Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far in our lectures, we have studied several types of functions of a real variable.
In particular, we saw that linear functions have graphs that are lines
and that quadratic functions have graphs that are parabolas.
Both of these functions are examples of a broader class of functions
called "polynomial functions".
Polynomial functions are powerful tools
that allow us to mathematically model many types of real world problems
in areas such as physics, economics and social science.
In the next several lectures we will explore these functions
so let's start by examining a typical example of a polynomial.
Polynomials are sums of one or more terms called "monomials".
A monomial is a single term
that may contain a variable
or a product of variables
with non-negative integer exponents.
These variables may be multiplied by a constant
called the monomial's "coefficient".
Although monomials can contain multiple variables
the monomials we will be discussing will contain only a single real variable, x.
Likewise, the coefficients of these monomials will be real numbers.
Therefore in the next few lectures, we will consider only "real polynomials".
In our polynomial example
each of the four monomials can be written as x with a non-negative integer exponent
and a constant coefficient.
The first monomial is already in this form
since it contains x with an exponent "five"
and a coefficient of "two".
The second monomial also contains x with a non-negative integer exponent
in this case "three".
And although no coefficient appears in this monomial
monomials with no coefficient have an implied coefficient of "one".
The third monomial has a coefficient of "nine"
and although x is not written with an exponent
variables with no exponent are understood to have an exponent of "one".
The fourth monomial consists of only a constant.
However, the constant "twelve"
can be thought of as the coefficient of "x to the zero power"
since in Algebra, variables with an exponent of "zero" are typically defined as "one".
So each of these monomials can be written as a constant coefficient
times x with a non-negative integer exponent.
As this example shows, when written in standard form
the terms of a polynomial expression are ordered
so that the exponents decrease going from left to right.
Likewise, if one of the terms is simply a constant
then that term goes last.
If a term contains a variable with no exponent
it is placed in the same order as if it had an exponent of "one".
And monomials with a coefficient of "one"
are normally written without a coefficient.
Since polynomials are ordered so that the exponents decrease going from left to right
the first term, called the "leading" term
always contains the largest exponent.
The value of that exponent determines the "degree" of the polynomial.
Therefore, this expression would be called a "fifth-degree" polynomial.
Since exponents in a polynomial are non-negative
variables cannot have exponents such as "negative one"
which is "one over x"
or "negative two"
which is "one over x-squared".
Likewise, since exponents must be integers
fractional exponents are not allowed such as "x to the one-half power"
which is "the square root of x"
or "x to the one-third power"
which is "the cube root of x".
Remember however, that the coefficients and constants in polynomials can be any real number.
Polynomials may contain any finite number of terms
including only a single term
in which case the polynomial would also be a monomial.
That monomial could contain a variable
or could simply be a constant
and that constant could be any number including "zero".
Polynomials consisting of one term, two terms, or three terms have special names.
These polynomials are referred to as "monomials"
"binomials"
and "trinomials" respectively.
Since the leading terms of all of these examples contain "x to the fifth power"
these polynomial expressions would be referred to as a "fifth-degree monomial"
a "fifth-degree binomial"
and a "fifth-degree trinomial".
You may occasionally see polynomial expressions represented in general notation such as this.
So let's see how to decipher this notation.
In this general representation of a polynomial expression
the a's with the little subscripts represent constants.
These constants can each be positive, negative, or zero.
To represent an nth-degree polynomial, the leading term
is written as "a sub n"
times "x to the nth power".
The next term in the polynomial
is then written as "a sub n-1" times "x to the n-1 power".
The term after that
is "a sub n-2" times "x to the n-2 power"
and so on
with the remaining terms following the same pattern.
The last two terms containing x are represented as "a sub two" times "x to the second power"
plus "a sub one" times "x to the first power"
followed by the constant term "a sub zero".
Any of the constants other than the leading coefficient may be zero
effectively eliminating those terms from the polynomial.
However, the leading coefficient "a sub n" must be non-zero
otherwise this would not represent an "nth degree polynomial".
So as an example, let's see how we would represent the polynomial
"three x to the fifth power, plus x-cubed, minus five x-squared, plus x, minus fifteen"
using this general notation.
Since the leading term in our polynomial "three x to the fifth power"
has an exponent of "five"
"n" in the general notation is "five".
"n-1" and "n-2" are then "four" and "three".
We now set the constants in the general notation
to the corresponding coefficient values of the polynomial.
The coefficient of the polynomial's "x to the fifth" term is "three"
so "a sub five" is "three".
Since there is no "x to the fourth" term in the polynomial, "a sub four" is "zero"
effectively eliminating that term.
Next, since the x-cubed term has an unwritten coefficient of "one"
"a sub three" is "one".
Likewise, "a sub two"
is "negative five"
"a sub one "
is "one"
and "a sub zero"
is "negative fifteen".
Since the coefficients of "one" do not need to be written, we can now eliminate them.
As we mentioned in the beginning of this lecture, polynomial expressions
can define polynomial functions.
A zero-degree polynomial function would be a constant function
whose graph is a horizontal line.
A first-degree polynomial function would be a linear function
whose graph is a sloped line.
And a second-degree polynomial function would be a quadratic function
whose graph is a parabola.
However, the graphs of higher-degree polynomial functions
can be much more interesting and varied.
In the next lecture, we will see how the shapes of those graphs
are affected by the various terms of the polynomial.