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Maths Query > Unit > Algebra > Polynomials

# Polynomial, its Terms, Degree and Types

## What is a polynomial?

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.

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

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

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

PolynomialNumber of termsExample
Monomial15x2
Binomial22x5 + 5x3
Trinomial39x8 + 3x5 – 12
Quadrinomial polynomial43x6 + x4 – 6x – 1
Quintrinomial polynomial53x9 – 2y7 – y4 – 2y2 – 4

## List of polynomials types on the basis of degree

PolynomialStandard formDegree
Constantax00
Linearax + b1
Cubicax3 + bx2 + cx + d3
Biquadraticax4 + bx3 + cx2 + dx + e4

### 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 _ _ _ _ + anxn, where a0, a1, a2, a3 _ _ _ _ _ _ _ _ _ _, an 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.

1. 3x3 - 2x2
2. 4y5 + 7y2
3. 11z4 + 7z2 + 2
4. 5t
5. x3 + 7x2 + 5

Solution

1. 3x3 - 2x2 is a binomial because there exists two terms which are 3x3 and 2x2.
2. 4y5 + 7y2 is a binomial because there exists two terms which are 4y5 and 7y2.
3. 11z4 + 7z2 + 2 is a trinomial because there exists three terms which are 11z4, 7z2 and 2.
4. 5t is a monomial because there exists one term which is 5t.
5. 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.

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

Solution

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

## Worksheet 1

### Fill in the blanks

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

Help box
$\sqrt{z}+7$
trinomial
7
variable
linear
zero
6x5
degree

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

## Worksheet 3

### Match the following.

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

## Worksheet 4

### Multiple choice questions

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

1. Yes
2. No
3. Maybe
4. None of these

#### 2) x5 + $\sqrt{7}$ is a polynomial.

1. Yes
2. No
3. Maybe
4. None of these

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

1. Yes
2. No
3. Maybe
4. None of these

1. -4
2. 6
3. 7
4. 2

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

1. -5
2. 1
3. $\sqrt{2}$
4. 0

#### 6) 10x3 is a

1. linear polynomial
3. cubic polynomial
4. trinomial

1. 6
2. 2
3. -3
4. 1

1. 1
2. 2
3. Not defined
4. 0

1. 35x
2. 3x5 + 35
3. 7x35 + 4
4. x35