above polynomial function is .


2. Solve the following polynomial equation

-2y + 9y - 8. find(4)


Solution: -2(4)² + 9(4) - 8

f(4) = - 32 +36 -8

f(4) = - 40 + 36

f(4) = - 4


Quiz Time

1. The zero of polynomial p(X) = 2y + 5 is

  1. 2

  2. 5/2

  3. -5/2


2. The graph of a polynomial function is tangent to its?

  1. Axis

  2. Y- Axis

  3. X- Axis

  4. Orbit

FAQ (Frequently Asked Questions)

1. Explain Polynomial Equations and also Mention its Types.

Polynomial equations are the equations formed with variables exponents and coefficients. The equation can have various distinct components , where the higher one is known as the degree of exponents. Polynomial Equations can be solved with respect to the degree and variables exist in the equation. A polynomial function is an equation which is made up of a single independent variable where the variable can appear in the equation more than once with a distinct degree of the exponent. The graph of the polynomial function can be drawn through turning points, intercepts, end behavior and the Intermediate Value theorem.


Different types of polynomial equations are:

  • Monomial Equation

  • Binomial equation

  • Trinomial or Cubic equation

  • Quadratic Polynomial Equation

  • Cubic polynomial equation

2. Explain the Degree of a Polynomial.

The degree of a polynomial in a single variable is the greatest power of the variable in an algebraic expression. In other words. Degree (for a polynomial for a single variable such as x) is the largest or greatest exponent of that variable.


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Some example of a polynomial functions with different degrees are given below:

4y = The degree is 1 ( A variable with no exponent has usually has an exponent of 1)

4y³ - y + 3 = The degree is 3 ( Largest exponent of y)

y² + 2y⁵ -y = The degree is 5 (Largest exponent of y)

x²- x + 3 = The degree is 2 (Largest exponent of x)