What is a polynomial?
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 those 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.
x,y,z,p,q,r,s etc.
Constant
In Math, a symbol having a fixed value is called a constant.
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.
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 –
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.
Consider the algebraic expression as 7 + 8x – 6x2y –
4
9
xy
So, the terms in this expression are 7, 8x, 6x2y, –
4
9
xy
Definition
An algebraic expression in which variables involved are having non negative integral powers is called a 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
For a polynomial involving one variable, the highest power of the variable is called degree of the 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.
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.
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.
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.
7x5 – 3x2 + 9x + 5
Quintrinomial
A polynomial containing five terms is called 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
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
5x2 + 2x – 7
Cubic polynomial
A polynomial of degree 3 is called a cubic polynomial. In one variable can have at most four
terms.
In variable x its general form is ax3 + bx2 + cx + d
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
2x4 + 5x3 – 3x2 + 7x – 4
What is a Constant Polynomial?
A polynomial of degree 0 is called a 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.
0x5 + 0x3 + 0x2 + 0x etc.
Degree of zero polynomial is not defined.
List of polynomials types
On the basis of 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
| 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 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.
- 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 + 1 7
- √2x + 1
- x2 + 1 x
- 1 - x3
Solution
- 7x2 + x - 5 is a quadratic polynomial as variable x has the highest power of 2.
- 9x3 + 1 7 is a cubic polynomial as variable x has the highest power of 3.
- √2 x + 1 is a linear polynomial as variable x has the highest power of 1.
-
x2 +
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
| Type: | Blanks |
| Count: | 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 √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 ___________.
Write True or False Worksheet
| Type: | True False |
| Count: | 1 |
| S.N. | Statement | ✓ or ✕ |
|---|---|---|
| 1) | 7x4 is a monomial. | |
| 2) | The degree of 3y is 0. | |
| 3) | x2 + 4x + 4 is a quadratic polynomial. | |
| 4) | √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 a non-negative integer. | |
| 10) | x3 + √x + 8 is a cubic polynomial. |
Match Columns Worksheet
| Type: | Matching |
| Count: | 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 |
Multiple Choice Questions Worksheet
| Type: | MCQ |
| Count: | 1 |
- Yes
- No
- Maybe
- None of these
- Yes
- No
- Maybe
- None of these
- Yes
- No
- Maybe
- None of these
- -4
- 6
- 7
- 2
- -5
- 1
- √2
- 0
- linear polynomial
- quadratic polynomial
- cubic polynomial
- trinomial
- 6
- 2
- -3
- 1
- 1
- 2
- Not defined
- 0
- 35x
- 3x5 + 35
- 7x35 + 4
- x35
- quadratic polynomial
- binomial
- cubic
- linear polynomial
