Polynomial - Introduction, Rules, Types, and Formula
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Table of Content
What is a Polynomial?
Rules for expressing a Polynomial
Important terminologies
Degree of a Polynomial
Types of Polynomial
Monomial
Binomial
Trinomial
Solve examples
Frequently Asked Questions
What is a Polynomial?
In Mathematics, a polynomial is an expression consisting of coefficients and variables which are also known as indeterminates.
As the name suggests poly means many and nominal means terms, hence a polynomial means many terms.
Polynomials are generally a sum or difference of variables and exponents.
Each part of the polynomial is known as a “term”.
Polynomial examples −−\[4x^{2} + 3x - 7\]
Rules for an Expression to be a Polynomial
An algebraic expression should not consist of –
Square root of variables.
Fractional powers on the variables.
Negative powers on the variables.
Variables in the denominators of any fractions.
Not a Polynomial example,
6x-2 is not a polynomial since there is negative power on the variable.
\[\sqrt{x}\] is not a polynomial, because the variable inside is a radical
\[\frac{1}{x^{2}}\] is not a polynomial, because the variable is in the denominator.
Terminologies in Polynomial Definition-
According to polynomial definition, \[4x^{2} + 3x - 7\]
In the polynomial given above,
\[4x^{2} \] = Leading term
7 = Constant
And
3x, and 7 are terms of the polynomial.
Now you Might Think about what A Constant is?
A constant is a part of the polynomial which does not consist of any variable.
As it does not consist of any variables, therefore its value never changes and it is known as a constant term.
For example −− 9x + 4,
where 4 is the constant as it does not contain any variable.
The degree of a polynomial can be defined as the highest degree of a monomial within a polynomial. The degree of the polynomial can be defined as a polynomial equation having a single variable that has the greatest exponent.
Degree of a Polynomial
Example: What is the degree of the given polynomial \[5x^{3} + 4x^{3} + 2x + 15x^{3} + 4 \times 3 + 2x +15x^{3} + 4x^{3} + 2x + 1\]
Solution: The degree of the given polynomial is 4.
Terms of a polynomial according to the polynomial definition-
The terms in a polynomial are the parts of the algebraic expression that are separated by “+” or “-” signs. Each part of the polynomial is known as a “term”.
For example, let us take a polynomial, say, \[4x^{2}+ 54x^{2}+ 54x^{2}\; + {\text{ }}5,\]
In the polynomial given below the number of terms is 2. We can classify a polynomial on the basis of the number of terms in the polynomial.
Types of Polynomial –
There are three types of polynomials.
Monomial
Binomial
Trinomial
What is a Monomial?
An algebraic expression that consists of only one term is known as a monomial. If the single term is a non-zero term then only the algebraic expression is known as a monomial. Here, are a few examples of monomials:
6x
2
\[2a^{4}\]
What is a Binomial?
An algebraic expression that consists of two terms is known as binomial. A binomial is generally represented as a sum or difference of two or more monomials. Here are a few examples of binomials:
– 9x+2,
\[6a^{4} + 11x\]
\[xy^{2} + 4xy\]
What is a Trinomial?
An algebraic expression that consists of three terms is known as trinomial Here are a few examples of trinomial expressions:
\[ 4a^{4} + 3x + 2\]
\[ 7x^{2} + 8x + 5\]
\[ 7x^{3} + 2x^{2} + 7\]
Degree of a Polynomial-
The highest or the greatest degree of a variable in any polynomial is known as the degree of the polynomial. The highest exponential power in the polynomial is the degree of the polynomial.
What do you mean by the degree of a Zero Polynomial?
If in a given polynomial all the coefficients are equal to zero, then the degree of the zero polynomial is either set equal to -1 or is undefined.
What do you mean by Constant Polynomials?
A constant polynomial contains no variables and the value of the polynomial does not change as there are no variables. In a constant polynomial, the power of the variable is equal to zero. Any constant can be expressed with a variable with its exponential power equal to zero. Constant term = 4, Polynomial form P(x)= 4x0
Solving Polynomials-
We can solve any polynomial using factorization and basic concepts of algebra. The first step to solve a polynomial is to set the right-hand side of the polynomial as 0.
There are two types of polynomials you need to know about!
Solving Linear Polynomials
Solving Quadratic Polynomials
How to Solve Linear Polynomials?
It is very easy to solve linear polynomials!
Step 1) Equate the given equation with 0, Make the equation equal to zero.
Step 2) Then solve the equation, using basic concepts of algebra.
How to Solve Linear Polynomials?
To solve a quadratic polynomial,
Step 1) We need to rewrite the given expression in the ascending order of degree.
Step 2) Equate the equation with zero.
Step 3) Use the factorization method to solve the equation.
Questions to be solved-
Question 1) What is the degree of the polynomial \[9x^{4} + 4x^{3} + 1\]?
Solution) The degree of the given polynomial \[9x^{4} + 4x^{3} + 1\] is 4 since the highest degree in the given polynomial is 4.
Question 2) What is the degree of the polynomial \[6x^{8} + 9x^{3}\]?
Solution) 8 is the degree of the polynomial \[6x^{8} + 9x^{3}\] since the highest degree in the given polynomial is 8.
Question 3) Solve the given polynomial 4x – 8.
Solution) Let’s put zero on the right-hand side,
4x – 8 = 0
\[ \Rightarrow 4x = 8\]
\[ \Rightarrow x = \frac{8}{4}\]
x= 2
Thus, the value of x= 2
Keep practicing the questions of polynomials from the available set of sample question papers and solved exercises of multiple textbooks. These study materials are available on the website or the mobile app of Vedantu.
FAQs on Polynomial
1. What is a Polynomial and what is not a Polynomial?
If an algebraic expression consists of a radical then it is not a polynomial. In order to a polynomial, all the exponents present in the algebraic expression should be non-negative integers.
2. What is a Polynomial with four terms known as?
A polynomial with four terms is generally known as quadrinomial. Since the number of terms is not important in a polynomial, therefore there is no need for such terms.
3. What is not a Polynomial?
If an algebraic expression consists of square roots of variables, fractional or negative powers on the variables, and variables in the denominators of any fractions, then it is not a polynomial.
Not a polynomial example, 6x-2 is not a polynomial since there is negative power on the variable. √x is not a polynomial, because the variable inside is radical.
4. What are Variables?
A Variable is a placeholder or a symbol for a quantity of any mathematical object. It may represent a number, a vector, a matrix, a set, or an element of the set. This symbol or term is manipulated without referring to its possible intuitive interpretation.
It can be further divided as an independent or dependent variable. A variable is called dependent when the value is determined on the basis of other factors. However, for an independent variable, the value is decided on its own without getting affected by other factors.
The topic is discussed in further detail by the subject experts at Vedantu. It is important to understand the concept of Variables and exponents to understand polynomial.
5. How can I get the confidence of solving questions related to polynomials?
Solving any mathematical questions requires a good amount of confidence and the confidence comes with clarity. To get clarity students should understand the given concept well from the available sources, it can be your textbooks or the free PDF notes and video lectures on Vedantu’s website. After which students should start solving questions related to the topic which are also compiled with solutions and made available on the website of Vedantu by the subject experts.
Once a student is done with this they will naturally get the confidence of solving any question related to the topic of polynomial,
For any further queries, you can get connected with our faculties through a 24*7 helpline facility.
6. How can we identify a polynomial Expression?
Polynomial equations can be identified by considering a few points -
A polynomial expression must not contain square roots of a variable
It must not contain any negative or fractional power on the variables
Fractions should not have any variables in their denominator
Also remember a polynomial expression is the one that has constants, variables, and exponents, and mathematical operations like addition, subtraction, etc are used to combine them.
Try to use these rules on expressions mentioned in the books and learn to identify any polymers.
7. What are exponents in polynomial expression?
To put it simply exponents are numbers that tell how many times a factor appears in a product. Let's take an example -
2
2 × 2 = 4
2 × 2 × 2 = 8
2 × 2 × 2 × 2 = 16
2 × 2 × 2 × 2 × 2 = 32
Now instead of writing 2 too many times, you can write it as 2n, where n is the exponent.
To understand more about exponents read the article on exponent from the website of Vedantu.
8. Can a variable be an exponent in a polynomial expression?
Exponents can be any real number and the variables are nothing but the set of real numbers. So, yes an exponent can be a variable.