Definition: A polynomial is an algebraic expression that is a sum of terms, where each term contains only variables with whole number exponents and integer coefficients.
Example: The following expressions are all considered polynomials:
x^{2} + 2x – 7
x^{4} – 7x^{3}
x
When we write a polynomial we follow the convention that says we write the terms in order of descending powers, from left to right.
The following are NOT polynomials:
_{}
_{}
x^{2} + 3x + 2x^{-2}
A polynomial can have any number of terms (“poly” means “many”). We have special names for polynomials that have one, two, or three terms:
A monomial has one term (“mono” means “one”). The following are monomials:
x
3x^{4}
2x^{3}
A binomial has two terms:
x + 1
5x^{2} – 3x
A trinomial has three terms:
x^{4} + 2x^{3} – 3x
2x^{2} – 4x + 1
The degree of an individual term in a polynomial is the sum of powers of all the variables in that term. We only have to use the plurals in this definition because of the possibility that there may be more than one variable. In practice, you will most often see polynomials that have only one variable (traditionally denoted by the letter ‘x’). In that case, the degree will simply be the power of the variable.
Examples:
2x^{3} |
Degree = 3 |
3x^{4} |
Degree = 4 |
x |
Degree = 1 |
3x^{2}y^{5} |
Degree = 7 (because 2 + 5 = 7) |
37 |
Degree = 0 |
Why is the last example, which is just a plain number, considered to be of degree zero? It is because of the fact that x^{0} = 1, and everything has a factor of 1. So we can say that 37 is the coefficient of x^{0}.
The degree of the entire polynomial is the degree of the highest-degree term that it contains, so
x^{2} + 2x – 7 is a second-degree trinomial, and x^{4} – 7x^{3} is a fourth-degree binomial.