Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Polynomials

A polynomial (in a variable ) is a function or an expression that can be evaluated by combining the variable and possibly some constants by a finite number of additions, subtractions, and multiplications. Note that the list excludes divisions (although a number like $\frac{1}{2}$ would be considered a constant). Also excluded are radicals like $\sqrt{x}$ although a number like $\sqrt{2}$ would be (again) considered a constant. When it matters we use the phrases polynomial expression or polynomial function, but more frequently we use just the word polynomial by itself.

A polynomial can always be written in standard form as

$\displaystyle p(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0\qquad(*)$

where the

are constants (called the coefficients ) of the polynomial. The integer $n\geq 0$ is called the degree of the polynomial. (In this context it is sometimes assumed that

$\displaystyle a_n\neq 0$

and we adopt that convention for the purposes of this class.) The form

indicates the origin of the word polynomial which is Greek for many terms.

The function is also a polynomial. It is called the zero polynomial (or the zero function.) Its degree is undefined, , or $-\infty$ , depending on the author. You don't have to worry about the degree of the zero polynomial in this class.

Some examples will illustrate these concepts:

is a polynomial of degree

. It is written in standard form with

, and

$p(x) = \pi x^5 + 2x^4 -3x^2 +4 x -\sqrt{2}$ is a polynomial of degree 5 with $a_5=\pi$ ,

, and $a_0=\sqrt{2}$ .

is also a polynomial of degree 2 which becomes apparent if it is rewritten (via the distributive law or the second binomial formula) as .
$p(x) = (x+1)^{10,000}$ is a polynomial of degree 10,000 although it would be tedious to write it explicitly in standard form.
$p(x) = \vert x\vert$ is not a polynomial. However, it is not easy to see why not. There is no obvious way to convert it to standard form but there might be a non-obvious way that we aren't able to see. However, the graph of has a corner at the origin. It turns out that the graph of a polynomial never has a corner, but that is a subject much beyond the scope of this class.
To show that appearances may be deceiving note that $p(x)= \vert x\vert^2$ is in fact a polynomial. This is true despite the appearance of the absolute value, because $\vert x\vert^2 = x^2$ (which you can easily check by considering the cases of positive and negative separately).
The function $p(x) = \frac{1}{1+x^2}$ is not a polynomial since the function value becomes arbitrarily close to zero as gets sufficiently large, and the only polynomial with that property is the zero polynomial.
On the other hand, the function $p(x) = \frac{x^4-1}{x^2+1}$ is a polynomial despite the apparent division because

$\displaystyle \frac{x^4-1}{x^2+1} = \frac{(x^2-1)(x^2+1)}{x^2+1} = x^2-1$
for all real numbers .
A more subtle example is provided by

$\displaystyle p(x) = \frac{x^2-1}{x-1}.$
This function is not defined when since we must not divide by zero. On the other hand, whenever $x\neq 1$ we have that

$\displaystyle \frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1.$
The last expression is in the standard form of a polynomial, and for most purposes one can brush over the fact that is not defined when or, better, define . However, technically, without that additional definition, is not a polynomial.
A more extreme example of a non-polynomial function is provided by

$\displaystyle p(x) = \frac{1}{x}.$
This function ``blows up'' (i.e., becomes arbitrarily large) for values of close to zero and no polynomial does that.

Don't get distracted by the subtlety of some of these examples. Usually it will be true that if you see a radical, an absolute value, or a division by an algebraic expression, then the expression or function in question is not a polynomial. But it is important to understand and appreciate the definition of the term polynomial with all its subtleties and ramifications.

Polynomials are important because they occur in applications and they have nice properties: they are defined for all values of the variable, and their graphs are smooth. Indeed, polynomials occur so frequently that there is a whole language associated with them. The terms involved are defined here, and you can also find their definitions in virtually any ordinary dictionary of the English language. Throughout let be defined as in above, with $a_n\neq 0$ .

The constant is the leading coefficient of , and is its constant term. A polynomial with only one non-zero coefficient (such as ) is a monomial, one with two such coefficients (like $x^{17}-x^{12}$ ) is a binomial, and one with three such terms (such as $x^{14} + x^2 + x$ but more likely and frequently a polynomial of degree 2 like ) is a trinomial. Polynomials of low degree occur so often that each degree has been given a special name as listed in the following table:

$\displaystyle \begin{array}{rll} \hbox{degree} & \hbox{name} &\hbox{Examples} \... ... \hbox{octic} & x^8 + 5x^2 \\ \\ 9 & \hbox{nonic} & x^9 \\ \\ \end{array}$

Combining Polynomials

You can add, subtract, and multiply polynomials and get a new polynomial. On the other hand, the ratio of two polynomials is usually not a polynomial. Rather than memorizing a number of rules for the various operations you should recognize that they are just the ordinary rules underlying all algebra, such as the distributive law and the commutative and associative laws of multiplication and addition. A few examples illustrate the idea.

Let

$\displaystyle p(x) = x^2 + 1\qquad\hbox{and}\qquad q(x) = x-1.$

Then new functions

can be defined as follows:

$\begin{displaymath} \begin{array}{ccccccl} h = p+q &\hbox{~where~}& h(x) &=& x^2... ...e~}& h(x) &=& (x^2+1)(x-1) &=& x^3 -x^2 +x -1 \\ \end{array} \end{displaymath}$

Remember that you multiply powers with the same base by adding the exponents. This implies that the degree of the product of two polynomials is the sum of the individual degrees. If you add (or subtract) two polynomials of different degrees then the degree of the sum (or difference) is the larger of the two individual degrees. If the two polynomials have the same degrees then the degree of the sum or difference is that same degree unless the leading coefficients cancel, in which case the degree of the sum or difference is less. Don't attempt to memorize these facts, instead make up some examples, think about the mechanism of the operation, and then work out the details from your understanding when you need them.