|
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:
x2 + 2x – 7
x4 – 7x3
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:
x2 + 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
3x4
2x3
A binomial has two terms:
x + 1
5x2 – 3x
A trinomial has three terms:
x4 + 2x3 – 3x
2x2 – 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:
|
2x3 |
Degree = 3 |
|
3x4 |
Degree = 4 |
|
x |
Degree = 1 |
|
3x2y5 |
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 x0 = 1,
and everything has a factor
of 1. So we can say that 37
is the coefficient of x0.
The degree of the entire
polynomial is the degree of
the highest-degree term that
it contains, so
x2 + 2x – 7 is
a second-degree trinomial,
and x4 – 7x3
is a fourth-degree binomial.
|
