Polynomial Functions: Definition, Graphing, Domain & Range

Polynomial Functions are the simplest, most used, and most important mathematical functions. These functions are primarily used in real-world models and are the building blocks of algebra. Polynomial functions also cover a vast number of other functions. One needs to study and understand polynomial functions due to their extensive applications. In this article, we will discuss everything about different types of polynomial functions.

Polynomial Function Definition

A polynomial function is a function, for example, a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of \(x\). We can give a general definition of a polynomial and define its degree.

Polynomial functions are expressions that might contain variables of differing degrees, non-zero coefficients, positive exponents, and constants. Constants are whole numbers that happen at the end of a polynomial expression. And, the first term of \(f(x)\) is \({a_n}{x^n}\), where \(n\) is the polynomial’s highest exponent.

This is the general form of a polynomial : \(f(x) = {a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + \ldots + {a_2}{x^2} + {a_1}x + {a_0}\). Here, this algebraic expression is known as the polynomial function in variable \(x\).

Here,

1. \({a_n},{a_{n – 1}}, \ldots {a_0}\) are real number constants
2. \({a_n}\) can’t be equal to zero
3. \(n\) is a non-negative integer
4. All the powers of a polynomial function must be a whole number.

If we break the general expression, we can identify the various components and common elements of a polynomial function. If the constant \({a_n}\) is non-zero, we will say this is a polynomial function of degree \(n\) and \({a_n}\) is considered the leading coefficient.

Graphing Polynomial Functions

It is easier to represent more specific polynomial functions like linear and quadratic, but how can we define a polynomial function with a degree greater than two? Let us learn to define a polynomial expression with a degree of more than two. We can explain everything about polynomial functions in the shape of a graph. The image below shows the graphs of different polynomial functions. An essential skill in coordinate geometry involves acknowledging the connection between equations as well as their graphs.

1. Linear polynomial functions are also known as first-degree polynomials, and they can be represented as \(y=ax+b\). The graph of a linear polynomial function shapes a straight line.
2. The graph of a second-degree or quadratic polynomial function is a curve referred to as a parabola. It may be represented as \(y = a{x^2} + bx + c\).
3. A cubic polynomial function of the third degree and can be represented as \(y = a{x^3} + b{x^2} + cx + d\)
4. A quartic polynomial function of the fourth degree and can be represented as \(y = a{x^4} + b{x^3} + c{x^2} + dx + e\).

Polynomial Function Types

There is a comprehensive number of polynomials and polynomial functions that one might encounter in algebra Now, we will learn how we can classify the most common types of polynomials depending on the number of variables used in the polynomial. The three most common polynomials we usually meet up with are monomials, binomials, and trinomials. The specifics of these three types of polynomials the following are

1. Zero Polynomial function \(f(y) = a{y^0} = a\)
2. Linear Polynomial function \(f(y) = ay + b\)
3. Quadratic Polynomial function \(f(y) = a{y^2} + by + c\)
4. Cubic Polynomial function \(f(y) = a{y^3} + b{y^2} + cy + d\)
5. Quartic polynomial function \(f(y) = a{y^4} + b{y^3} + c{y^2} + dy + e\)

Polynomial Function Examples

A polynomial function has just positive integers as exponents. We can even perform different arithmetic operations for such functions as addition, subtraction, multiplication, and division.

Several of the examples of polynomial functions are \({y^7} + 4{y^3} + 16,{x^3} + 2{x^2} + 5{x^4} + 1\) and \( – 2{x^5} – 3{x^4} + 7{x^2} + 8\)

Domain and Range of Polynomial Functions

The domain of Polynomial Functions

A domain refers to “all the values” that go into a function. The domain of a function is the set of all possible inputs for the function.

Example:

When the function \(f(y) = {y^2}\) is given, and the values \(y = 1,2,3,4, \ldots \) then the domain is simply the set of natural numbers, and the output values are called the range.

\({\rm{Domain }} \to {\rm{ Function }} \to {\rm{ Range}}\)

Range of Polynomial Functions

The range of a function is the set of all its outputs.

Example: Let us consider the function \(f : A \to A\) where \(A = \left\{ {1,2,3,4} \right\}\)

The domain elements are called pre-images, and the elements of the codomain which are mapped are called the images.

Here, the range of the function \(f\) is the set of all images of the elements of the domain (or) the set of all the outputs of the function.

Thus, the range of \(f = \left\{ {2,3} \right\}\)

Polynomial Functions Notes

Following is a list of some points that you must remember at the time of examining polynomial functions:

1. Constant functions is yet another term used to refer to zero-degree polynomial functions, and they are represented as \(y=a\)
2. First-degree polynomials is yet another term used to refer to linear polynomial functions, and they are represented as \(y=ax+b\)
3. The graph of a linear polynomial function constantly forms a straight line.
4. The highest power of the variable determines the degree of the polynomial function it is raised to.

Solved examples – Polynomial Functions

Q.1. Find \(p(0),{\rm{ }}p(1)\) and \(p(2)\) for each of the following polynomial functions:
(i) \(p(y) = {y^2} – y + 1\)
(ii) \(p(t) = 2 + t + 2{t^2} – {t^3}\)
Ans: We need to find \(p(0),{\rm{ }}p(1)\) and \(p(2)\)
For the polynomial function \(p(y) = {y^2} – y + 1\)
\(p(0) = {0^2} – 0 + 1\)
\(=1\)
\(p(1) = {1^2} – 1 + 1\)
\(=1\)
\(p(2) = {2^2} – 2 + 1\)
\(=4-1=3\)
For the polynomial function \(p(t) = 2 + t + 2{t^2} – {t^3}\)
\(p(0) = 2 + 0 + 2{(0)^2} – {(0)^3}\)
\(=2\)
\(p(1) = 2 + 1 + 2{(1)^2} – {(1)^3}\)
\(=4\)
\(p(2) = 2 + 1 + 2{(2)^2} – {(2)^3}\)
\(=3\)

Q.2. Find the solutions of the quadratic function \(f(x) = {x^2} + 8x + 15.\)
Ans: Finding the solution of the quadratic function means \(f(x) = 0\)
So, \({x^2} + 8x + 15 = 0\)
Find two factors whose sum \(=8\) and product \(=15\).
The factors are \(5\) and \(3\).
\(\therefore\, {x^2} + 8x + 15 = 0 \Rightarrow {x^2} + 5x + 3x + 15 = 0\)
\( \Rightarrow x(x + 5) + 3(x + 5) = 0\)
\( \Rightarrow (x + 5)(x + 3) = 0\)
\( \Rightarrow (x + 5) = 0,(x + 3) = 0\)
\( \Rightarrow x = – 5,x = – 3\)
Hence, the solutions of \(f(x) = {x^2} + 8x + 15\) are \(x=-5\) and \(x=-3\).

Q.3. Find a zero of the polynomial function \(p(x) = 2x + 1\)
Ans: Finding a zero of \(p(x)\), is the same as solving the equation \(p(x)=0\)
Now, \(2x+1=0\) gives us \(x = \frac{{ – 1}}{2}\)
So, \(\frac{{ – 1}}{2}\) is a zero of the polynomial \(2x+1.\)

Q.4. Find the solution of cubic polynomial function \(p(x) = {x^3} – 2{x^2} – x + 2.\)
Ans: Finding a solution of \(p(x)\), is the same as solving the equation \(p(x)=0\)
\({x^3} – 2{x^2} – x + 2 = 0 \Rightarrow {x^2}(x – 2) – 1(x – 2) = 0\)
\( \Rightarrow \left( {{x^2} – 1} \right)(x – 2) = 0\)
\( \Rightarrow (x – 1)(x + 1)(x – 2) = 0\)
\( \Rightarrow (x – 1) = 0,(x + 1) = 0,(x – 2) = 0\)
\( \Rightarrow x = 1,x = – 1,x = 2\)
Hence, the solutions of cubic polynomial function \(p(x) = {x^3} – 2{x^2} – x + 2\) are \(x = 1,{\rm{ }}x = – 1,{\rm{ }}x = 2\)

Q.5. Find the degree of the polynomial taking into account polynomial function \(f(x) = 16{x^5} + 5{x^4} – 2{x^7} + {x^2}\)
Ans: The highest exponent found is \(7\) from \( – 2{x^7}.\) This implies that the degree of this polynomial is \(7.\)

Summary

In this article, we learned about polynomial function definition, graphing polynomial functions, polynomial function types, polynomial function examples, polynomial functions notes, solved examples on polynomial functions, and FAQs on Polynomial Functions.

The learning outcome of this article is helpful in advanced mathematics, and polynomials are used to construct polynomial rings and algebraic varieties.

Frequently Asked Questions – Polynomial Functions

Q.1. How to determine a polynomial function?
Ans: To determine if the function is polynomial or does not, the function needs to be validated against certain conditions for the exponent of the variables. These conditions are as follows:
1. The degree of the variable in the function must only be a positive integer and should not contain any fractional or negative powers.
2. The function variable must not be within a radical, i.e.; it should not contain square roots or cube roots.
3. The variable must not be in the denominator.

Q.2. How do you solve polynomial functions?
Ans: 1. To solve an equation, put it in standard form with \(0\) on one side and simplify.  
2. Know how many roots to expect. 
3. If you can be able to reduce the given polynomial into a linear or quadratic equation (degree \(1\) or \(2\)), solve by inspection or the quadratic formula.

Q.3. How to graph polynomial functions?
Ans: 1. Linear polynomial functions are sometimes referred to as first-degree polynomials, and they can be represented as \(y=ax+b\). The graph of a linear polynomial function constantly forms a straight line.
2. The graph of a second-degree or quadratic polynomial function is a curve referred to as a parabola. It may be represented as \(y = a{x^2} + bx + c.\)
3. A cubic polynomial function of the third degree and can be represented as \(y = a{x^3} + b{x^2} + cx + d.\)
4. A quartic polynomial function of the fourth degree and can be represented as \(y = a{x^4} + b{x^3} + c{x^2} + dx + e.\)

Q.4. How to find the degree of a polynomial function?
Ans: Degrees are very useful to predict the behaviour of polynomials, and they also help us to group the polynomials better. The highest power of the variable determines the degree of the polynomial function it is raised to. Consider this polynomial function \(f(x) = 2{x^4} + 5{x^2} + 9\), the highest exponent found is \(4\) from \(2{x^4}\). This means that the degree of this polynomial is \(4\). The term which contains the highest degree is called the polynomial’s leading coefficient, in this case, it is \(2{x^4}\).

Q.5. What is a polynomial function? Give examples.
Ans: A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc. For example, \(2x+5\) is a polynomial that has an exponent equal to \(1\).

Q.6. What are the types of polynomial functions?
Ans: There is a comprehensive number of polynomials and polynomial functions that one might encounter in algebra. Now, we will learn how we can classify the most common types of polynomials depending on the number of variables used in the polynomial. The three most common polynomials we normally meet up with are monomials, binomials, and trinomials. The specifics of these three types of polynomials the following are
1. Zero Polynomial function \(f(y) = a{y^0} = a\)
2. Linear Polynomial function \(f(y) = ay + b\)
3. Quadratic Polynomial function \(f(y) = a{y^2} + by + c\)
4. Cubic Polynomial function \(f(y) = a{y^3} + b{y^2} + cy + d\)
5. Quartic polynomial function \(f(y) = a{y^4} + b{y^3} + c{y^2} + dy + e\)

We hope this detailed article on polynomial functions has helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment section. We will be more than happy to assist you. Happy learning!