Problem 46 - Algebra 2

Question

The number of regions formed by connecting \(n\) points of a circle can be described by the function \(f(n)=\frac{1}{24}\left(n^{4}-6 n^{3}+23 n^{2}-18 n+24\right) .\) What is the degree of this polynomial function?

Step-by-Step Solution

Answer

The degree of the polynomial is 4.

Step 1: Identify the Polynomial

The given function is \(f(n)=\frac{1}{24}(n^{4}-6n^{3}+23n^{2}-18n+24)\). This is a polynomial function inside the parentheses.

Step 2: Determine the Highest Degree Term

Observe the polynomial inside the parentheses: \(n^{4}-6n^{3}+23n^{2}-18n+24\). The term with the highest power of \(n\) is \(n^4\).

Step 3: Conclude the Degree

The degree of a polynomial is determined by the term with the highest power of the variable. Since the term \(n^4\) is the term with the highest power, the degree of the polynomial is 4.

Key Concepts

Polynomial Function

A polynomial function is a type of algebraic expression that consists of variables elevated to whole number powers, combined using addition, subtraction, and multiplication. These functions are fundamental in algebra and are encountered frequently in mathematics.
Key characteristics of polynomial functions include:

• They do not have variables in the denominator, nor do they involve roots or negative exponents of variables.
• Each element of the function, known as a term, includes a coefficient and a variable raised to a non-negative integer power.
• They are often written in standard form, which arranges the terms in descending order of degrees, or from the highest power to the lowest power.

In the given exercise, the polynomial function inside the function is represented as a quadratic expression inside the parentheses, which is essential to determine the degree of the polynomial.

Highest Degree Term

The highest degree term in a polynomial plays a crucial role in defining the degree of the polynomial. It is determined by finding the term with the greatest exponent in the polynomial. The exponent indicates how many times the variable is multiplied by itself, which is central to understanding polynomials.
To identify the highest degree term, follow these steps:

• Look for the variable with the largest exponent.
• This term's exponent is the degree of the entire polynomial.

For the polynomial function from the exercise, evaluating the expression inside the parentheses, we distinguish the term \(n^4\) as having the greatest exponent; hence, it is the highest degree term. This helps us to conclude that the degree of the polynomial is 4.

Algebraic Expression

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. It does not include the equal sign that is part of an equation.
Important aspects of algebraic expressions include:

• They can consist of one or more terms, where each term is a product of numbers and variables raised to powers.
• The terms are separated by addition or subtraction operations within the expression.
• They act as building blocks for more complex mathematical statements like equations or functions.

The expression in the original exercise, \(n^{4}-6n^{3}+23n^{2}-18n+24\), serves as an example. It consists of multiple terms, each with a variable \(n\) elevated to different powers. Recognizing these terms and their operations is crucial for understanding how polynomial functions are formed and analyzed.