Power, Polynomial & Rational Functions

Power, Polynomial & Rational Functions#

Definitions#

Definition

A power function is a function that can be written in the form

\[ax^r\]

where \(a\) and \(r\) are constants.

Definition

A polynomial function of degree \(n\) is a function that can be written in the form

\[a_0 + a_1x + a_2x^2 + \cdots + a_nx^n\]

where \(n\) is a nonnegative integer and the constants \(a_0, a_1, \ldots, a_n\) (\(a_n\neq 0\)) are called the coefficients of the polynomial.

Definition

A rational function is a function that can be written as the ratio of two polynomials.

By definition, every polynomial is also considered to be a rational function.

Domain Considerations

  • The domain of a polynomial function is all real numbers.

  • The domain of a rational function, \(f(x)/g(x)\), is all real numbers excluding values of \(x\) such that \(g(x) = 0\).

Example 1#

The square root function

The square root function, \(\sqrt{x}\), is

  • a power function since \(\sqrt{x} = x^{1/2}\), but

  • is neither a polynomial nor a rational function since it includes a fractional power of \(x\).

Example 2#

The reciprocal function

The reciprocal function, \(\dfrac{1}{x}\), is

  • a power function since \(\dfrac{1}{x} = x^{-1}\), and

  • is also a rational function since it is the ratio of two polynomials, but

  • is not a polynomial since it includes a negative power of \(x\).

Example 3#

The volume of a sphere

The volume of a sphere with radius \(r\) is \(\dfrac{4}{3}\pi r^3\), which is a power function, a polynomial, and a rational function in the variable \(r\).

Example 4#

The product of polynomials

Show that \(f(x) = (x^2 + 7)(x^3 - 1)\) is a polynomial (and therefore also a rational function) and determine its degree.

Step 1:   Expand the product using the FOIL technique.
\[(x^2 + 7)(x^3 - 1) = x^5 + 7x^3 - x^2 - 7\]

Since \(f\) can be written as a sum of power functions where every power of \(x\) is a nonnegative integer, \(f\) is a polynomial.

Step 2:   Determine the degree of   \(f\).

The degree of \(f\) is \(5\) since \(f\) is a polynomial and the largest power of \(x\) is \(5\).

An Observation about the Degree of a Product of Polynomials

In the previous example, the degree of the polynomial could have been determined by adding the degrees of the individual factors. More specifically, \(f\) is the product of a polynomial of degree 2 (i.e., \(x^2 + 7\)) and a polynomial of degree 3 (i.e., \(x^3-1\)). Therefore, the degree of \(f\) is the sum of \(2\) and \(3\), which is \(5\).

In general, a product of polynomials is also a polynomial and its degree is the sum of the degrees of the polynomial factors.

Example 5#

A rational function

Show that \(g(x) = \dfrac{4}{x-1} + \dfrac{5}{x}\) is a rational function (but not a polynomial) and determine its domain.

Step 1:   Get a common denominator.
\[\dfrac{4}{x-1} + \dfrac{5}{x} = \frac{4x}{x(x-1)} + \frac{5(x-1)}{x(x-1)}\]
Step 2:   Add numerators and simplify.
\[\frac{4x + 5(x-1)}{x(x-1)} = \frac{9x-5}{x(x-1)}\]

Since \(g(x)\) can be written as a ratio of polynomials, it is also a rational function.

Step 3:   Determine the domain of   \(g\).

Since \(g\) is the sum of two functions, we begin by considering the domain of each of the two functions. Specifically, the domain of \(4/(x-1)\) consists of all real numbers except \(x=1\) and the domain of \(5/x\) consists of all real numbers except \(x=0\).

Therefore, the domain of \(g\) consists of all real numbers except \(x=0\) and \(x=1\). In interval notation, this can be written as

\[(-\infty,0)\cup (0,1) \cup (1,\infty).\]
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