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.

Definition of a polynomial quiz

Recent Articles

  1. Quadratic Formula: Easy To Follow Steps

    Jan 26, 23 11:44 AM

    Quadratic formula
    Learn about the quadratic formula, the discriminant, important definitions related to the formula, and applications.

    Read More

  2. Area Formula - List of Important Formulas

    Jan 25, 23 05:54 AM

    Frequently used area formulas
    What is the area formula for a two-dimensional figure? Here is a list of the ones that you must know!

    Read More

Tough algebra word problems

100 Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Math quizzes

 Recommended

Math vocabulary quizzes