Chapter Contents

Polynomial, its Terms, Degree and Types

Introduction

In Maths, Polynomial is a concept which is studied under the Algebra branch. Polynomials are a particular type of algebraic expressions, when all variables involved in an algebraic expression have powers with whole numbers only.
Algebraic expressions can be studied in details in the chapter Algebraic Expression and its Operations with Examples.
Let’s first read the basic terms that comprise the polynomials before moving to the definition and various types of polynomial.

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

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

Constant

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

Example

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

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 – $\frac{4}{9}$ 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

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.

Example

We can learn polynomial with two examples:
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.
So, that makes them polynomials.

Degree of polynomial

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

Example

3x⁴ + 5x³ + 7x² + 8
This polynomial 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 of polynomial?

Polynomials have different types depending upon the degree of polynomial and number of terms involved in the polynomial.

On the basis of number of terms

Polynomials are classified and named on the basis of the number of terms it has.
In general, the naming of type of polynomial 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 the various types of polynomials with their examples.

Monomial

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

Example

8x⁵
In this polynomial, 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

7x⁶-3x⁴
In this polynomial, 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

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 polynomial

A polynomial containing four terms is called a quadrinomial polynomial.

Example

7x⁵ – 3x² + 9x + 5

Quintrinomial polynomial

A polynomial containing five terms is called quintrinomial polynomial.

Example

y⁶ + 8y⁵ + 9y⁴ + 9y² + 7

On the basis of degree

Linear polynomial

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

Example

x¹ + 8

Quadratic polynomial

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

Example

5x² + 2x – 7

Cubic polynomial

A polynomial of degree 3 is called a cubic polynomial. A cubic polynomial in one variable can have at most four terms.
Cubic polynomial in variable x can be in general form of ax³ + bx² + cx + d

Example

2x³ + 3x² – 5x + 7

Biquadratic polynomial

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

Example

2x⁴ + 5x³ – 3x² + 7x – 4

What is a Constant Polynomial?

A polynomial of degree 0 is called a constant polynomial.

Example

4, -7 etc. Why?
∵ 4 can be written as 4y⁰, where the degree of this polynomial is zero.
Also, -7 can be written as -7y⁰, where the degree of this polynomial is zero.
Hence, 4 and -7 are examples of constant polynomials.

What is a Zero Polynomial?

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

Example

0x⁵ + 0x³ + 0x² + 0x etc.

Note

Degree of zero polynomial is not defined.

List of polynomials types on the basis of number of terms

Polynomial	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

List of polynomials types on the basis of degree

Polynomial	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 polynomial?

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

2) What is degree of a polynomial?

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

3) What is degree of 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 as constant polynomial.

5) What is standard form of polynomial?

A polynomial is in which all variables are either in ascending order or descending order is referred to as 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 leading coefficient and its example?

The coefficient of the highest power of 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 leading term in polynomial and its example?

In polynomial, the term with highest exponent is called leading term. Example of 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

Solution

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 is 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

Solution

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.

Worksheet 1 on Polynomial and Types of Polynomial

Fill in the blanks

Help box

\sqrt{z} + 7

trinomial

variable

linear

quadrinomial

zero

6x⁵

degree

quadratic

1) 6x² + 7x + 5 is a polynomial in one ___________.

2) 5x is a ___________ polynomial.

3) ___________ is a leading term of 6x⁵ + 7x + 8.

4) A polynomial of degree 2 is called ___________ polynomial.

5) The degree of polynomial $\sqrt{7}$ is ___________.

6) ___________ is not a polynomial.

7) The coefficient of x⁴ in 7x⁴ + 8x³ + 3x² + 8 is ___________.

8) 4x⁹ + 6x⁸ + 7x⁴ + 5 is an example of ___________.

9) ___________ of zero polynomial is not defined.

10) A polynomial contains three non zero terms is called ___________.

Worksheet 2 on Polynomial and Types of Polynomial

Write True or False in the boxes for the following questions.

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

Worksheet 3 on Polynomial and Types of Polynomial

Match the following.

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

Worksheet 4 on Polynomial and Types of Polynomial

Multiple choice questions

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

3) 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 - $\sqrt{2}$ is

-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 Answer Key Hide Show

Last updated on: 02-07-2024

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Zeros of Polynomial and its Geometrical Meaning

Relationship of Zeros and Coefficients of Polynomials

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Algebraic Expression and its Operations with Examples

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Polynomial, its Terms, Degree and Types

Introduction

Variable

Constant

What is an Algebraic Expression?

What are Terms in algebraic expression?

Definition of Polynomial

Degree of polynomial

What are the types of polynomial?

On the basis of number of terms

Monomial

Binomial

Trinomial

Quadrinomial polynomial

Quintrinomial polynomial

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 on the basis of number of terms

List of polynomials types on the basis of degree

Frequently Asked Questions

1) What is polynomial?

2) What is degree of a polynomial?

3) What is degree of constant polynomial?

4) Is 100 a constant polynomial?

5) What is standard form of polynomial?

6) What is leading coefficient and its example?

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

Solved Examples

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

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

Fill in the blanks

1) 6x2 + 7x + 5 is a polynomial in one ___________.

2) 5x is a ___________ polynomial.

3) ___________ is a leading term of 6x5 + 7x + 8.

4) A polynomial of degree 2 is called ___________ polynomial.

5) The degree of polynomial 7 is ___________.

6) ___________ is not a polynomial.

7) The coefficient of x4 in 7x4 + 8x3 + 3x2 + 8 is ___________.

8) 4x9 + 6x8 + 7x4 + 5 is an example of ___________.

9) ___________ of zero polynomial is not defined.

10) A polynomial contains three non zero terms is called ___________.

Write True or False in the boxes for the following questions.

7x4 is a monomial.

The degree of 3y is 0.

x2 + 4x + 4 is a quadratic polynomial.

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 biquadratic polynomial is 2.

The exponent of any variable in a polynomial can be non-negative integer.

x3 + x + 8 is a cubic polynomial.

Match the following.

Multiple choice questions

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

2) x5 + 7 is a polynomial.

3) 4x + 3 is a polynomial.

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

5) The degree of polynomial 5z - 2 is

6) 10x3 is a

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

8) The degree of constant polynomial is

9) Example of binomial with degree 35 is

10) πx2 + 7x - 5 is a

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1) 6x² + 7x + 5 is a polynomial in one ___________.

3) ___________ is a leading term of 6x⁵ + 7x + 8.

5) The degree of polynomial $\sqrt{7}$ is ___________.

7) The coefficient of x⁴ in 7x⁴ + 8x³ + 3x² + 8 is ___________.

8) 4x⁹ + 6x⁸ + 7x⁴ + 5 is an example of ___________.

7x⁴ is a monomial.

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.

x³ + $\sqrt{x}$ + 8 is a cubic polynomial.

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

2) x⁵ + $\sqrt{7}$ is a polynomial.

3) 4 $\sqrt{x}$ + 3 is a polynomial.

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

5) The degree of polynomial 5z - $\sqrt{2}$ is

6) 10x³ is a

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

10) πx² + 7x - 5 is a