Polynomial Definition

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What is a Polynomial?

Polynomial is an important chapter of Mathematics that contains variables that are also known as indeterminates and coefficients that mainly consist of the operations of addition, multiplication, etc.


In order to elaborate on this topic of Polynomial definition, it can be said that it is a numeric expression consisting of one or more terms with a non-zero coefficient. An instance of Polynomial of a single indeterminate x is x2 - 4x + 7. You can also consider the real numbers as polynomials; however, it has to be a whole number when it comes to an exponent of the Polynomial.


This particular part can be defined as the highest degree of a monomial along with a Polynomial. It is the reason why a polynomial equation possesses the largest exponent is defined as a degree of Polynomial.


Here is a table students can also refer to under the basic concept of the degree of a Polynomial definition.


Degree of Polynomial

Polynomial

Degree

Example

Constant or Zero Polynomial

0

6

Linear Polynomial

1

3x + 1

Quadratic Polynomial

2

4x2 + 1x + 1

Cubic Polynomial

3

6x3 + 4x3+ 3x + 1

Quartic Polynomial

4

6x4 + 3x3 + 3x2 + 2x + 1


Types of Polynomial

This particular chapter has three different types, which includes:

  • Monomial implies an expression with a single term. The key aspect for this one is that the single term should be a non-zero term here. Example: 5x, -3xy, etc.

  • Binomial is an expression with two terms. This is mainly a difference or a sum of two or more polynomials. Example: x2 + x, y + 2, etc.

  • Trinomial that means expression with three terms. Example: xy2 + xy.

One of the essential factors to keep in mind is that different subtypes of polynomials can be combined using subtraction, addition, division; however, it should never be divided by a variable.


Polynomial Equations

When it comes to Polynomial Definition in math, the equations play a significant role. These equations contain multiple variables and constants and should be practised properly to attain a decent score. The standard form of stating a polynomial equation that students can refer to is putting the highest degree first, and the constant term in the end. An example of that will be, b = a4 +3a3 - 2a2 + a + 1.


Polynomial Functions

'Poly' signifies many, and 'nominal' means terms, therefore, it is quite self-explanatory that gives away the fact that it is constructed with one or more terms.

Here is an idea of how the function with one variable and degree will look like:

f(x) = a0xn + a1xn-1 + a2xn-2 + ….. + an-2x2 + an-1x + an.


Polynomial Operations

Students can find mainly four sub-types of Polynomial operations, such as Addition of Polynomials, Subtraction of Polynomials, Division of Polynomials, and Multiplication of Polynomials.


The first one mainly results in a polynomial of the same degree and consists of terms like variable and power. The second one is almost similar to the first one; however, the difference is in the type of operation. Even though the result can be or not be a polynomial in case of the third one, when it comes to the multiplication of Polynomials, the result always turns out to be a polynomial of higher degree.


The basic meaning of Polynomial in Math should be explained by an algebraic expression that has variables and coefficients. Students usually find the core areas of Polynomials Definition fascinating, especially those inclined to pursue the subject in their higher education.


The Polynomial Definition is a subject that should be appropriately grasped in order to excel in Maths exam. Since the chapter contains different terms, degrees, properties, etc., students can master it by while solving these equations.


Moreover, they can take help of Vedantu - India’s leading e-learning platform to better their preparation. With exam notes, study material, live online classes, and doubt clearing sessions, individuals can now understand complicated topics like this without any difficulties. Additionally, the availability of the Vedantu app makes this process more convenient.

FAQ (Frequently Asked Questions)

1. What are Some Important Polynomial Theorems?

Ans. Polynomial consists of some essential properties and theorems, such as:

The first property starts with the Division theorem which means that a polynomial P(x) is divided by another polynomial and the result has turned out to be in quotient Q(x) with remainder R(x). The second theorem is Bezout's theorem where a polynomial P(x) is capable of being divided by binomial only if P(a) = 0. The third property is Factor theorem, where P(x) is divided by a Quotient Q(x) and provides the result R (x).

Apart from these, intermediate value theorems, Descartes' Rule of Sign, etc. are equally important factors.

2. How to Explain the Polynomial Division Steps?

Ans. The Polynomial chapter contains different rules, and this one is one of them. As a part of Polynomial operation, the long division method is used for a polynomial containing more than one term.


The steps for this method start with stating the Polynomial in descending order and then finding the highest power and dividing all the terms by that. Then you are required to subtract it and bring down the next term. Students need to keep repeating these steps until there are zero terms left. The final result, i.e. the last subtracted term, has to be in the fraction form.

3. What is the Process of Solving Quadratic Polynomials?

Ans. In order to solve the quadratic Polynomial, students are required to start by writing expressions in the descending order of degree. In the next step, the equation has to be equated, and they need to perform polynomial factorization as well to get to the result.


For instance, 3x2 - 6x + x3 - 18 is a quadratic polynomial that needs to be solved. And the solution will begin with arranging it in the descending order of degree and then equated to zero. ⇒ x3 + 3x2 - 6x - 18 = 0.


In the next step, focus on the common terms and take them out. 

⇒ -6(x + 3) + x2(x + 3) = 0

⇒ (x + 3)(x2 - 6) 0

Your final solutions will be x2 = 6, and x= -3, or x = √6.