Problem 43 - Beginning and Intermediate Algebra

Question

Fill in the blank with the correct term. Some of the given choices will not be used. Descartes' rule of signs a vertical asymptote the leading-term test \(\quad\) an oblique the intermediate \(\quad\) asymptote value theorem direct variation the fundamental \(\quad\) inverse variation theorem of algebra a horizontal line a polynomial function a vertical line a rational function \(\quad\) parallel a one-to-one function \(\quad\) perpendicular a constant function a horizontal asymptote _____is a function that is a quotient of two polynomials.

Step-by-Step Solution

Answer

A \(\textbf{rational function}\) is a function that is a quotient of two polynomials.

Step 1: Define the concept

A function that is the quotient of two polynomials \(\frac{p(x)}{q(x)}\) where \(q(x) \neq 0\).

Step 2: Answer

This is called a \(\textbf{rational function}\).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers of one or more variables multiplied by coefficients. It is written in the form:

• \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where \( a_n, a_{n-1}, \ldots, a_0 \) are constants called coefficients, and \( n \) is a non-negative integer representing the degree of the polynomial.

Polynomial functions are classified based on their degree:

• Linear polynomial: Degree 1, e.g., \( f(x) = 3x + 2 \)
• Quadratic polynomial: Degree 2, e.g., \( f(x) = x^2 - 4x + 4 \)
• Cubic polynomial: Degree 3, e.g., \( f(x) = x^3 + 3x^2 + 3x + 1 \)

Understanding polynomial functions is vital as they form the building blocks for other types of functions such as rational functions.

Quotient

The quotient in mathematics refers to the result obtained when you divide one number by another. In the context of functions, the quotient is essential in forming rational functions.

For instance, if we have two polynomial functions \( P(x) \) and \( Q(x) \), the quotient \( \frac{P(x)}{Q(x)} \) defines a new function provided \( Q(x) eq 0 \). This quotient represents the division of one polynomial by another and is crucial for creating rational functions.

While performing division:

• The dividend is the polynomial being divided, \( P(x) \).
• The divisor is the polynomial \( Q(x) \).

The quotient obtained may sometimes not be a polynomial itself if the division does not result in a whole number, unlike polynomial long division which aims for whole polynomial remainders.

Polynomials

Polynomials are mathematical expressions involving variables and coefficients, constructed in a specific format involving powers. A polynomial can have any number of terms, and each term consists of a constant coefficient and a variable raised to a non-negative integer power.

Key characteristics of polynomials include:

• Terms: Individual elements like \( ax^n \) in the expression.
• Degree: The highest power of the variable in the polynomial (e.g., \( n \) in \( ax^n \)).
• Coefficients: The constant factors preceding the variable part in each term.

Polynomials have smooth, continuous curves when graphed and can model a variety of phenomena ranging from simple linear trends to more complex natural occurrences like quadratic or cubic behavior.

Function

In mathematics, a function represents a special kind of relation where each element of a set, called the domain, is associated with exactly one element of another set, called the codomain. Functions are fundamental in describing any kind of consistent and predictable relationship between two variables.

The general form of a function is denoted as:

• \( f: A \rightarrow B \)
• \( y = f(x) \)

Here, \( A \) is the domain, \( B \) is the codomain, \( x \) represents the input values, and \( y \) represents the output values.

Functions can be classified in various ways, such as:

• One-to-One Function: Each element of the domain is mapped to a unique element in the codomain.
• Onto Function: Every element of the codomain is mapped from some element in the domain.
• Rational Function: Formed as the quotient of two polynomials, i.e., \( \frac{P(x)}{Q(x)} \) where \( Q(x) eq 0 \).

Understanding functions is critical in solving equations and modeling real-world phenomena.