Welcome to our introduction to polynomials. A polynomial is a mathematical expression that consists of variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents. Let's look at an example: three x squared plus two x minus five. Here we can see the coefficients in blue, the variables with their exponents in red, and the constant term in green.
It's important to understand what is NOT a polynomial. Expressions with division by variables, variables under square roots, negative exponents, or variables in the exponent position are not polynomials. For example, one over x plus two x contains division by a variable. Square root of x plus three has a variable under a square root. However, four x cubed minus two x squared plus seven x minus one is a valid polynomial because it only uses non-negative integer exponents.
Polynomials can be classified by the number of terms they contain. A monomial has exactly one term, like five x cubed. A binomial has two terms, such as three x squared plus seven. A trinomial has three terms, for example x squared plus four x minus nine. When a polynomial has four or more terms, we simply call it a polynomial, like two x to the fourth minus x cubed plus five x minus one.
The degree of a polynomial is the highest exponent of the variable. For example, in seven x to the fifth plus three x squared minus x plus four, the highest exponent is five, so the degree is five. In two x cubed minus six x plus one, the degree is three. For nine x squared plus four x minus seven, the degree is two. A linear polynomial like five x plus eight has degree one. Even a constant like twelve is considered a polynomial of degree zero.