Polynomial Functions
Polynomial functions are considered to be the simplest, most commonly used, and most important mathematical functions. These functions are mostly used in realworld models and are considered to be the building blocks of Algebra. Polynomial functions also cover a wide number of other functions. It is essential for one to study and understand polynomial functions due to their extensive applications.
In this article, let's learn about the definition of polynomial functions, their types, and graphs with solved examples.
What Is a Polynomial Function?
In this article, we will be learning about the different aspects of polynomial functions. If we break down the word polynomial into two, we get poly and nomial. Poly means many, and nomial means the term, and hence when they are combined, we can say that polynomials are algebraic expressions with many terms. Let’s go ahead and start with the definition of polynomial functions and their types.
Polynomial Function Definition
Polynomial functions are expressions that may contain variables of varying degrees, nonzero coefficients, positive exponents, and constants. Constants are whole numbers that occur at the end of a polynomial expression. And, the first term of f(x) is \(a_{n}\)xn, where n is the polynomial’s highest exponent.
This is the general form of a polynomial: f(x) = \(a_{n}\)xn + \(a_{n1}\)xn1 + ... + \(a_{2}\)x2+ \(a_{1}\)x + \(a_{0}\). This algebraic expression is called a polynomial function in variable x.
Here,
 \(a_{n}\), \(a_{n1}\), … \(a_{0}\) are real number constants
 \(a_{n}\) can’t be equal to zero
 n is a nonnegative integer
 All the exponents of a polynomial function should be a whole number
In the figure, if we break down the general expression, we can identify the various components and common elements a polynomial function is made of. If the constant \(a_n\) is nonzero, we say this is a polynomial function of degree \(n\) and \(a_n\) is the leading coefficient.
Degree of a Polynomial Function
Degrees are very useful to predict the behavior of polynomials and they also help us to group the polynomials better. The degree of the polynomial function is determined by the highest power of the variable it is raised to. Consider this polynomial function f(x) = 7x^{3} + 6x^{2} + 11x – 19, the highest exponent found is 3 from 7x^{3}. This means that the degree of this particular polynomial is 3. The term containing the highest degree is called the polynomial’s leading coefficient, in this case, it’s 7x^{3}.
What Are the Types of Polynomial Functions?
There is an extensive number of polynomials and polynomial functions that one might encounter in algebra and now we are going to learn how we can classify the most common types of polynomial based on the number of variables used in a polynomial. The three most common polynomials we usually encounter are monomials, binomials, and trinomials. The details of these three types of polynomials are as follows.
 Monomials are polynomials that contain only one term. Examples: 15x^{2}, 3b, and 12y^{4}
 Binomials are polynomials that contain only two terms. Examples: x + y, 4x – 7, and 9x + 2
 Trinomials are polynomials that contain only three terms. Examples: x^{2} – 3 + 5x, z^{4} + 45 + 3z, and x^{2} – 12x + 15x
Further, the polynomials are also classified in another way based on their degrees along with their examples. The four most common types of polynomials that are used in precalculus and algebra are cubic, quadratic, linear, and quartic.
Type of the polynomial Function  Degree  Example 
Zero Polynomial Function or constant function  0  
Linear Polynomial Function  1  x + 3, 25x + 4, and 8y – 3 
Quadratic Polynomial Function  2  5m^{2} – 12m + 4, 14x^{2} – 6, and x^{2} + 4x 
Cubic Polynomial Function  3  4y^{3}, 15y^{3} – y^{2} + 10, and 3a + a^{3} 
How Do You Determine a Polynomial Function?
In order to determine if a function is polynomial or not, the function needs to be checked against certain conditions for the exponent of the variables. These conditions are as follows:
 The degree of the variable in the function must only be a positive whole number and should not contain any fractional or negative powers.
 The variable of the function should not be inside a radical i.e, it should not contain any square roots or cube roots.
 The variable should not be in the denominator.
The belowgiven table shows an example and some nonexamples of polynomial functions:
Functions  Variable  Exponent  Constant  Terms  Polynomial Function or Not? 

f(b) = 4b^{2} – 6b + b^{3} – 15  a  2 in b^{2} and 3 in b^{3}  15  4b^{2}, 6b, b^{3}, and 15  Yes 
f(x) = x^{2/3}+ 2x  x  2/3 in x^{2/3 }and 1 in 2x^{ }  None  x^{2/3 }and 2x  No 
f(y) = 1/y^{3}  y  3 in 1/y^{3}  None  1/y^{3}  No 
Some nonexamples of Polynomial Functions:
 In the polynomial function f(x), the exponent 2/3 of the term x^{2/3} is not a whole number.
 In the polynomial function f(y), the variable y is in the denominator.
How to Represent Polynomial Function on Graph?
It is easier to represent simpler polynomial functions like linear and quadratic but how can we represent a polynomial function with a degree greater than two? Let us learn to represent a polynomial expression with a degree of more than two. We can represent all the polynomial functions in the form of a graph. The belowgiven image shows the graphs of different polynomial functions. An important skill in coördinate geometry consists in recognizing the relationship between equations and their graphs.
 Linear polynomial functions are also known as firstdegree polynomials and they can be represented as y = ax + b. The graph of a linear polynomial function always forms a straight line.
 The graph of a seconddegree or quadratic polynomial function is a curve known as a parabola. It can be represented as y = ax^{2} + bx + c.
 A cubic polynomial function of the third degree has the form shown on the right and it can be represented as y = ax^{3} + bx^{2} + cx + d.
Consider this example: Draw the graph for the quadratic polynomial function f(x) = x^{2}
First, for the polynomial function given here, f(x) = x^{2}, let us complete the table by finding the domain and the range of the function. Let's fill this table by populating the values for the function f(x) by substituting the corresponding values of the variable x.
x  4  3  2  1  0  1  2  3  4 

f(x) = x^{2}  16  9  4  1  0  1  4  9  16 
Let's draw the graph of the function.
So, the abovegiven graph is the graph for the quadratic polynomial function f(x) = x^{2}
How to Solve Polynomial Functions?
Polynomial functions are expressions that are constructed with variables and constants. These are certain steps that are to be followed to solve any polynomial function. The basic concept is to equate the given polynomial expression to 0 and solve for x. The main aim of solving a polynomial function is to find the value of x in the polynomial expression
For any given function, p(y), its zeros are found by setting the function p(y) to zero. The values of y that represent the set equation are the zeroes of the function p(y). To find the zeros of a function, find the values of y where p(y) = 0.
Linear Polynomial Function
Consider the belowmentioned polynomial function f(y). Let us solve this function by first putting all the terms on one side and 0 on the other as shown below.
16y  4 = 0
Simplifying it further, we get:
4y 1 = 0 ⇒ 4y = 1
Thus, y = 1/4 = [2 ± 2√6]/2
Quadratic Polynomial Function
Any quadratic function has the standard form: ax^{2} + bx + c, where a, b are coefficients and c is the constant. Consider the belowmentioned quadratic polynomial function f(x). Let us solve this function by first putting all the terms on one side and 0 on the other as shown below.
x^{2} + 2x 5. Here a = 1, b = 2 and c = 5
Let us use the quadratic formula to find the quadratic roots, x = [b ± √(b^{2}  4ac)]/2a
x = [2 ± √(2^{2}  4(1)(5))]/(2)(1)
= [2 ± √(4 +20)]/2
= [2 ± √(24)]/2
= [2 ± 2√6]/2
= 1 ± √6
Hence, 1 ± √6 is the solution for the polynomial function f(x)
Cubic Polynomial Function
Cubic polynomials can be solved in the same way as quadratic equations. But to make it to a much simpler form, we can use some of these special products:
 Perfect cube (2 forms): a^{3} ± 3a^{2}b + 3ab^{2} ± b^{3} = (a ± b)^{3}
 Difference of the cubes: a^{3} − b^{3} = (a − b)(a^{2} + ab +b^{2})
 Sum of the cubes: a^{3} + b^{3} = (a + b)(a^{2} − ab + b^{2})
Let us solve this cubic polynomial function y^{3} – 2y^{2} – y + 2. We will start by factorizing the equation:
y^{3} – 2y^{2} – y + 2 = y^{2}(y – 2) – (y – 2)
= (y^{2} – 1) (y – 2)
= (y + 1) (y – 1) (y– 2)
y = 1, 1 and 2.
Related Articles on Polynomial Functions
Check out the following pages related to polynomial functions:
 Multiplying Polynomials
 Multiplying Polynomials Calculator
 Multiplying Binomials Calculator
 Polynomial Calculator
 Multiplying Exponents
Important Notes on Polynomial Functions
Here is a list of a few points that should be remembered while studying polynomial functions:
 Constant functions is another term used to denote zerodegree polynomial functions, and they are represented as y = a
 Firstdegree polynomials is another term used to denote linear polynomial functions, and they are represented as y = ax + b
 The graph of a linear polynomial function always forms a straight line.
 The degree of the polynomial function is determined by the highest power of the variable it is raised to.
Examples of Polynomial Functions

Example 1: Solve the Polynomial Function f(x) = 4x  8
Solution:
f(x) = 4x  8
Let us, set the equation to 0.
4x  8 = 0
4x = 8
x = 8/4 = 2
The solution for the polynomial function 4x  8 is 2.

Example 2: Find the degree of the polynomial function f(y) = 16y^{5} + 5y^{4}− 2y^{7 }+ y^{2}
Solution:
In the given example, the highest exponent found is 7 from 2y^{7}. This means that the degree of this particular polynomial is 7.
FAQs on Polynomial Functions
What Are Polynomial Functions?
Polynomial functions are expressions that may contain variables of varying degrees, nonzero coefficients, positive exponents, and constants. For example, f(b) = 4b^{2} – 6 is a polynomial having a variable 'b' and the degree is 2.
What Are the Types of Polynomial Functions?
These are the various types of polynomial functions that are classified based on their degrees. Among these, the four most common types of polynomials that are used in precalculus and algebra are cubic, quadratic, linear, and quartic.
 Zero Polynomial Function or constant function.(F(x) = k)
 Linear Polynomial Function.(f(x) = ax + b)
 Quadratic Polynomial Function.(f(x) = ax^{2} + bx + c )
 Cubic Polynomial Function.(f(x) = ax^{3} + bx^{2} + cx + d)
 Quartic Polynomial Function. (f(x) = ax^{4} + bx^{3} + cx^{2}+ dx + e)
How Do You Write a Polynomial Function?
This is how a polynomial function is written f(x) = \(a_{n}\)x^{n} + \(a_{n1}\)x^{n1} + ... + \(a_{2}\)x^{2}+ \(a_{1}\)x + \(a_{0}\). This algebraic expression is called a polynomial function in variable x. The first term of f(x) is \(a_{n}\)x^{n}, where n is the polynomial’s highest exponent.
Here,
 \(a_{n}\), \(a_{n1}\), … \(a_{0}\) are real number constants
 \(a_{n}\) can’t be equal to zero
 n is a nonnegative integer
 All the exponents of a polynomial function should be a whole number
How to Find the Degree of Polynomial Functions?
The degree of the polynomial function is determined by the highest power of the variable it is raised to. Consider the polynomial function f(y) = 4y^{3} + 6y^{4} + 11y – 10, the highest exponent found is 4 from the term 6y^{4}. Hence the degree of this particular polynomial is 4.
How to Find the Number of Roots of a Polynomial Function?
To find the number of roots of a polynomial function, we first need to find the highestdegree term of the polynomial which is the containing the highest exponent. The highest exponent is equal to the number of roots the polynomial function will have. If the highest exponent of the given polynomial function is 2, then the function will have two roots; if the highest exponent of the function is 3, then it'll have three roots; and so on. Consider the example of a polynomial function f(u) = u^{6}  5u^{2} +2. Here, the highest exponent of the given polynomial function is 6, and hence it'll have six roots.
What Are Not Polynomial Functions? Is 0 a Polynomial Function?
For any function to be considered as a polynomial function, the function needs to follow all these conditions. These conditions are as follows:
 The degree of the variable in the function must only be a positive whole number and should not contain any fractional or negative powers.
 The variable of the function should not be inside a radical i.e, it should not contain any square roots or cube roots.
 The variable should not be in the denominator.
Any function that does not follow any of the conditions listed above cannot be considered a polynomial function. Example: The function f(a) = 2a  6a^{2} is not a polynomial function since the degree of the variable a is 2 which is a negative integer. The function f(y) = 0 is also a polynomial, but its degree is not defined.
What Are the Examples of Polynomial Functions?
Here are a few examples of polynomial functions:
 Zero Polynomial Function or constant function. Eg: y = 1
 Linear Polynomial Function. Eg: 5y + 10
 Quadratic Polynomial Function. Eg: 14x^{2} + 2x – 6
 Cubic Polynomial Function. Eg: 4y^{3} + 5y^{2} + 2
 Quartic Polynomial Function. Eg: 3y^{4} + 5