What is Polynomial, Definition, Terms, Degree & Types?

In Maths, Polynomial is a concept which is studied under the Algebra branch. These are a particular type of algebraic expressions whose all variables have powers of whole numbers only.
Let's first read the basic terms of polynomials that comprise the polynomials before moving to its definition and various types.

Variable

Variables are defined and used in algebra. Its definition states that a symbol which can be used to assign different numerical values is known as a variable.
Variables are represented by any small case English letters.

Example of variables

x,y,z,p,q,r,s etc.

Constant

In Math, a symbol having a fixed value is called a constant.

Example of constants

8, 5, 9, π etc.

What is an Algebraic Expression?

A combination of constants and variables connected by some or all basic operations +, -, X, ÷ is called an algebraic expression.

Example of algebraic expression

7 + 8x
So, here we have made an algebraic expression connected by + operation by combining a constant 7 and a constant 8 with variable x.
7 + 8x - 6x²y - $\frac{14}{9}$ xy is another example of algebraic expression.

What are Terms in algebraic expression?

Various parts of an algebraic expression separated by + or - operations are called terms.

Example of terms

Consider the algebraic expression as 7 + 8x - 6x²y - $\frac{4}{9}$ xy
So, the terms in this expression are 7, 8x, 6x²y, - $\frac{4}{9}$ xy

Definition of polynomial

An algebraic expression in which variables involved are having non negative integral powers is called a polynomial.

Examples of polynomial

Example 1: x³ + 2x² + 5x + 7
Variables involved in the expression are only x.
The power of x in each term is:
x³, x has power of 3
2x², x has power of 2
5x, x has power of 1
7, 7 has power of 0
So, we can see x variable in terms of the expression has powers as 3, 2, 1 and 0; which are non negative integrals.
Example 2: x²y + xy² + 7y
So, variables involved in the expression are x and y.
The powers of x and y in each term are:
x²y, x has power of 2 and y has 1
xy², x has power of 1 and y has 2
7y, y has power of 1
So, we can see x and y variables in terms of the expression have powers as 2 or 1 which are non negative integrals also.

Degree of polynomial

For a polynomial involving one variable, the highest power of the variable is called degree of the polynomial.

Example of degree of polynomial

3x⁴ + 5x³ + 7x² + 8
This is one variable polynomial i.e. x.
The powers of the x variable are 4, 3 and 2
∴ the highest power is 4.
Hence, the degree of the polynomial is 4.

What are the types?

Polynomials have different types depending upon its degree and number of terms in it.

On the basis of number of terms

These are classified and named on the basis of the number of terms it has.
In general, the naming of its type is written by prefixing the words mono, bi and tri to "nomial".
Where mono refers to one, bi refers to two and tri refers to three.
The types are monomial, binomial and trinomial.
Let's have a look at its various types with their examples.

Monomial

A polynomial containing one non zero term is called a monomial.

Example of monomial

8x⁵
There is only one non zero term i.e. 8x⁵. Therefore, it is an example of a monomial.
-7y³
Here, -7y³ is only one non zero term.Therefore, it is also an example of a monomial.

Binomial

A polynomial containing two non zero terms is called a binomial.

Example of binomial

7x⁶-3x⁴
There are two non zero terms i.e. 7x⁶ and 3x⁴. Therefore, it is an example of binomial.

Trinomial

A polynomial containing three non zero terms is called a trinomial.

Example of trinomial

5x⁴ + 3x² - 8
Here we have three non zero terms i.e. 5x⁴, 3x² and 8. Therefore, it is an example of a trinomial.

Quadrinomial

A polynomial containing four terms is called a quadrinomial.

Example of quadrinomial

7x⁵ - 3x² + 9x + 5

Quintrinomial

A polynomial containing five terms is called a quintrinomial.

Example of quintrinomial

y⁶ + 8y⁵ + 9y⁴ + 9y² + 7

On the basis of degree

Linear polynomial

A polynomial of degree 1 is called a linear polynomial. In one variable it can have at most two terms.
In variable x its general form is ax + b

Example of linear

x¹ + 8

Quadratic polynomial

A polynomial of degree 2 is called a quadratic polynomial. In one variable it can have at most three terms.
In variable x its general form is ax² + bx + c

Example of quadratic

5x² + 2x - 7

Cubic polynomial

A polynomial of degree 3 is called a cubic polynomial. One variable can have at most four terms.
In variable x its general form is ax³ + bx² + cx + d

Example of cubic

2x³ + 3x² - 5x + 7

Biquadratic polynomial

A polynomial of degree 4 is called a biquadratic polynomial. In one variable it can have at most five terms.
In variable x its general form is ax⁴ + bx³ + cx² + dx + e

Example of biquadratic

2x⁴ + 5x³ - 3x² + 7x - 4

What is a Constant Polynomial?

A polynomial of degree 0 is called a constant polynomial.

Example of constant polynomial

4, -7 etc. Why?
∵ 4 can be written as 4y⁰, where its degree is zero.
Also, -7 can be written as -7y⁰, where its degree is zero.

What is a Zero Polynomial?

A polynomial is said to be zero polynomial if all coefficients are equal to zero.

Example of zero polynomial

0x⁵ + 0x³ + 0x² + 0x etc.

Note

The degree of zero polynomial is not defined.

List of polynomials types

On the basis of number of terms

Table of polynomial with number of terms
Name	Number of terms	Example
Monomial	1	5x²
Binomial	2	2x⁵ + 5x³
Trinomial	3	9x⁸ + 3x⁵ - 12
Quadrinomial polynomial	4	3x⁶ + x⁴ - 6x - 1
Quintrinomial polynomial	5	3x⁹ - 2y⁷ - y⁴ - 2y² - 4

On the basis of degree

Table of polynomial with its degree and standard form
Name	Standard form	Degree
Constant	ax⁰	0
Linear	ax + b	1
Quadratic	ax² + bx + c	2
Cubic	ax³ + bx² + cx + d	3
Biquadratic	ax⁴ + bx³ + cx² + dx + e	4

Frequently Asked Questions

1) What is a polynomial?

An algebraic expression in which variables have only whole numbers as its exponents, is called polynomial.

2) What is the degree of a polynomial?

In a polynomial, the highest power of a variable is called degree of polynomial.

3) What is the degree of a constant polynomial?

The degree of constant polynomial is always zero.

4) Is 100 a constant polynomial?

Yes, because we can write 100 as 100x⁰, so the highest degree is zero, which is called a constant polynomial.

5) What is the standard form of a polynomial?

A polynomial in which all variables are either in ascending order or descending order is referred to as the standard form of polynomial. i.e. the polynomials 8x⁴ + 3x³ + 7x² + 3x + 5 and 5 + 3x + 7x² + 3x³ + 8x⁴ are in standard form of polynomial. In general, the standard form of polynomial is a₀ + a₁x + a₂x² + a₃x³ _ _ _ _ + a_nxⁿ, where a₀, a₁, a₂, a₃ _ _ _ _ _ _ _ _ _ _, a_n are all real numbers and n is any whole number.

6) What is the leading coefficient and its example?

The coefficient of the highest power of the term is called its leading coefficient. Example of leading coefficient is 8x⁴ + 6x³ + 8x² - 4x + 2 and its leading coefficient is 8 as the highest coefficient power of x is 4.

7) What is the leading term in a polynomial and its example?

In polynomials, the term with the highest exponent is called the leading term. An example of a leading term is 8x⁴ + 3x³ + 2x² - 7x + 6. Here, 8x⁴ is the leading term.

Solved Examples

1) Find out which of the following polynomials are monomial, binomial and trinomial.

3x³ - 2x²
4y⁵ + 7y²
11z⁴ + 7z² + 2
5t
x³ + 7x² + 5

Solutions:

3x³ - 2x² is a binomial because there exists two terms which are 3x³ and 2x².
4y⁵ + 7y² is a binomial because there exists two terms which are 4y⁵ and 7y².
11z⁴ + 7z² + 2 is a trinomial because there exists three terms which are 11z⁴, 7z² and 2.
5t is a monomial because there exists one term which is 5t.
x³ + 7x² + 5 is a trinomial because there exists three terms which are x³, 7x² and 5.

2) Classify the following polynomials as linear, quadratic, cubic and biquadratic.

7x² + x - 5
9x³ + $\frac{1}{7}$
$\sqrt{2}$ x + 1
x² + $\frac{1}{x}$
1 - x³

Solutions:

7x² + x - 5 is a quadratic polynomial as variable x has the highest power of 2.
9x³ + $\frac{1}{7}$ is a cubic polynomial as variable x has the highest power of 3.
$\sqrt{2}$ x + 1 is a linear polynomial as variable x has the highest power of 1.
x² + $\frac{1}{x}$ = x² + x^-1
Hence, it is not a polynomial because x has power of -1, which is a negative integer.
1 - x³ is a cubic polynomial as variable x has the highest power of 3.

Fill in Blanks Worksheet

Blanks - 1

6x² + 7x + 5 is a polynomial in one ___.
5x is a ___ polynomial.
___ is a leading term of 6x⁵ + 7x + 8.
A polynomial of degree 2 is called ___ polynomial.
The degree of polynomial $\sqrt{7}$ is ___.
___ is not a polynomial.
The coefficient of x⁴ in 7x⁴ + 8x³ + 3x² + 8 is ___.
4x⁹ + 6x⁸ + 7x⁴ + 5 is an example of ___.
___ of zero polynomial is not defined.
A polynomial containing three non zero terms is called ___.

Help box

\sqrt{z}

+ 7

trinomial

variable

linear

quadrinomial

zero

6x⁵

degree

quadratic

Blanks PDF Worksheet

Write True or False Worksheet

True False - 1

7x⁴ is a monomial.
The degree of 3y is 0.
x² + 4x + 4 is a quadratic polynomial.
$\sqrt{x} + 8$ is a polynomial.
5x² + 4x + 20 is a trinomial.
The coefficient of x⁵ in x⁵ + 4x⁴ + x³ + 1 is 5.
x - x³ is a cubic polynomial.
The degree of a biquadratic polynomial is 2.
The exponent of any variable in a polynomial can be a non-negative integer.
$x^{3} + \sqrt{x} + 8$ is a cubic polynomial.

True False PDF Worksheet

Match Columns Worksheet

Matching - 1

1)	Degree of x² + 7x	a)	Binomial
2)	Leading coefficient of 6x⁴ + 7x² + 5	b)	Constant polynomial
3)	x + 5	c)	6
4)	x³ + x² + x + 1	d)	2
5)	100	e)	Cubic polynomial

Matching PDF Worksheet

Multiple Choice Questions Worksheet

MCQ - 1

1) 7x⁵ + 3x² - 5x + 2 is a polynomial.

Yes
No
Maybe
None of these

2) x⁵ +

\sqrt{7}

is a polynomial.

Yes
No
Maybe
None of these

4 \sqrt{x} + 3

is a polynomial.

Yes
No
Maybe
None of these

4) The coefficient of x³ in -4x³ + 6x² + 7x + 2

-4
6
7
2

5) The degree of polynomial

5z - \sqrt{2}

-5
1
$\sqrt{2}$
0

6) 10x³ is a

linear polynomial
quadratic polynomial
cubic polynomial
trinomial

7) The leading coefficient in polynomial x⁵ + 6x⁴ - 3x³ + 2x² + 3

6
2
-3
1

8) The degree of constant polynomial is

1
2
Not defined
0

9) Example of binomial with degree 35 is

35x
3x⁵ + 35
7x³⁵ + 4
x³⁵

10) πx² + 7x - 5 is a

quadratic polynomial
binomial
cubic
linear polynomial

MCQ PDF Worksheet

Polynomial, its Terms, Degree and Types

Variable

Constant

What is an Algebraic Expression?

What are Terms in algebraic expression?

Definition of polynomial

Degree of polynomial

What are the types?

On the basis of number of terms

Monomial

Binomial

Trinomial

Quadrinomial

Quintrinomial

On the basis of degree

Linear polynomial

Quadratic polynomial

Cubic polynomial

Biquadratic polynomial

What is a Constant Polynomial?

What is a Zero Polynomial?

List of polynomials types

Related chapters

Zeros of Polynomial and its Geometrical Meaning

Relationship of Zeros and Coefficients of Polynomials

Linear Equation with Examples and Methods to Solve

Quadratic Equation. Find Roots and Methods to Solve

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