Definition of a Polynomial

Before giving you the definition of a polynomial, it is important to provide the definition of a monomial. 

Definition of a monomial:

A monomial is a variable, a real number, or a multiplication of one or more variables and a real number with whole-number exponents.

Examples of monomials and non-monomials


Monomials
9
x
9x
6xy
0.60x4y
Not monomials
y - 6
x-1 or 1/x
√(x) or x1/2
6 + x
a/x


Polynomial definition:

A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of the polynomial.

Important observation!

Terms are separated by addition signs and subtraction signs, but never by multiplication signs.

  • A polynomial with one term is called a monomial.
  • A polynomial with two terms is called a binomial.
  • A polynomial with three terms is called a trinomial.

Examples of polynomials:

Polynomial
Number of terms
Some examples
Monomial
1
2, x, 5x3
Binomial
2
2x + 5, x2 - x, x - 5
Trinomial
3
x2 + 5x + 6, x5 - 3x + 8


Similarity and difference between a monomial and a polynomial.:

A polynomial may have more than one variable.

For example, x + y and x2 + 5y + 6 are still polynomials although they have two different variables x and y.

By the same token, a monomial can have more than one variable. For example, 2 × x × y × z is a monomial.

A monomial will never have an addition or a subtraction sign.

Take a close look at the Venn diagram below showing the difference between a monomial and a polynomial

definition of a polynomial
Exercises

For all expressions below, look for all expressions that are polynomials.

For those that are polynomials, state whether the polynomial is a monomial, a binomial, or a trinomial.

1) 3.4 + 3.4x

2) z2 + 5z-1 + 6

3) -8

4) 2c2 + 5b + 6

5) 14 + x

6) 5x - 2-1

7) 4 b2 - 2 b-2

8) f2 + 5f + 6

Answer: 1), 3), 4), 5), 6), and 8) are polynomials. 1), 5), and 6) are binomials. 3) is a monomial. 4) and 8) are trinomials.

2) and 7) are not because they have negative exponents

Notice that 6) is still a polynomial although it has a negative exponent. It is because it is the exponent of a real number, not a variable

In fact, 5x - 2-1 = 5x - 1/2 = 5x - 0.5

It is subtle, but if you have any questions about the definition of a polynomial, feel free to contact me.

Degree of a polynomial