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Ellipse: Definition, Properties, Applications, Equation, Formulas
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Ellipse: Definition, Properties, Applications, Equation, Formulas
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April 8, 2025Polynomial 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.
A polynomial function is a function, for example, a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of
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
This is the general form of a polynomial :
Here,
If we break the general expression, we can identify the various components and common elements of a polynomial function. If the constant
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.
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
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
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
The range of a function is the set of all its outputs.
Example: Let us consider the function
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
Thus, the range of
Following is a list of some points that you must remember at the time of examining polynomial functions:
Q.1. Find
(i)
(ii)
Ans: We need to find
For the polynomial function
For the polynomial function
Q.2. Find the solutions of the quadratic function
Ans: Finding the solution of the quadratic function means
So,
Find two factors whose sum
The factors are
Hence, the solutions of
Q.3. Find a zero of the polynomial function
Ans: Finding a zero of
Now,
So,
Q.4. Find the solution of cubic polynomial function
Ans: Finding a solution of
Hence, the solutions of cubic polynomial function
Q.5. Find the degree of the polynomial taking into account polynomial function
Ans: The highest exponent found is
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.
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
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
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
2. The graph of a second-degree or quadratic polynomial function is a curve referred to as a parabola. It may be represented as
3. A cubic polynomial function of the third degree and can be represented as
4. A quartic polynomial function of the fourth degree and can be represented as
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
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,
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
2. Linear Polynomial function
3. Quadratic Polynomial function
4. Cubic Polynomial function
5. Quartic polynomial function
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!
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