Introduction and Terminology

Define Basic Polynomial Terms

Polynomials are algebraic expressions that include real numbers and variables. Division and square roots cannot be involved in the variables. The variables can only include addition, subtraction and multiplication.

Polynomials contain more than one term. Polynomials are the sums of monomials.

A monomial has one term: 5y or –8x2 or 3.
A binomial has two terms: –3x2 + 2, or 9y – 2y2
A trinomial has 3 terms: –3x2 + 2 +3x, or 9y – 2y2 + y

The degree of the term is the exponent of the variable: 3x2 has a degree of 2.
When the variable does not have an exponent – always understand that there's a '1' e.g., x = x 1

Polynomials are usually written in decreasing order of terms. The largest term or the term with the highest exponent in the polynomial is usually written first. The first term in a polynomial is called a leading term. When a term contains an exponent, it tells you the degree of the term.

Here's an example of a three term polynomial:

6x 2– 4xy + 2xy2 – This three term polynomial has a leading term to the second degree. It is called a second degree polynomial and often referred to as a trinomial.

9x5– 2x + 3x4– 2 – This 4 term polynomial has a leading term to the fifth degree and a term to the fourth degree. It is called a fifth degree polynomial.

3x3– This is a one term algebraic expression which is actually referred to as a monomial.

Example 1. Determine if each expression is a polynomial. If it is, classify each polynomial by the appropriate type (monomial, binomial, or trinomial) and give the degree.

a) 3x –2 – 5x + 2

b) x2 – 5

c) 8x5– 3x3– 2x2 + 6

d) a 4– 16b4

Solution.

a) 3x –2 – 5x + 2 does not represent a polynomial since the first exponent, –1, is not a whole number. b)

x2 – 5 is a second degree binomial.

c)

8x5– 3x3– 2x2 + 6 represents a fifth degree polynomial.

d) a 4– 16b4 is a fourth degree binomial.

Indicate which is not a polynomial: 12x 9 , 2x+1, \sqrt{x-1}
Write the polynomial in descending powers of the variable:

y-6y3+8y5-9y4+12

Identify the polynomial as a monomial, binomial, trinomial, or none. Also, indicate the degree of the polynomial.

7m9+18m14