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Monomials & Polynomials

Monomials & Polynomials


This lesson covers Monomials, Polynomials, and degrees in Polynomials.  

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  • Monomials
  • Polynomials
  • Degrees in Polynomials
  • Ordering Terms in a Polynomial


The prefix "mono" means "one."  A monopoly is a market that is controlled by a single company.  A monocle is eyewear that consists of only one lens.  Someone who speaks in a monotone voice tends to put people to sleep because the speak with a singular tone, with no variation in pitch. 

In mathematics, a monomial is a single-term expression.  

Monomial: an expression containing a single term.

A term is made up of a combination of number, variables, and powers.  Below are some examples of monomials:

Note that there can be several variables in a monomial, as well as no variable at all.  Coefficients can be fractional, positive or negative.  The important thing to notice is that there is no addition or subtraction.  Addition and subtract separate terms, and since monomials only have one term, there is no need to separate multiple terms with addition or subtraction. 

There are a few other special things to note about monomials:

  • Variables must never be in the denominator. If there is a variable in the denominator, we are not dealing with a monomial. 
  • Exponents to the variables must not be negative.  This is because variables with negative exponents can be written as fractions with the variable in the denominator, and this violates the previous statement that variables must not be in denominators.
  • Exponents to the variables must not be not be fractional. For example,  is not a monomial.   


The prefix "poly" means "many."  A polygon has many sides.  A polytheistic religion believes in many deities.  A polypeptide is a chain made up of several amino acids.  A polynomial, then, is an expression with several terms.

Polynomial: an expression with several terms. 

Below are some examples of polynomials.

Expression containing two terms are called binomials, and expression with three terms are called trinomials.


Degree in Polynomials

Each term in a polynomial can be described by its degree, which is related to the exponent powers attached to variables in the term:

degree (of a term): the sum of all variable exponent powers in the term. 

To find the degree of a term, simply find the sum of all exponents.  For example:

  • In the first example, the degree is 1 because the variable x has an implied exponent of 1. 
  • In the second example, the degree is 5, because the variable x has an exponent of 5. 
  • In the third example, the degree is 3, because the variable x has an exponent of 2, and the variable y has an implied exponent of 1.  The sum of 2 and 1 is 3. 


Ordering Terms in a Polynomial

It is standard to write terms in a polynomial by order of its degree, from highest to lowest.  A reason for this is because we can also describe the degree of a polynomial.  The degree of a polynomial is the same as the highest degree of all the terms.  So when we have a polynomial written in order of descending degree, the first term also describes the degree of the polynomial. 

Degree (of a polynomial): also called order, the highest degree of the terms in a polynomial expression

Take, for example, this polynomial expression that is not written in proper order:

Here, we see a first degree term, followed by a second degree term, followed by a third degree term.  We need to write the terms in order of descending degree (highest to lowest):

Notice that once the polynomial was written in standard order, I could name the degree of the entire polynomial, simply by looking at the degree of the first term.