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
|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
4x – 3xy + 4z2
The terms are 4x, -3xy, and 4z2. Note that to determine the terms of an expression, subtraction is
re-written 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.
commutative and Associative
properties of Addition to re-arrange 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
|Combine the terms in each column|
Subtraction of polynomials
The opposite of the polynomial (3x2 – 7x + 8) is –(3x2 – 7x + 8)
To simplify the opposite of a polynomial, change the sign of each term inside of the
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.
opposite of the second
polynomial to the first
|Combine like terms|
Simplify: Use a vertical
The opposite of 2y2 + 4y – 21 is –2y2– 4y + 21
terms of each polynomial in descending
order with like terms in the same column
|Note that 4y – 4y = 0, but 0 is not written.|