Polynomial

A polynomial is a mathematical expression that have variables, coefficients and exponent. Polynomials are sums of these variables, coefficients and exponents. But polynomial does not contain a negative exponent.

An example of polynomial equation is:

x² - x - 1 = 0

In the word polynomial, poly  means 'many' and nomial means 'name' but here it will be used as term. So polynomial means "many terms". 

A polynomial in one variable with constant coefficients is given by

a_nxⁿ + a_n-1x^n-1 +.............. +a₂x² +a₁x +a₀

where a₀,  … ,a_n are constants and x is the variables.

Examples of polynomial:

• 6x 
• x - 4 
• 3x²  - 5x + 4
• -4y² - (³/₄)x 
• 2x⁵ – 5x³ – 10x + 9
• 6

6 could be considered as polynomial with 1 term.  Technically, the term polynomial should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms. However, the shorter polynomials do have their own names.

A one-term polynomial, such as 5x or 16x², may also be called a "monomial".

A two-term polynomial, such as 3x + y or x² – 4, may also be called a "binomial".

A three-term polynomial, such as 2a + b + c or 3x + 5y² – 3, may also be called a "trinomial".

In the polynomial expression, there should not be any negative exponent  (i.e x ^–2), the variable cannot be in the denominator (i.e 1/x), or any square root of variable. That's why below expression cannot be considered as polynomials:

4xy ^–2  -  because dividing by a variable is not allowed
3/(x+3) -  because dividing by a variable is not allowed
1/x - same as above. Variable is in denominator
√x - because square root is not allowed

Degree of terms:

The degree of a term is the sum of the exponents of all the variables in that term.

3x²  - 5x + 4

The above expression has 3 terms. First term has degree 2, second term has degree 1 and third term has no degree or zero degree or can be said constant term.

3yx² - 2x

Here, the first term has two variables and its degree is 3 because variable 'y' has power 1 and variable 'x' has power 2. So, 1+2=3. The second term has degree 1.

Naming of degree:

Degree	Name	Example
0	Constant	7
1	Linear	x + 7
2	Quadratic	x2− x + 2
3	Cubic	x3 − x2 + 5
4	Quartic	6x4 − x3 + x − 2
5	Quintic	x5− 3x3 + x2 + 8

This naming table is collected from Math is Fun.

The general form of a polynomial shows the terms of all possible degree.  Here, for example, is the general form of a polynomial of the third degree:  

3x³ + 2x² + 5x + 8

An equation that is in polynomial form is called Polynomial Equation.  Polynomials is used widely in the areas of mathematics and science.