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Chapter 13: Problem 23

Find the degree of each polynomial. $$2-8 n$$

Short Answer

Expert verified
The degree of the polynomial is 1.

Step by step solution

01

Identify the terms of the polynomial

A polynomial is composed of terms, each consisting of a coefficient and a variable raised to an exponent. In the polynomial \(2 - 8n\), there are two terms: \(2\) and \(-8n\).
02

Determine the degree of each term

The degree of a term in a polynomial is determined by the exponent of the variable in that term. For the term \(2\), the variable \(n\) does not appear, meaning its exponent is \(0\). For the term \(-8n\), the exponent of \(n\) is \(1\).
03

Identify the highest degree of the polynomial

The degree of a polynomial is the highest degree among its terms. Here, the terms have degrees \(0\) and \(1\). The highest of these is \(1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Polynomial Terms
Polynomials are expressions made up of terms, where each term is a combination of a coefficient and a variable raised to an exponent. The terms are usually separated by addition or subtraction, and each has its unique characteristics.
For example, in the polynomial expression \(2 - 8n\), we have two distinct terms:
  • \(2\) - a constant term, which means there is no variable associated with it.
  • \(-8n\) - a linear term, which includes the variable \(n\).
Each term plays a role in determining the polynomial's degree, as we will explore in the next sections. The number of terms and the presence of variables define the complexity of the polynomial.
Exploring the Exponent of a Variable
The exponent of a variable is a key factor in understanding polynomials. It tells us how many times the variable is used as a factor in its term. In simpler forms, it's the power to which the variable is raised.
In our example \(2 - 8n\):
  • The constant term \(2\) has no variable, meaning its exponent is deemed to be \(0\). This is because any number can be viewed as being multiplied by the variable raised to 0; \(n^0 = 1\).
  • In the term \(-8n\), the variable \(n\) has an exponent of \(1\). This implies a linear factor as any variable raised to the power of \(1\) maintains its basic value.
Understanding the exponents helps determine the structure of each term and eventually the entire polynomial.
Determining the Highest Degree in Polynomials
The highest degree of a polynomial is determined by the term with the largest exponent on the variable. This measure signifies the polynomial's overall degree and is crucial for understanding the behavior of the polynomial expression.
Consider the polynomial \(2 - 8n\):
  • For the term \(2\), the effective exponent on the non-existent variable is \(0\).
  • For the term \(-8n\), the exponent is \(1\).
The highest of these exponents is \(1\). Therefore, the degree of our polynomial is \(1\). This suggests that the polynomial is linear, indicating that its graph will be a straight line. Knowing the degree of a polynomial helps predict its growth rate and the number of potential roots.

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