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 wholenumber exponents.
Examples of monomials and nonmonomials
Monomials

9

x

9x

6xy

0.60x^{4}y

Not monomials

y  6

x^{1} or 1/x

√(x) or x^{1/2}

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, 5x^{3}

Binomial

2

2x + 5, x^{2}  x, x  5

Trinomial

3

x^{2} + 5x + 6, x^{5}  3x + 8

Similarity and difference between a monomial and a polynomial.:
A polynomial may have more than one variable.
For example, x + y and x
^{2} + 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
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) z
^{2} + 5z
^{1} + 6
3) 8
4) 2c
^{2} + 5b + 6
5) 14 + x
6) 5x  2
^{1}
7) 4 b
^{2}  2 b
^{2}
8) f
^{2} + 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.
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