A polynomial is an expression that contains variables and coefficients. Polynomials only involve addition, subtraction, multiplication, and variables raised to non-negative, positive integers. Expressions involving other operations on variables, such as division, are not polynomials. However, polynomials can include constants that are divided, such as fractions like ⅓.
Determine whether the following expressions are polynomials.
This is a polynomial. Although 3 is more likely to be described as a constant, it is technically 0th degree polynomial: 3x0.
2. (x + 1)(x + 2) - 7x5 + ⅓x - 3
This is a polynomial because it only contains coefficients and variables, and none of the variables are undergoing any operations other than multiplication, addition, subtraction, or being raised to non-negative integer exponents.
3. + 3x2 - 4
This is not a polynomial. The term involves division of a variable, so the expression is not a polynomial. If this term were removed, then the expression would be a polynomial.
Polynomials can be further specified as monomials, binomials, and trinomials. The prefixes refer to the number of terms in the polynomial. A monomial has only 1 term, a binomial has 2 terms, and a trinomial has 3 terms. Polynomials with more than 3 terms are simply referred to as polynomials.
- Monomial: x2
- Binomial: x2 + 1
- Trinomial: x3 + ⅔x + 3
- Polynomial: (x + 1)3 + 4x2 + 7x - 4
Standard form of a polynomial
Polynomials are typically written in order of highest degree to lowest degree terms. The degree of a polynomial is the exponent on its highest term. For example, a polynomial where the highest degree term is x3 has a degree of 3, and can be referred to as a third-degree polynomial.
Write 4x2 + 5x7 - x3 in standard form:
5x7 - x3 + 4x2
Writing polynomials in standard form allows us to quickly identify the degree of the polynomial and generally makes it easier to work with. It is not necessary, but is how polynomials are most commonly expressed.