A monomial is a number, a variable, or a product of
numbers and variables. The following
examples are monomials. The degree of a monomial is the sum of the exponents of
the
variables.
Degree = 1  
Degree = 2  
Degree = 3  
Degree = 6 
The degree of a nonzero constant is zero.
Ex: the degree of 7 is 0
Each of the addends of a variable expression is called a term. For example, the
following
variable expression:
4x – 3xy + 4z^{2}
The terms are 4x, 3xy, and 4z^{2}. Note that to determine the terms of an
expression, subtraction is
rewritten as addition of the opposite.
A polynomial is a variable expression in which the terms are monomials.
A polynomial of one term is a monomial  Example:  
A polynomial of two terms is a binomial  Example:  
A polynomial of three terms is a trinomial  Example:  
A polynomial of four or more terms is simply  
called a polynomial  Example: 
The terms of a polynomial in one variable are usually
arranged so that the exponents of the
variable decrease from left to right. This
is called descending order.
The degree of a polynomial is the degree of the term of largest degree.
The degree of is 3.
The degree of is 4
Polynomials can be added, using either a horizontal or
vertical format, by combining like terms.
Simplify: Use a horizontal format.
Use the
commutative and Associative properties of Addition to rearrange and group like terms. 

Then combine like terms. 
Simplify: Use a vertical format.
Arrange the
terms of each polynomial in descending order with like terms in the same column 

Combine the terms in each column 
Subtraction of polynomials
The opposite of the polynomial (3x^{2} – 7x + 8) is –(3x^{2} – 7x + 8)
To simplify the opposite of a polynomial, change the sign of each term inside of
the
parentheses.
Polynomials can be subtracted using either a horizontal or vertical format. To
subtract, add the
opposite of the second polynomial to the first.
Simplify: . Use a horizontal format.
Add the
opposite of the second polynomial to the first 

Combine like terms 
Simplify: Use a vertical
format.
The opposite of 2y^{2} + 4y – 21 is –2y^{2}– 4y + 21
Arrange the
terms of each polynomial in descending order with like terms in the same column 

Note that 4y – 4y = 0, but 0 is not written. 